Theorem sge0tsms 45759

Description: Σ^{^} applied to a nonnegative function (its meaningful domain) is the same as the infinite group sum (that's always convergent, in this case). (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
sge0tsms.g	⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞))
sge0tsms.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
sge0tsms.f	⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))

Assertion

Ref	Expression
sge0tsms	⊢ (𝜑 → (Σ^‘𝐹) ∈ (𝐺 tsums 𝐹))

Proof of Theorem sge0tsms

Dummy variables 𝑠 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2728	. . . 4 ⊢ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )
2	1	a1i 11	. . 3 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ))
3		xrltso 13147	. . . . . 6 ⊢ < Or ℝ*
4	3	supex 9481	. . . . 5 ⊢ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ V)
6		elsng 4639	. . . 4 ⊢ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ V → (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )} ↔ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )))
7	5, 6	syl 17	. . 3 ⊢ (𝜑 → (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )} ↔ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )))
8	2, 7	mpbird 257	. 2 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )})
9		sge0tsms.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
10	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝑋 ∈ 𝑉)
11		sge0tsms.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))
12	11	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
13		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
14	10, 12, 13	sge0pnfval 45752	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = +∞)
15	11	ffnd 6718	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn 𝑋)
16	15	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹 Fn 𝑋)
17		fvelrnb 6954	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 → (+∞ ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞))
18	16, 17	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (+∞ ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞))
19	13, 18	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → ∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞)
20		iccssxr 13434	. . . . . . . . . . . . . 14 ⊢ (0[,]+∞) ⊆ ℝ*
21		sge0tsms.g	. . . . . . . . . . . . . . 15 ⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞))
22		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ (𝒫 𝑋 ∩ Fin))
23	11	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,]+∞))
24		elinel1 4192	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ 𝒫 𝑋)
25		elpwi 4606	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
26	24, 25	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ⊆ 𝑋)
27	26	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ⊆ 𝑋)
28		fssres 6758	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 ⊆ 𝑋) → (𝐹 ↾ 𝑥):𝑥⟶(0[,]+∞))
29	23, 27, 28	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑥):𝑥⟶(0[,]+∞))
30		elinel2 4193	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
31	30	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
32		0red 11242	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 0 ∈ ℝ)
33	29, 31, 32	fdmfifsupp 9393	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑥) finSupp 0)
34	21, 22, 29, 33	gsumge0cl 45750	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹 ↾ 𝑥)) ∈ (0[,]+∞))
35	20, 34	sselid 3977	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹 ↾ 𝑥)) ∈ ℝ*)
36	35	ralrimiva 3142	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹 ↾ 𝑥)) ∈ ℝ*)
37	36	3ad2ant1 1131	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹 ↾ 𝑥)) ∈ ℝ*)
38		eqid 2728	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥)))
