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Theorem tsmsgsum 24056

Description: The convergent points of a finite topological group sum are the closure of the finite group sum operation. (Contributed by Mario Carneiro, 19-Sep-2015.) (Revised by AV, 24-Jul-2019.)

**Hypotheses**
Ref	Expression
tsmsid.b	⊢ 𝐵 = (Base‘𝐺)
tsmsid.z	⊢ 0 = (0_g‘𝐺)
tsmsid.1	⊢ (𝜑 → 𝐺 ∈ CMnd)
tsmsid.2	⊢ (𝜑 → 𝐺 ∈ TopSp)
tsmsid.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
tsmsid.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
tsmsid.w	⊢ (𝜑 → 𝐹 finSupp 0 )
tsmsgsum.j	⊢ 𝐽 = (TopOpen‘𝐺)

**Assertion**
Ref	Expression
tsmsgsum	⊢ (𝜑 → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{(𝐺 Σ_g 𝐹)}))

Proof of Theorem tsmsgsum Dummy variables 𝑦 𝑧 𝑢 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tsmsid.2	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TopSp)
2		tsmsid.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
3		tsmsgsum.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝐺)
4	2, 3	istps 22849	. . . . . . 7 ⊢ (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐵))
5	1, 4	sylib 217	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐵))
6		toponuni 22829	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽)
7	5, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝐵 = ∪ 𝐽)
8	7	eleq2d 2815	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ ∪ 𝐽))
9		elfpw 9379	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑦 ⊆ 𝐴 ∧ 𝑦 ∈ Fin))
10	9	simplbi 497	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ⊆ 𝐴)
11	10	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ⊆ 𝐴)
12		suppssdm 8182	. . . . . . . . . . . . . . 15 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
13		tsmsid.f	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
14	12, 13	fssdm 6742	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴)
15	14	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 supp 0 ) ⊆ 𝐴)
16	11, 15	unssd 4186	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑦 ∪ (𝐹 supp 0 )) ⊆ 𝐴)
17		elinel2 4196	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ∈ Fin)
18	17	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ Fin)
19		tsmsid.w	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 finSupp 0 )
20	19	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐹 finSupp 0 )
21	20	fsuppimpd 9394	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 supp 0 ) ∈ Fin)
22		unfi 9197	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ Fin ∧ (𝐹 supp 0 ) ∈ Fin) → (𝑦 ∪ (𝐹 supp 0 )) ∈ Fin)
23	18, 21, 22	syl2anc 583	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑦 ∪ (𝐹 supp 0 )) ∈ Fin)
24		elfpw 9379	. . . . . . . . . . . 12 ⊢ ((𝑦 ∪ (𝐹 supp 0 )) ∈ (𝒫 𝐴 ∩ Fin) ↔ ((𝑦 ∪ (𝐹 supp 0 )) ⊆ 𝐴 ∧ (𝑦 ∪ (𝐹 supp 0 )) ∈ Fin))
25	16, 23, 24	sylanbrc 582	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑦 ∪ (𝐹 supp 0 )) ∈ (𝒫 𝐴 ∩ Fin))
26		ssun1 4172	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ⊆ (𝑦 ∪ (𝐹 supp 0 ))
27		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑦 ∪ (𝐹 supp 0 )) → 𝑧 = (𝑦 ∪ (𝐹 supp 0 )))
28	26, 27	sseqtrrid 4033	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑦 ∪ (𝐹 supp 0 )) → 𝑦 ⊆ 𝑧)
29		pm5.5 361	. . . . . . . . . . . . . 14 ⊢ (𝑦 ⊆ 𝑧 → ((𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢) ↔ (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢))
30	28, 29	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑦 ∪ (𝐹 supp 0 )) → ((𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢) ↔ (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢))
31		reseq2 5980	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑦 ∪ (𝐹 supp 0 )) → (𝐹 ↾ 𝑧) = (𝐹 ↾ (𝑦 ∪ (𝐹 supp 0 ))))
