xlimmnfvlem2 - Mathbox for Glauco Siliprandi

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Theorem xlimmnfvlem2 40059

Description: The "if" part of the biconditional in xlimpnf 40068. (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: 𝑥,𝑘 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 21009	. . . . . . 7 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*)
2	1	a1i 11	. . . . . 6 ⊢ (𝜑 → (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*))
3	2	elfvexd 6222	. . . . 5 ⊢ (𝜑 → ℝ^* ∈ V)
4		cnex 10017	. . . . . 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 39708	. . . . . 6 ⊢ 𝑍 ⊆ ℂ
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑍 ⊆ ℂ)
10		elpm2r 7875	. . . . 5 ⊢ (((ℝ^* ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝑍⟶ℝ^* ∧ 𝑍 ⊆ ℂ)) → 𝐹 ∈ (ℝ^* ↑_pm ℂ))
11	3, 5, 6, 9, 10	syl22anc 1327	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (ℝ^* ↑_pm ℂ))
12		mnfxr 10096	. . . . 5 ⊢ -∞ ∈ ℝ^*
13	12	a1i 11	. . . 4 ⊢ (𝜑 → -∞ ∈ ℝ^*)
14		mnfnei 21025	. . . . . . . 8 ⊢ ((𝑢 ∈ (ordTop‘ ≤ ) ∧ -∞ ∈ 𝑢) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝑢)
15	14	adantll 750	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ (ordTop‘ ≤ )) ∧ -∞ ∈ 𝑢) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝑢)
16		xlimmnfvlem2.j	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝜑
17		nfv 1843	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 𝑥 ∈ ℝ
18	16, 17	nfan 1828	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑥 ∈ ℝ)
19		nfv 1843	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(-∞[,)𝑥) ⊆ 𝑢
20	18, 19	nfan 1828	. . . . . . . . . . 11 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢)
21		simprr 796	. . . . . . . . . . . . 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 1843	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘 𝑥 ∈ ℝ
24	22, 23	nfan 1828	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ ℝ)
25		nfv 1843	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(-∞[,)𝑥) ⊆ 𝑢
26	24, 25	nfan 1828	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢)
27		nfv 1843	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
28	26, 27	nfan 1828	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍)
29	7	uztrn2 11705	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
30	29	3adant1 1079	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
31	6	fdmd 39420	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → dom 𝐹 = 𝑍)
32	31	3ad2ant1 1082	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → dom 𝐹 = 𝑍)
33	30, 32	eleqtrrd 2704	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ dom 𝐹)
34	33	ad5ant134 1313	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → 𝑘 ∈ dom 𝐹)
35	34	adantl4r 787	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → 𝑘 ∈ dom 𝐹)
36		simp-4r 807	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → (-∞[,)𝑥) ⊆ 𝑢)
37	36	adantl4r 787	. . . . . . . . . . . . . . . . . 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 807	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → 𝑥 ∈ ℝ)
40		rexr 10085	. . . . . . . . . . . . . . . . . . . . 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 806	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → 𝜑)
43	29	ad4ant23 1297	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → 𝑘 ∈ 𝑍)
44	6	ffvelrnda 6359	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ^*)
45	42, 43, 44	syl2anc 693	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → (𝐹‘𝑘) ∈ ℝ^*)
46	45	mnfled 39609	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → -∞ ≤ (𝐹‘𝑘))
47		simpr 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → (𝐹‘𝑘) < 𝑥)
48	38, 41, 45, 46, 47	elicod 12224	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → (𝐹‘𝑘) ∈ (-∞[,)𝑥))
49	48	ad5ant1345 1316	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → (𝐹‘𝑘) ∈ (-∞[,)𝑥))
50	37, 49	sseldd 3604	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → (𝐹‘𝑘) ∈ 𝑢)
51	35, 50	jca 554	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (𝐹‘𝑘) < 𝑥) → (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))
52	51	ex 450	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) < 𝑥 → (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
53	28, 52	ralimda 39326	. . . . . . . . . . . . . 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 753	. . . . . . . . . . . . 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 1260	. . . . . . . . . . 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 2932	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) < 𝑥)
59	58	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) < 𝑥)
60	20, 56, 59	reximdd 39344	. . . . . . . . . 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 14088	. . . . . . . . . . . 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 762	. . . . . . . . . 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 222	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (-∞[,)𝑥) ⊆ 𝑢) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))
66	65	rexlimdva2 39339	. . . . . . . 8 ⊢ (𝜑 → (∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
67	66	ad2antrr 762	. . . . . . 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 450	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (ordTop‘ ≤ )) → (-∞ ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
70	69	ralrimiva 2966	. . . 4 ⊢ (𝜑 → ∀𝑢 ∈ (ordTop‘ ≤ )(-∞ ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
71	11, 13, 70	3jca 1242	. . 3 ⊢ (𝜑 → (𝐹 ∈ (ℝ^* ↑_pm ℂ) ∧ -∞ ∈ ℝ^* ∧ ∀𝑢 ∈ (ordTop‘ ≤ )(-∞ ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))))
72		nfcv 2764	. . . 4 ⊢ Ⅎ𝑘𝐹
73	72, 2	lmbr3 39979	. . 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 247	. 2 ⊢ (𝜑 → 𝐹(⇝_𝑡‘(ordTop‘ ≤ ))-∞)
75		df-xlim 40045	. . . 4 ⊢ ~~>* = (⇝_𝑡‘(ordTop‘ ≤ ))
76	75	breqi 4659	. . 3 ⊢ (𝐹~~>*-∞ ↔ 𝐹(⇝_𝑡‘(ordTop‘ ≤ ))-∞)
77	76	a1i 11	. 2 ⊢ (𝜑 → (𝐹~~>*-∞ ↔ 𝐹(⇝_𝑡‘(ordTop‘ ≤ ))-∞))
78	74, 77	mpbird 247	1 ⊢ (𝜑 → 𝐹~~>*-∞)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 Vcvv 3200 ⊆ wss 3574 class class class wbr 4653 dom cdm 5114 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑_pm cpm 7858 ℂcc 9934 ℝcr 9935 -∞cmnf 10072 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 ℤcz 11377 ℤ_≥cuz 11687 [,)cico 12177 ordTopcordt 16159 TopOnctopon 20715 ⇝_𝑡clm 21030 ~~>*clsxlim 40044

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fi 8317 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-z 11378 df-uz 11688 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-topgen 16104 df-ordt 16161 df-ps 17200 df-tsr 17201 df-top 20699 df-topon 20716 df-bases 20750 df-lm 21033 df-xlim 40045

This theorem is referenced by: xlimmnfv 40060

W3C validator