39	38	rnmptss 7128	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹 ↾ 𝑥)) ∈ ℝ* → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))) ⊆ ℝ*)
40	37, 39	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))) ⊆ ℝ*)
41		snelpwi 5440	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝑋 → {𝑦} ∈ 𝒫 𝑋)
42		snfi 9063	. . . . . . . . . . . . . . 15 ⊢ {𝑦} ∈ Fin
43	42	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝑋 → {𝑦} ∈ Fin)
44	41, 43	elind 4191	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑋 → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
45	44	3ad2ant2 1132	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
46	11	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝐹:𝑋⟶(0[,]+∞))
47		snssi 4808	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ 𝑋 → {𝑦} ⊆ 𝑋)
48	47	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → {𝑦} ⊆ 𝑋)
49	46, 48	fssresd 6759	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹 ↾ {𝑦}):{𝑦}⟶(0[,]+∞))
50	49	feqmptd 6962	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹 ↾ {𝑦}) = (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)))
51		fvres 6911	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ {𝑦} → ((𝐹 ↾ {𝑦})‘𝑥) = (𝐹‘𝑥))
52	51	mpteq2ia 5246	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)) = (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥))
53	52	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)) = (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥)))
54	50, 53	eqtrd 2768	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹 ↾ {𝑦}) = (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥)))
55	54	oveq2d 7431	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐺 Σg (𝐹 ↾ {𝑦})) = (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥))))
56	55	3adant3 1130	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (𝐺 Σg (𝐹 ↾ {𝑦})) = (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥))))
57		xrge0cmn 21335	. . . . . . . . . . . . . . . . 17 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd
58	21, 57	eqeltri 2825	. . . . . . . . . . . . . . . 16 ⊢ 𝐺 ∈ CMnd
59		cmnmnd 19746	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
60	58, 59	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 𝐺 ∈ Mnd
61	60	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → 𝐺 ∈ Mnd)
62		simp2 1135	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → 𝑦 ∈ 𝑋)
63	11	ffvelcdmda 7089	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ∈ (0[,]+∞))
64	63	3adant3 1130	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (𝐹‘𝑦) ∈ (0[,]+∞))
65		df-ss 3962	. . . . . . . . . . . . . . . . . 18 ⊢ ((0[,]+∞) ⊆ ℝ* ↔ ((0[,]+∞) ∩ ℝ*) = (0[,]+∞))
66	20, 65	mpbi 229	. . . . . . . . . . . . . . . . 17 ⊢ ((0[,]+∞) ∩ ℝ*) = (0[,]+∞)
67	66	eqcomi 2737	. . . . . . . . . . . . . . . 16 ⊢ (0[,]+∞) = ((0[,]+∞) ∩ ℝ*)
68		ovex 7448	. . . . . . . . . . . . . . . . 17 ⊢ (0[,]+∞) ∈ V
69		xrsbas 21305	. . . . . . . . . . . . . . . . . 18 ⊢ ℝ* = (Base‘ℝ*𝑠)
70	21, 69	ressbas 17209	. . . . . . . . . . . . . . . . 17 ⊢ ((0[,]+∞) ∈ V → ((0[,]+∞) ∩ ℝ*) = (Base‘𝐺))
71	68, 70	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ((0[,]+∞) ∩ ℝ*) = (Base‘𝐺)
72	67, 71	eqtri 2756	. . . . . . . . . . . . . . 15 ⊢ (0[,]+∞) = (Base‘𝐺)
73		fveq2 6892	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
74	72, 73	gsumsn 19903	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Mnd ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) ∈ (0[,]+∞)) → (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥))) = (𝐹‘𝑦))
75	61, 62, 64, 74	syl3anc 1369	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥))) = (𝐹‘𝑦))
76		simp3 1136	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (𝐹‘𝑦) = +∞)
77	56, 75, 76	3eqtrrd 2773	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → +∞ = (𝐺 Σg (𝐹 ↾ {𝑦})))