32	31	oveq2d 7436	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑦 ∪ (𝐹 supp 0 )) → (𝐺 Σ_g (𝐹 ↾ 𝑧)) = (𝐺 Σ_g (𝐹 ↾ (𝑦 ∪ (𝐹 supp 0 )))))
33	32	eleq1d 2814	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑦 ∪ (𝐹 supp 0 )) → ((𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢 ↔ (𝐺 Σ_g (𝐹 ↾ (𝑦 ∪ (𝐹 supp 0 )))) ∈ 𝑢))
34	30, 33	bitrd 279	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑦 ∪ (𝐹 supp 0 )) → ((𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢) ↔ (𝐺 Σ_g (𝐹 ↾ (𝑦 ∪ (𝐹 supp 0 )))) ∈ 𝑢))
35	34	rspcv 3605	. . . . . . . . . . 11 ⊢ ((𝑦 ∪ (𝐹 supp 0 )) ∈ (𝒫 𝐴 ∩ Fin) → (∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢) → (𝐺 Σ_g (𝐹 ↾ (𝑦 ∪ (𝐹 supp 0 )))) ∈ 𝑢))
36	25, 35	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢) → (𝐺 Σ_g (𝐹 ↾ (𝑦 ∪ (𝐹 supp 0 )))) ∈ 𝑢))
37		tsmsid.z	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘𝐺)
38		tsmsid.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ CMnd)
39	38	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐺 ∈ CMnd)
40		tsmsid.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
41	40	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐴 ∈ 𝑉)
42	13	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐹:𝐴⟶𝐵)
43		ssun2 4173	. . . . . . . . . . . . 13 ⊢ (𝐹 supp 0 ) ⊆ (𝑦 ∪ (𝐹 supp 0 ))
44	43	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 supp 0 ) ⊆ (𝑦 ∪ (𝐹 supp 0 )))
45	2, 37, 39, 41, 42, 44, 20	gsumres 19868	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐺 Σ_g (𝐹 ↾ (𝑦 ∪ (𝐹 supp 0 )))) = (𝐺 Σ_g 𝐹))
46	45	eleq1d 2814	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ((𝐺 Σ_g (𝐹 ↾ (𝑦 ∪ (𝐹 supp 0 )))) ∈ 𝑢 ↔ (𝐺 Σ_g 𝐹) ∈ 𝑢))
47	36, 46	sylibd 238	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢) → (𝐺 Σ_g 𝐹) ∈ 𝑢))
48	47	rexlimdva 3152	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐽) → (∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢) → (𝐺 Σ_g 𝐹) ∈ 𝑢))
49	19	fsuppimpd 9394	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
50		elfpw 9379	. . . . . . . . . . 11 ⊢ ((𝐹 supp 0 ) ∈ (𝒫 𝐴 ∩ Fin) ↔ ((𝐹 supp 0 ) ⊆ 𝐴 ∧ (𝐹 supp 0 ) ∈ Fin))
51	14, 49, 50	sylanbrc 582	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ (𝒫 𝐴 ∩ Fin))
52	38	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐽 ∧ (𝐺 Σ_g 𝐹) ∈ 𝑢)) ∧ (𝑧 ∈ (𝒫 𝐴 ∩ Fin) ∧ (𝐹 supp 0 ) ⊆ 𝑧)) → 𝐺 ∈ CMnd)
53	40	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐽 ∧ (𝐺 Σ_g 𝐹) ∈ 𝑢)) ∧ (𝑧 ∈ (𝒫 𝐴 ∩ Fin) ∧ (𝐹 supp 0 ) ⊆ 𝑧)) → 𝐴 ∈ 𝑉)
54	13	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐽 ∧ (𝐺 Σ_g 𝐹) ∈ 𝑢)) ∧ (𝑧 ∈ (𝒫 𝐴 ∩ Fin) ∧ (𝐹 supp 0 ) ⊆ 𝑧)) → 𝐹:𝐴⟶𝐵)
55		simprr 772	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐽 ∧ (𝐺 Σ_g 𝐹) ∈ 𝑢)) ∧ (𝑧 ∈ (𝒫 𝐴 ∩ Fin) ∧ (𝐹 supp 0 ) ⊆ 𝑧)) → (𝐹 supp 0 ) ⊆ 𝑧)
56	19	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐽 ∧ (𝐺 Σ_g 𝐹) ∈ 𝑢)) ∧ (𝑧 ∈ (𝒫 𝐴 ∩ Fin) ∧ (𝐹 supp 0 ) ⊆ 𝑧)) → 𝐹 finSupp 0 )
57	2, 37, 52, 53, 54, 55, 56	gsumres 19868	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐽 ∧ (𝐺 Σ_g 𝐹) ∈ 𝑢)) ∧ (𝑧 ∈ (𝒫 𝐴 ∩ Fin) ∧ (𝐹 supp 0 ) ⊆ 𝑧)) → (𝐺 Σ_g (𝐹 ↾ 𝑧)) = (𝐺 Σ_g 𝐹))
58		simplrr 777	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐽 ∧ (𝐺 Σ_g 𝐹) ∈ 𝑢)) ∧ (𝑧 ∈ (𝒫 𝐴 ∩ Fin) ∧ (𝐹 supp 0 ) ⊆ 𝑧)) → (𝐺 Σ_g 𝐹) ∈ 𝑢)
59	57, 58	eqeltrd 2829	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐽 ∧ (𝐺 Σ_g 𝐹) ∈ 𝑢)) ∧ (𝑧 ∈ (𝒫 𝐴 ∩ Fin) ∧ (𝐹 supp 0 ) ⊆ 𝑧)) → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢)
60	59	expr 456	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐽 ∧ (𝐺 Σ_g 𝐹) ∈ 𝑢)) ∧ 𝑧 ∈ (𝒫 𝐴 ∩ Fin)) → ((𝐹 supp 0 ) ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢))
61	60	ralrimiva 3143	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐽 ∧ (𝐺 Σ_g 𝐹) ∈ 𝑢)) → ∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)((𝐹 supp 0 ) ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢))
