xlimmnfvlem2 - Mathbox for Glauco Siliprandi

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Theorem xlimmnfvlem2 45215

Description: Lemma for xlimmnf 45223: the "if" part of the biconditional. (Contributed by Glauco Siliprandi, 5-Feb-2022.)

**Hypotheses**
Ref	Expression
xlimmnfvlem2.k	⊢ Ⅎ𝑘𝜑
xlimmnfvlem2.j	⊢ Ⅎ𝑗𝜑
xlimmnfvlem2.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
xlimmnfvlem2.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
xlimmnfvlem2.f	⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
xlimmnfvlem2.g	⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) < 𝑥)

**Assertion**
Ref	Expression
xlimmnfvlem2	⊢ (𝜑 → 𝐹~~>*-∞)

Distinct variable groups: 𝑗,𝐹,𝑘,𝑥 𝑗,𝑀 𝑗,𝑍,𝑘 𝜑,𝑥 Allowed substitution hints: 𝜑(𝑗,𝑘) 𝑀(𝑥,𝑘) 𝑍(𝑥)

Proof of Theorem xlimmnfvlem2 Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		letopon 23102	. . . . . . 7 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*)
2	1	a1i 11	. . . . . 6 ⊢ (𝜑 → (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*))
3	2	elfvexd 6930	. . . . 5 ⊢ (𝜑 → ℝ^* ∈ V)
4		cnex 11213	. . . . . 6 ⊢ ℂ ∈ V
5	4	a1i 11	. . . . 5 ⊢ (𝜑 → ℂ ∈ V)
6		xlimmnfvlem2.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
7		xlimmnfvlem2.z	. . . . . . 7 ⊢ 𝑍 = (ℤ_≥‘𝑀)
8	7	uzsscn2 44854	. . . . . 6 ⊢ 𝑍 ⊆ ℂ
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑍 ⊆ ℂ)
10		elpm2r 8857	. . . . 5 ⊢ (((ℝ^* ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝑍⟶ℝ^* ∧ 𝑍 ⊆ ℂ)) → 𝐹 ∈ (ℝ^* ↑_pm ℂ))
11	3, 5, 6, 9, 10	syl22anc 838	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (ℝ^* ↑_pm ℂ))
12		mnfxr 11295	. . . . 5 ⊢ -∞ ∈ ℝ^*
13	12	a1i 11	. . . 4 ⊢ (𝜑 → -∞ ∈ ℝ^*)
14		mnfnei 23118	. . . . . . . 8 ⊢ ((𝑢 ∈ (ordTop‘ ≤ ) ∧ -∞ ∈ 𝑢) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝑢)
15	14	adantll 713	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ (ordTop‘ ≤ )) ∧ -∞ ∈ 𝑢) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝑢)
16		xlimmnfvlem2.j	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝜑
17		nfv 1910	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 𝑥 ∈ ℝ
18	16, 17	nfan 1895	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑥 ∈ ℝ)
19		nfv 1910	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(-∞[,)𝑥) ⊆ 𝑢
20	18, 19	nfan 1895	. . . . . . . . . . 11 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢)
21		simprr 772	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) < 𝑥)) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) < 𝑥)
22		xlimmnfvlem2.k	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘𝜑
23		nfv 1910	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘 𝑥 ∈ ℝ
24	22, 23	nfan 1895	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ ℝ)
25		nfv 1910	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(-∞[,)𝑥) ⊆ 𝑢
26	24, 25	nfan 1895	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢)
27		nfv 1910	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
28	26, 27	nfan 1895	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍)
29	7	uztrn2 12865	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
30	29	3adant1 1128	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
31	6	fdmd 6727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → dom 𝐹 = 𝑍)
32	31	3ad2ant1 1131	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → dom 𝐹 = 𝑍)
33	30, 32	eleqtrrd 2832	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ dom 𝐹)
34	33	ad5ant134 1365	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → 𝑘 ∈ dom 𝐹)
35	34	adantl4r 754	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → 𝑘 ∈ dom 𝐹)
36		simp-4r 783	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → (-∞[,)𝑥) ⊆ 𝑢)
37	36	adantl4r 754	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → (-∞[,)𝑥) ⊆ 𝑢)
38	12	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → -∞ ∈ ℝ^*)
39		simp-4r 783	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → 𝑥 ∈ ℝ)
40		rexr 11284	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
41	39, 40	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → 𝑥 ∈ ℝ^*)
42		simp-4l 782	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → 𝜑)
43	29	ad4ant23 752	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → 𝑘 ∈ 𝑍)
44	6	ffvelcdmda 7088	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ^*)