78		reseq2 5975	. . . . . . . . . . . . . 14 ⊢ (𝑥 = {𝑦} → (𝐹 ↾ 𝑥) = (𝐹 ↾ {𝑦}))
79	78	oveq2d 7431	. . . . . . . . . . . . 13 ⊢ (𝑥 = {𝑦} → (𝐺 Σg (𝐹 ↾ 𝑥)) = (𝐺 Σg (𝐹 ↾ {𝑦})))
80	79	rspceeqv 3630	. . . . . . . . . . . 12 ⊢ (({𝑦} ∈ (𝒫 𝑋 ∩ Fin) ∧ +∞ = (𝐺 Σg (𝐹 ↾ {𝑦}))) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹 ↾ 𝑥)))
81	45, 77, 80	syl2anc 583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹 ↾ 𝑥)))
82		pnfxr 11293	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ*
83	82	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → +∞ ∈ ℝ*)
84	38	elrnmpt 5953	. . . . . . . . . . . 12 ⊢ (+∞ ∈ ℝ* → (+∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹 ↾ 𝑥))))
85	83, 84	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (+∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹 ↾ 𝑥))))
86	81, 85	mpbird 257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))))
87		supxrpnf 13324	. . . . . . . . . 10 ⊢ ((ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))) ⊆ ℝ* ∧ +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥)))) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = +∞)
88	40, 86, 87	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = +∞)
89	88	3exp 1117	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝑋 → ((𝐹‘𝑦) = +∞ → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = +∞)))
90	89	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (𝑦 ∈ 𝑋 → ((𝐹‘𝑦) = +∞ → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = +∞)))
91	90	rexlimdv 3149	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞ → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = +∞))
92	19, 91	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = +∞)
93	14, 92	eqtr4d 2771	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ))
94	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝑋 ∈ 𝑉)
95	11	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
96		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
97	95, 96	fge0iccico 45749	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,)+∞))
98	94, 97	sge0reval 45751	. . . . 5 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ))
99	23, 27	feqresmpt 6963	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑥) = (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)))
100	99	adantlr 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑥) = (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)))
101	100	oveq2d 7431	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹 ↾ 𝑥)) = (𝐺 Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))))
102	21	fveq2i 6895	. . . . . . . . . . 11 ⊢ (+g‘𝐺) = (+g‘(ℝ*𝑠 ↾s (0[,]+∞)))
103		eqid 2728	. . . . . . . . . . . . . 14 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s (0[,]+∞))
104		xrsadd 21306	. . . . . . . . . . . . . 14 ⊢ +𝑒 = (+g‘ℝ*𝑠)
105	103, 104	ressplusg 17265	. . . . . . . . . . . . 13 ⊢ ((0[,]+∞) ∈ V → +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞))))
106	68, 105	ax-mp 5	. . . . . . . . . . . 12 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞)))
107	106	eqcomi 2737	. . . . . . . . . . 11 ⊢ (+g‘(ℝ*𝑠 ↾s (0[,]+∞))) = +𝑒
108	102, 107	eqtr2i 2757	. . . . . . . . . 10 ⊢ +𝑒 = (+g‘𝐺)
109	21	oveq1i 7425	. . . . . . . . . . 11 ⊢ (𝐺 ↾s (0[,)+∞)) = ((ℝ*𝑠 ↾s (0[,]+∞)) ↾s (0[,)+∞))
110		icossicc 13440	. . . . . . . . . . . . 13 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
111	68, 110	pm3.2i 470	. . . . . . . . . . . 12 ⊢ ((0[,]+∞) ∈ V ∧ (0[,)+∞) ⊆ (0[,]+∞))
112		ressabs 17224	. . . . . . . . . . . 12 ⊢ (((0[,]+∞) ∈ V ∧ (0[,)+∞) ⊆ (0[,]+∞)) → ((ℝ*𝑠 ↾s (0[,]+∞)) ↾s (0[,)+∞)) = (ℝ*𝑠 ↾s (0[,)+∞)))
113	111, 112	ax-mp 5	. . . . . . . . . . 11 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) ↾s (0[,)+∞)) = (ℝ*𝑠 ↾s (0[,)+∞))