62		sseq1 4005	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹 supp 0 ) → (𝑦 ⊆ 𝑧 ↔ (𝐹 supp 0 ) ⊆ 𝑧))
63	62	rspceaimv 3615	. . . . . . . . . 10 ⊢ (((𝐹 supp 0 ) ∈ (𝒫 𝐴 ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)((𝐹 supp 0 ) ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢)) → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢))
64	51, 61, 63	syl2an2r 684	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐽 ∧ (𝐺 Σ_g 𝐹) ∈ 𝑢)) → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢))
65	64	expr 456	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐽) → ((𝐺 Σ_g 𝐹) ∈ 𝑢 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢)))
66	48, 65	impbid 211	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐽) → (∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢) ↔ (𝐺 Σ_g 𝐹) ∈ 𝑢))
67		disjsn 4716	. . . . . . . 8 ⊢ ((𝑢 ∩ {(𝐺 Σ_g 𝐹)}) = ∅ ↔ ¬ (𝐺 Σ_g 𝐹) ∈ 𝑢)
68	67	necon2abii 2988	. . . . . . 7 ⊢ ((𝐺 Σ_g 𝐹) ∈ 𝑢 ↔ (𝑢 ∩ {(𝐺 Σ_g 𝐹)}) ≠ ∅)
69	66, 68	bitrdi 287	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐽) → (∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢) ↔ (𝑢 ∩ {(𝐺 Σ_g 𝐹)}) ≠ ∅))
70	69	imbi2d 340	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐽) → ((𝑥 ∈ 𝑢 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢)) ↔ (𝑥 ∈ 𝑢 → (𝑢 ∩ {(𝐺 Σ_g 𝐹)}) ≠ ∅)))
71	70	ralbidva 3172	. . . 4 ⊢ (𝜑 → (∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢)) ↔ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → (𝑢 ∩ {(𝐺 Σ_g 𝐹)}) ≠ ∅)))
72	8, 71	anbi12d 631	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢))) ↔ (𝑥 ∈ ∪ 𝐽 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → (𝑢 ∩ {(𝐺 Σ_g 𝐹)}) ≠ ∅))))
73		eqid 2728	. . . 4 ⊢ (𝒫 𝐴 ∩ Fin) = (𝒫 𝐴 ∩ Fin)
74	2, 3, 73, 38, 1, 40, 13	eltsms 24050	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢)))))
75		topontop 22828	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
76	5, 75	syl 17	. . . 4 ⊢ (𝜑 → 𝐽 ∈ Top)
77	2, 37, 38, 40, 13, 19	gsumcl 19870	. . . . . 6 ⊢ (𝜑 → (𝐺 Σ_g 𝐹) ∈ 𝐵)
78	77	snssd 4813	. . . . 5 ⊢ (𝜑 → {(𝐺 Σ_g 𝐹)} ⊆ 𝐵)
79	78, 7	sseqtrd 4020	. . . 4 ⊢ (𝜑 → {(𝐺 Σ_g 𝐹)} ⊆ ∪ 𝐽)
80		eqid 2728	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
81	80	elcls2 22991	. . . 4 ⊢ ((𝐽 ∈ Top ∧ {(𝐺 Σ_g 𝐹)} ⊆ ∪ 𝐽) → (𝑥 ∈ ((cls‘𝐽)‘{(𝐺 Σ_g 𝐹)}) ↔ (𝑥 ∈ ∪ 𝐽 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → (𝑢 ∩ {(𝐺 Σ_g 𝐹)}) ≠ ∅))))
82	76, 79, 81	syl2anc 583	. . 3 ⊢ (𝜑 → (𝑥 ∈ ((cls‘𝐽)‘{(𝐺 Σ_g 𝐹)}) ↔ (𝑥 ∈ ∪ 𝐽 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → (𝑢 ∩ {(𝐺 Σ_g 𝐹)}) ≠ ∅))))
83	72, 74, 82	3bitr4d 311	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) ↔ 𝑥 ∈ ((cls‘𝐽)‘{(𝐺 Σ_g 𝐹)})))
84	83	eqrdv 2726	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{(𝐺 Σ_g 𝐹)}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ∪ cun 3945 ∩ cin 3946 ⊆ wss 3947 ∅c0 4323 𝒫 cpw 4603 {csn 4629 ∪ cuni 4908 class class class wbr 5148 ↾ cres 5680 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 supp csupp 8165 Fincfn 8964 finSupp cfsupp 9386 Basecbs 17180 TopOpenctopn 17403 0_gc0g 17421 Σ_g cgsu 17422 CMndccmn 19735 Topctop 22808 TopOnctopon 22825 TopSpctps 22847 clsccl 22935 tsums ctsu 24043

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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216

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 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-fzo 13661 df-seq 14000 df-hash 14323 df-0g 17423 df-gsum 17424 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-cntz 19268 df-cmn 19737 df-fbas 21276 df-fg 21277 df-top 22809 df-topon 22826 df-topsp 22848 df-cld 22936 df-ntr 22937 df-cls 22938 df-nei 23015 df-fil 23763 df-fm 23855 df-flim 23856 df-flf 23857 df-tsms 24044

This theorem is referenced by: tsmsid 24057 tgptsmscls 24067

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