45	42, 43, 44	syl2anc 583	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → (𝐹‘𝑘) ∈ ℝ^*)
46	45	mnfled 13141	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → -∞ ≤ (𝐹‘𝑘))
47		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → (𝐹‘𝑘) < 𝑥)
48	38, 41, 45, 46, 47	elicod 13400	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → (𝐹‘𝑘) ∈ (-∞[,)𝑥))
49	48	adantl3r 749	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → (𝐹‘𝑘) ∈ (-∞[,)𝑥))
50	37, 49	sseldd 3979	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → (𝐹‘𝑘) ∈ 𝑢)
51	35, 50	jca 511	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))
52	51	ex 412	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) < 𝑥 → (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
53	28, 52	ralimdaa 3253	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) < 𝑥 → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
54	53	adantrr 716	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) < 𝑥)) → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) < 𝑥 → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
55	21, 54	mpd 15	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) < 𝑥)) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))
56	55	3impb 1113	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) < 𝑥) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))
57		xlimmnfvlem2.g	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) < 𝑥)
58	57	r19.21bi 3244	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) < 𝑥)
59	58	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) < 𝑥)
60	20, 56, 59	reximdd 44512	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))
61		xlimmnfvlem2.m	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
62	7	rexuz3 15321	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
63	61, 62	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
64	63	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
65	60, 64	mpbid 231	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))
66	65	rexlimdva2 3153	. . . . . . . 8 ⊢ (𝜑 → (∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
67	66	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ (ordTop‘ ≤ )) ∧ -∞ ∈ 𝑢) → (∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
68	15, 67	mpd 15	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ (ordTop‘ ≤ )) ∧ -∞ ∈ 𝑢) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))
69	68	ex 412	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (ordTop‘ ≤ )) → (-∞ ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
70	69	ralrimiva 3142	. . . 4 ⊢ (𝜑 → ∀𝑢 ∈ (ordTop‘ ≤ )(-∞ ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
71	11, 13, 70	3jca 1126	. . 3 ⊢ (𝜑 → (𝐹 ∈ (ℝ^* ↑_pm ℂ) ∧ -∞ ∈ ℝ^* ∧ ∀𝑢 ∈ (ordTop‘ ≤ )(-∞ ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))))
72		nfcv 2899	. . . 4 ⊢ Ⅎ𝑘𝐹
73	72, 2	lmbr3 45129	. . 3 ⊢ (𝜑 → (𝐹(⇝_𝑡‘(ordTop‘ ≤ ))-∞ ↔ (𝐹 ∈ (ℝ^* ↑_pm ℂ) ∧ -∞ ∈ ℝ^* ∧ ∀𝑢 ∈ (ordTop‘ ≤ )(-∞ ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))))
74	71, 73	mpbird 257	. 2 ⊢ (𝜑 → 𝐹(⇝_𝑡‘(ordTop‘ ≤ ))-∞)
75		df-xlim 45201	. . . 4 ⊢ ~~>* = (⇝_𝑡‘(ordTop‘ ≤ ))
76	75	breqi 5148	. . 3 ⊢ (𝐹~~>*-∞ ↔ 𝐹(⇝_𝑡‘(ordTop‘ ≤ ))-∞)
77	76	a1i 11	. 2 ⊢ (𝜑 → (𝐹~~>*-∞ ↔ 𝐹(⇝_𝑡‘(ordTop‘ ≤ ))-∞))
78	74, 77	mpbird 257	1 ⊢ (𝜑 → 𝐹~~>*-∞)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 ∀wral 3057 ∃wrex 3066 Vcvv 3470 ⊆ wss 3945 class class class wbr 5142 dom cdm 5672 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ↑_pm cpm 8839 ℂcc 11130 ℝcr 11131 -∞cmnf 11270 ℝ^*cxr 11271 < clt 11272 ≤ cle 11273 ℤcz 12582 ℤ_≥cuz 12846 [,)cico 13352 ordTopcordt 17474 TopOnctopon 22805 ⇝_𝑡clm 23123 ~~>*clsxlim 45200

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-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209

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-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 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-1o 8480 df-er 8718 df-pm 8841 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fi 9428 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-z 12583 df-uz 12847 df-ioo 13354 df-ioc 13355 df-ico 13356 df-icc 13357 df-topgen 17418 df-ordt 17476 df-ps 18551 df-tsr 18552 df-top 22789 df-topon 22806 df-bases 22842 df-lm 23126 df-xlim 45201

This theorem is referenced by: xlimmnfv 45216

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