114	109, 113	eqtr2i 2757	. . . . . . . . . 10 ⊢ (ℝ*𝑠 ↾s (0[,)+∞)) = (𝐺 ↾s (0[,)+∞))
115	58	elexi 3490	. . . . . . . . . . 11 ⊢ 𝐺 ∈ V
116	115	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐺 ∈ V)
117		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ (𝒫 𝑋 ∩ Fin))
118	110	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (0[,)+∞) ⊆ (0[,]+∞))
119		0xr 11286	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ*
120	119	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 0 ∈ ℝ*)
121	82	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → +∞ ∈ ℝ*)
122	95	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝐹:𝑋⟶(0[,]+∞))
123	26	sselda 3979	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑋)
124	123	adantll 713	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑋)
125	122, 124	ffvelcdmd 7090	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (0[,]+∞))
126	20, 125	sselid 3977	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ℝ*)
127		iccgelb 13407	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝐹‘𝑦) ∈ (0[,]+∞)) → 0 ≤ (𝐹‘𝑦))
128	120, 121, 125, 127	syl3anc 1369	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 0 ≤ (𝐹‘𝑦))
129		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝑦) = +∞ → (𝐹‘𝑦) = +∞)
130	129	eqcomd 2734	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑦) = +∞ → +∞ = (𝐹‘𝑦))
131	130	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ (𝐹‘𝑦) = +∞) → +∞ = (𝐹‘𝑦))
132	11	ffund 6721	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → Fun 𝐹)
133	132	ad2antrr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → Fun 𝐹)
134	22, 123	sylan 579	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑋)
135	11	fdmd 6728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → dom 𝐹 = 𝑋)
136	135	eqcomd 2734	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑋 = dom 𝐹)
137	136	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑋 = dom 𝐹)
138	134, 137	eleqtrd 2831	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ dom 𝐹)
139		fvelrn 7081	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) ∈ ran 𝐹)
140	133, 138, 139	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ran 𝐹)
141	140	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ (𝐹‘𝑦) = +∞) → (𝐹‘𝑦) ∈ ran 𝐹)
142	131, 141	eqeltrd 2829	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ (𝐹‘𝑦) = +∞) → +∞ ∈ ran 𝐹)
143	142	adantl3r 749	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ (𝐹‘𝑦) = +∞) → +∞ ∈ ran 𝐹)
144	96	ad3antrrr 729	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ (𝐹‘𝑦) = +∞) → ¬ +∞ ∈ ran 𝐹)
145	143, 144	pm2.65da 816	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑦) = +∞)
146	145	neqned 2943	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ≠ +∞)
147		ge0xrre 44907	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑦) ∈ (0[,]+∞) ∧ (𝐹‘𝑦) ≠ +∞) → (𝐹‘𝑦) ∈ ℝ)
148	125, 146, 147	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ℝ)
149	148	ltpnfd 13128	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) < +∞)
150	120, 121, 126, 128, 149	elicod 13401	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (0[,)+∞))
151		eqid 2728	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)) = (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))
152	150, 151	fmptd 7119	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)):𝑥⟶(0[,)+∞))
153		0e0icopnf 13462	. . . . . . . . . . 11 ⊢ 0 ∈ (0[,)+∞)
154	153	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 0 ∈ (0[,)+∞))
155		eliccxr 13439	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0[,]+∞) → 𝑦 ∈ ℝ*)
156		xaddlid 13248	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ* → (0 +𝑒 𝑦) = 𝑦)
157		xaddrid 13247	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ* → (𝑦 +𝑒 0) = 𝑦)
