xlimmnfvlem2 - Mathbox for Glauco Siliprandi

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Theorem xlimmnfvlem2 46662

Description: Lemma for xlimmnf 46670: 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 23431	. . . . . . 7 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*)
2	1	a1i 11	. . . . . 6 ⊢ (𝜑 → (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*))
3	2	elfvexd 6915	. . . . 5 ⊢ (𝜑 → ℝ^* ∈ V)
4		cnex 11206	. . . . . 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 46306	. . . . . 6 ⊢ 𝑍 ⊆ ℂ
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑍 ⊆ ℂ)
10		elpm2r 8845	. . . . 5 ⊢ (((ℝ^* ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝑍⟶ℝ^* ∧ 𝑍 ⊆ ℂ)) → 𝐹 ∈ (ℝ^* ↑_pm ℂ))
11	3, 5, 6, 9, 10	syl22anc 852	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (ℝ^* ↑_pm ℂ))
12		mnfxr 11291	. . . . 5 ⊢ -∞ ∈ ℝ^*
13	12	a1i 11	. . . 4 ⊢ (𝜑 → -∞ ∈ ℝ^*)
14		mnfnei 23447	. . . . . . . 8 ⊢ ((𝑢 ∈ (ordTop‘ ≤ ) ∧ -∞ ∈ 𝑢) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝑢)
15	14	adantll 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ (ordTop‘ ≤ )) ∧ -∞ ∈ 𝑢) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝑢)
16		xlimmnfvlem2.j	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝜑
17		nfv 1947	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 𝑥 ∈ ℝ
18	16, 17	nfan 1932	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑥 ∈ ℝ)
19		nfv 1947	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(-∞[,)𝑥) ⊆ 𝑢
20	18, 19	nfan 1932	. . . . . . . . . . 11 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢)
21		simprr 785	. . . . . . . . . . . . 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 1947	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘 𝑥 ∈ ℝ
24	22, 23	nfan 1932	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ ℝ)
25		nfv 1947	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(-∞[,)𝑥) ⊆ 𝑢
26	24, 25	nfan 1932	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢)
27		nfv 1947	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
28	26, 27	nfan 1932	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍)
29	7	uztrn2 12907	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
30	29	3adant1 1148	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
31	6	fdmd 6714	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → dom 𝐹 = 𝑍)
32	31	3ad2ant1 1151	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → dom 𝐹 = 𝑍)
33	30, 32	eleqtrrd 2863	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ dom 𝐹)
34	33	ad5ant134 1392	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → 𝑘 ∈ dom 𝐹)
35	34	adantl4r 768	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → 𝑘 ∈ dom 𝐹)
36		simp-4r 796	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → (-∞[,)𝑥) ⊆ 𝑢)
37	36	adantl4r 768	. . . . . . . . . . . . . . . . . 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 796	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → 𝑥 ∈ ℝ)
40		rexr 11280	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
41	39, 40	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → 𝑥 ∈ ℝ^*)
42		simp-4l 795	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → 𝜑)
43	29	ad4ant23 766	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → 𝑘 ∈ 𝑍)
44	6	ffvelcdmda 7078	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ^*)
45	42, 43, 44	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → (𝐹‘𝑘) ∈ ℝ^*)
46	45	mnfled 13188	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → -∞ ≤ (𝐹‘𝑘))
47		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → (𝐹‘𝑘) < 𝑥)
48	38, 41, 45, 46, 47	elicod 13449	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → (𝐹‘𝑘) ∈ (-∞[,)𝑥))
49	48	adantl3r 763	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → (𝐹‘𝑘) ∈ (-∞[,)𝑥))
50	37, 49	sseldd 3932	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → (𝐹‘𝑘) ∈ 𝑢)
51	35, 50	jca 521	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))
52	51	ex 418	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) < 𝑥 → (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
53	28, 52	ralimdaa 3263	. . . . . . . . . . . . . 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 730	. . . . . . . . . . . . 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 16	. . . . . . . . . . . 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 1132	. . . . . . . . . . 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 3254	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) < 𝑥)
59	58	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) < 𝑥)
60	20, 56, 59	reximdd 45981	. . . . . . . . . 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 15437	. . . . . . . . . . . 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 18	. . . . . . . . . . 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 739	. . . . . . . . . 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 235	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))
66	65	rexlimdva2 3165	. . . . . . . 8 ⊢ (𝜑 → (∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
67	66	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ (ordTop‘ ≤ )) ∧ -∞ ∈ 𝑢) → (∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
68	15, 67	mpd 16	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ (ordTop‘ ≤ )) ∧ -∞ ∈ 𝑢) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))
69	68	ex 418	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (ordTop‘ ≤ )) → (-∞ ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
70	69	ralrimiva 3154	. . . 4 ⊢ (𝜑 → ∀𝑢 ∈ (ordTop‘ ≤ )(-∞ ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
71	11, 13, 70	3jca 1146	. . 3 ⊢ (𝜑 → (𝐹 ∈ (ℝ^* ↑_pm ℂ) ∧ -∞ ∈ ℝ^* ∧ ∀𝑢 ∈ (ordTop‘ ≤ )(-∞ ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))))
72		nfcv 2922	. . . 4 ⊢ Ⅎ𝑘𝐹
73	72, 2	lmbr3 46576	. . 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 260	. 2 ⊢ (𝜑 → 𝐹(⇝_𝑡‘(ordTop‘ ≤ ))-∞)
75		df-xlim 46648	. . . 4 ⊢ ~~>* = (⇝_𝑡‘(ordTop‘ ≤ ))
76	75	breqi 5109	. . 3 ⊢ (𝐹~~>*-∞ ↔ 𝐹(⇝_𝑡‘(ordTop‘ ≤ ))-∞)
77	76	a1i 11	. 2 ⊢ (𝜑 → (𝐹~~>*-∞ ↔ 𝐹(⇝_𝑡‘(ordTop‘ ≤ ))-∞))
78	74, 77	mpbird 260	1 ⊢ (𝜑 → 𝐹~~>*-∞)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 Ⅎwnf 1816 ∈ wcel 2145 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ⊆ wss 3899 class class class wbr 5103 dom cdm 5655 ⟶wf 6529 ‘cfv 6533 (class class class)co 7414 ↑_pm cpm 8828 ℂcc 11123 ℝcr 11124 -∞cmnf 11266 ℝ^*cxr 11267 < clt 11268 ≤ cle 11269 ℤcz 12616 ℤ_≥cuz 12888 [,)cico 13401 ordTopcordt 17586 TopOnctopon 23136 ⇝_𝑡clm 23452 ~~>*clsxlim 46647

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-1st 7987 df-2nd 7988 df-1o 8456 df-2o 8457 df-er 8697 df-pm 8830 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fi 9382 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-z 12617 df-uz 12889 df-ioo 13403 df-ioc 13404 df-ico 13405 df-icc 13406 df-topgen 17529 df-ordt 17588 df-ps 18655 df-tsr 18656 df-top 23120 df-topon 23137 df-bases 23172 df-lm 23455 df-xlim 46648

This theorem is used by: xlimmnfv 46663

W3C validator