158	156, 157	jca 511	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ* → ((0 +𝑒 𝑦) = 𝑦 ∧ (𝑦 +𝑒 0) = 𝑦))
159	155, 158	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0[,]+∞) → ((0 +𝑒 𝑦) = 𝑦 ∧ (𝑦 +𝑒 0) = 𝑦))
160	159	adantl 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ (0[,]+∞)) → ((0 +𝑒 𝑦) = 𝑦 ∧ (𝑦 +𝑒 0) = 𝑦))
161	72, 108, 114, 116, 117, 118, 152, 154, 160	gsumress 18636	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = ((ℝ*𝑠 ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))))
162		rege0subm 21350	. . . . . . . . . . . . 13 ⊢ (0[,)+∞) ∈ (SubMnd‘ℂfld)
163	162	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (0[,)+∞) ∈ (SubMnd‘ℂfld))
164		eqid 2728	. . . . . . . . . . . 12 ⊢ (ℂfld ↾s (0[,)+∞)) = (ℂfld ↾s (0[,)+∞))
165	117, 163, 152, 164	gsumsubm 18781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = ((ℂfld ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))))
166		eqidd 2729	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℂfld ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = ((ℂfld ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))))
167		vex 3474	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
168	167	mptex 7230	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)) ∈ V
169	168	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)) ∈ V)
170		ovexd 7450	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld ↾s (0[,)+∞)) ∈ V)
171		ovexd 7450	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℝ*𝑠 ↾s (0[,)+∞)) ∈ V)
172		rge0ssre 13460	. . . . . . . . . . . . . . . . 17 ⊢ (0[,)+∞) ⊆ ℝ
173		ax-resscn 11190	. . . . . . . . . . . . . . . . 17 ⊢ ℝ ⊆ ℂ
174	172, 173	sstri 3988	. . . . . . . . . . . . . . . 16 ⊢ (0[,)+∞) ⊆ ℂ
175		cnfldbas 21277	. . . . . . . . . . . . . . . . 17 ⊢ ℂ = (Base‘ℂfld)
176	164, 175	ressbas2 17212	. . . . . . . . . . . . . . . 16 ⊢ ((0[,)+∞) ⊆ ℂ → (0[,)+∞) = (Base‘(ℂfld ↾s (0[,)+∞))))
177	174, 176	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (0[,)+∞) = (Base‘(ℂfld ↾s (0[,)+∞)))
178	177	eqcomi 2737	. . . . . . . . . . . . . 14 ⊢ (Base‘(ℂfld ↾s (0[,)+∞))) = (0[,)+∞)
179	110, 20	sstri 3988	. . . . . . . . . . . . . . 15 ⊢ (0[,)+∞) ⊆ ℝ*
180		eqid 2728	. . . . . . . . . . . . . . . 16 ⊢ (ℝ*𝑠 ↾s (0[,)+∞)) = (ℝ*𝑠 ↾s (0[,)+∞))
181	180, 69	ressbas2 17212	. . . . . . . . . . . . . . 15 ⊢ ((0[,)+∞) ⊆ ℝ* → (0[,)+∞) = (Base‘(ℝ*𝑠 ↾s (0[,)+∞))))
182	179, 181	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (0[,)+∞) = (Base‘(ℝ*𝑠 ↾s (0[,)+∞)))
183	178, 182	eqtri 2756	. . . . . . . . . . . . 13 ⊢ (Base‘(ℂfld ↾s (0[,)+∞))) = (Base‘(ℝ*𝑠 ↾s (0[,)+∞)))
184	183	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Base‘(ℂfld ↾s (0[,)+∞))) = (Base‘(ℝ*𝑠 ↾s (0[,)+∞))))
185		rge0srg 21365	. . . . . . . . . . . . . . 15 ⊢ (ℂfld ↾s (0[,)+∞)) ∈ SRing
186	185	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → (ℂfld ↾s (0[,)+∞)) ∈ SRing)
187		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → 𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
188		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
189		eqid 2728	. . . . . . . . . . . . . . 15 ⊢ (Base‘(ℂfld ↾s (0[,)+∞))) = (Base‘(ℂfld ↾s (0[,)+∞)))
190		eqid 2728	. . . . . . . . . . . . . . 15 ⊢ (+g‘(ℂfld ↾s (0[,)+∞))) = (+g‘(ℂfld ↾s (0[,)+∞)))
191	189, 190	srgacl 20139	. . . . . . . . . . . . . 14 ⊢ (((ℂfld ↾s (0[,)+∞)) ∈ SRing ∧ 𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
192	186, 187, 188, 191	syl3anc 1369	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
193	192	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))))) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
194	172	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → (0[,)+∞) ⊆ ℝ)
195		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → 𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
196	195, 178	eleqtrdi 2839	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → 𝑠 ∈ (0[,)+∞))
197	194, 196	sseldd 3980	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → 𝑠 ∈ ℝ)
198	197	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → 𝑠 ∈ ℝ)
199	172	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → (0[,)+∞) ⊆ ℝ)
200		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
201	200, 178	eleqtrdi 2839	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → 𝑡 ∈ (0[,)+∞))
202	199, 201	sseldd 3980	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → 𝑡 ∈ ℝ)
203	202	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → 𝑡 ∈ ℝ)
204		rexadd 13238	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠 +𝑒 𝑡) = (𝑠 + 𝑡))
205	204	eqcomd 2734	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠 + 𝑡) = (𝑠 +𝑒 𝑡))
206	162	elexi 3490	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0[,)+∞) ∈ V
207		cnfldadd 21279	. . . . . . . . . . . . . . . . . . . . 21 ⊢ + = (+g‘ℂfld)
208	164, 207	ressplusg 17265	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0[,)+∞) ∈ V → + = (+g‘(ℂfld ↾s (0[,)+∞))))
209	206, 208	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ + = (+g‘(ℂfld ↾s (0[,)+∞)))
210	209, 207	eqtr3i 2758	. . . . . . . . . . . . . . . . . 18 ⊢ (+g‘(ℂfld ↾s (0[,)+∞))) = (+g‘ℂfld)
211	210, 207	eqtr4i 2759	. . . . . . . . . . . . . . . . 17 ⊢ (+g‘(ℂfld ↾s (0[,)+∞))) = +
212	211	oveqi 7428	. . . . . . . . . . . . . . . 16 ⊢ (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) = (𝑠 + 𝑡)
213	212	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) = (𝑠 + 𝑡))
214	180, 104	ressplusg 17265	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0[,)+∞) ∈ V → +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,)+∞))))
215	206, 214	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,)+∞)))
216	215	eqcomi 2737	. . . . . . . . . . . . . . . . 17 ⊢ (+g‘(ℝ*𝑠 ↾s (0[,)+∞))) = +𝑒
217	216	oveqi 7428	. . . . . . . . . . . . . . . 16 ⊢ (𝑠(+g‘(ℝ*𝑠 ↾s (0[,)+∞)))𝑡) = (𝑠 +𝑒 𝑡)
218	217	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℝ*𝑠 ↾s (0[,)+∞)))𝑡) = (𝑠 +𝑒 𝑡))
219	205, 213, 218	3eqtr4d 2778	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠 ↾s (0[,)+∞)))𝑡))
220	198, 203, 219	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠 ↾s (0[,)+∞)))𝑡))
221	220	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))))) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠 ↾s (0[,)+∞)))𝑡))
222		funmpt 6586	. . . . . . . . . . . . 13 ⊢ Fun (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))
223	222	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Fun (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)))
224	150, 177	eleqtrdi 2839	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
225	224	ralrimiva 3142	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
226	151	rnmptss 7128	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → ran (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)) ⊆ (Base‘(ℂfld ↾s (0[,)+∞))))
227	225, 226	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ran (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)) ⊆ (Base‘(ℂfld ↾s (0[,)+∞))))
228	169, 170, 171, 184, 193, 221, 223, 227	gsumpropd2 18634	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℂfld ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = ((ℝ*𝑠 ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))))
229	165, 166, 228	3eqtrd 2772	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = ((ℝ*𝑠 ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))))
230	30	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
231	148	recnd 11267	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ℂ)
232	230, 231	gsumfsum 21361	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
233	229, 232	eqtr3d 2770	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℝ*𝑠 ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
234	101, 161, 233	3eqtrrd 2773	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) = (𝐺 Σg (𝐹 ↾ 𝑥)))
235	234	mpteq2dva 5243	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))))
236	235	rneqd 5935	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) = ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))))
237	236	supeq1d 9464	. . . . 5 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ))
238	98, 237	eqtrd 2768	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ))
239	93, 238	pm2.61dan 812	. . 3 ⊢ (𝜑 → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ))
240	21, 9, 11, 1	xrge0tsms 24744	. . 3 ⊢ (𝜑 → (𝐺 tsums 𝐹) = {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )})
241	239, 240	eleq12d 2823	. 2 ⊢ (𝜑 → ((Σ^‘𝐹) ∈ (𝐺 tsums 𝐹) ↔ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )}))
242	8, 241	mpbird 257	1 ⊢ (𝜑 → (Σ^‘𝐹) ∈ (𝐺 tsums 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ≠ wne 2936  ∀wral 3057  ∃wrex 3066  Vcvv 3470   ∩ cin 3944   ⊆ wss 3945  𝒫 cpw 4599  {csn 4625   class class class wbr 5143   ↦ cmpt 5226  dom cdm 5673  ran crn 5674   ↾ cres 5675  Fun wfun 6537   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7415  Fincfn 8958  supcsup 9458  ℂcc 11131  ℝcr 11132  0cc0 11133   + caddc 11136  +∞cpnf 11270  ℝ^*cxr 11272   < clt 11273   ≤ cle 11274   +_𝑒 cxad 13117  [,)cico 13353  [,]cicc 13354  Σcsu 15659  Basecbs 17174   ↾_s cress 17203  +_gcplusg 17227   Σ_g cgsu 17416  ℝ^*_𝑠cxrs 17476  Mndcmnd 18688  SubMndcsubmnd 18733  CMndccmn 19729  SRingcsrg 20120  ℂ_fldccnfld 21273   tsums ctsu 24024  Σ^{^}csumge0 45741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-inf2 9659  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210  ax-pre-sup 11211  ax-addf 11212  ax-mulf 11213

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-iin 4995  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-se 5629  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-of 7680  df-om 7866  df-1st 7988  df-2nd 7989  df-supp 8161  df-frecs 8281  df-wrecs 8312  df-recs 8386  df-rdg 8425  df-1o 8481  df-er 8719  df-map 8841  df-en 8959  df-dom 8960  df-sdom 8961  df-fin 8962  df-fsupp 9381  df-fi 9429  df-sup 9460  df-inf 9461  df-oi 9528  df-card 9957  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-div 11897  df-nn 12238  df-2 12300  df-3 12301  df-4 12302  df-5 12303  df-6 12304  df-7 12305  df-8 12306  df-9 12307  df-n0 12498  df-z 12584  df-dec 12703  df-uz 12848  df-q 12958  df-rp 13002  df-xadd 13120  df-ioo 13355  df-ioc 13356  df-ico 13357  df-icc 13358  df-fz 13512  df-fzo 13655  df-seq 13994  df-exp 14054  df-hash 14317  df-cj 15073  df-re 15074  df-im 15075  df-sqrt 15209  df-abs 15210  df-clim 15459  df-sum 15660  df-struct 17110  df-sets 17127  df-slot 17145  df-ndx 17157  df-base 17175  df-ress 17204  df-plusg 17240  df-mulr 17241  df-starv 17242  df-tset 17246  df-ple 17247  df-ds 17249  df-unif 17250  df-rest 17398  df-topn 17399  df-0g 17417  df-gsum 17418  df-topgen 17419  df-ordt 17477  df-xrs 17478  df-mre 17560  df-mrc 17561  df-acs 17563  df-ps 18552  df-tsr 18553  df-mgm 18594  df-sgrp 18673  df-mnd 18689  df-submnd 18735  df-grp 18887  df-minusg 18888  df-mulg 19018  df-cntz 19262  df-cmn 19731  df-abl 19732  df-mgp 20069  df-ur 20116  df-srg 20121  df-ring 20169  df-cring 20170  df-fbas 21270  df-fg 21271  df-cnfld 21274  df-top 22790  df-topon 22807  df-topsp 22829  df-bases 22843  df-ntr 22918  df-nei 22996  df-cn 23125  df-haus 23213  df-fil 23744  df-fm 23836  df-flim 23837  df-flf 23838  df-tsms 24025  df-sumge0 45742

This theorem is referenced by: (None)

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