xlimpnfvlem2 - Mathbox for Glauco Siliprandi

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Theorem xlimpnfvlem2 45758

Description: Lemma for xlimpnfv 45759: the "if" part of the biconditional. (Contributed by Glauco Siliprandi, 5-Feb-2022.)

**Hypotheses**
Ref	Expression
xlimpnfvlem2.k	⊢ Ⅎ𝑘𝜑
xlimpnfvlem2.j	⊢ Ⅎ𝑗𝜑
xlimpnfvlem2.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
xlimpnfvlem2.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
xlimpnfvlem2.f	⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
xlimpnfvlem2.g	⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑥 < (𝐹‘𝑘))

**Assertion**
Ref	Expression
xlimpnfvlem2	⊢ (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 xlimpnfvlem2 Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		letopon 23234	. . . . . . 7 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*)
2	1	a1i 11	. . . . . 6 ⊢ (𝜑 → (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*))
3	2	elfvexd 6959	. . . . 5 ⊢ (𝜑 → ℝ^* ∈ V)
4		cnex 11265	. . . . . 6 ⊢ ℂ ∈ V
5	4	a1i 11	. . . . 5 ⊢ (𝜑 → ℂ ∈ V)
6		xlimpnfvlem2.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
7		xlimpnfvlem2.z	. . . . . . 7 ⊢ 𝑍 = (ℤ_≥‘𝑀)
8	7	uzsscn2 45393	. . . . . 6 ⊢ 𝑍 ⊆ ℂ
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑍 ⊆ ℂ)
10		elpm2r 8903	. . . . 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		pnfxr 11344	. . . . 5 ⊢ +∞ ∈ ℝ^*
13	12	a1i 11	. . . 4 ⊢ (𝜑 → +∞ ∈ ℝ^*)
14		pnfnei 23249	. . . . . . . 8 ⊢ ((𝑢 ∈ (ordTop‘ ≤ ) ∧ +∞ ∈ 𝑢) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝑢)
15	14	adantll 713	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ (ordTop‘ ≤ )) ∧ +∞ ∈ 𝑢) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝑢)
16		xlimpnfvlem2.j	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝜑
17		nfv 1913	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 𝑥 ∈ ℝ
18	16, 17	nfan 1898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑥 ∈ ℝ)
19		nfv 1913	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝑥(,]+∞) ⊆ 𝑢
20	18, 19	nfan 1898	. . . . . . . . . . 11 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑢)
21		simprr 772	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑢) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑥 < (𝐹‘𝑘))) → ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑥 < (𝐹‘𝑘))
22		xlimpnfvlem2.k	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘𝜑
23		nfv 1913	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘 𝑥 ∈ ℝ
24	22, 23	nfan 1898	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ ℝ)
25		nfv 1913	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝑥(,]+∞) ⊆ 𝑢
26	24, 25	nfan 1898	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑢)
27		nfv 1913	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
28	26, 27	nfan 1898	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍)
29	7	uztrn2 12922	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
30	29	3adant1 1130	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
31	6	fdmd 6757	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → dom 𝐹 = 𝑍)
32	31	3ad2ant1 1133	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → dom 𝐹 = 𝑍)
33	30, 32	eleqtrrd 2847	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ dom 𝐹)
34	33	ad5ant134 1367	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (𝑥(,]+∞) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 < (𝐹‘𝑘)) → 𝑘 ∈ dom 𝐹)
35	34	adantl4r 754	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 < (𝐹‘𝑘)) → 𝑘 ∈ dom 𝐹)
36		simp-4r 783	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ (𝑥(,]+∞) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 < (𝐹‘𝑘)) → (𝑥(,]+∞) ⊆ 𝑢)
37	36	adantl4r 754	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 < (𝐹‘𝑘)) → (𝑥(,]+∞) ⊆ 𝑢)
38		simp-4r 783	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 < (𝐹‘𝑘)) → 𝑥 ∈ ℝ)
39		rexr 11336	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
40	38, 39	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 < (𝐹‘𝑘)) → 𝑥 ∈ ℝ^*)
41	12	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 < (𝐹‘𝑘)) → +∞ ∈ ℝ^*)
42		simp-4l 782	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 < (𝐹‘𝑘)) → 𝜑)
43	29	ad4ant23 752	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 < (𝐹‘𝑘)) → 𝑘 ∈ 𝑍)
44	6	ffvelcdmda 7118	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ^*)
45	42, 43, 44	syl2anc 583	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 < (𝐹‘𝑘)) → (𝐹‘𝑘) ∈ ℝ^*)
46		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 < (𝐹‘𝑘)) → 𝑥 < (𝐹‘𝑘))
47	6	3ad2ant1 1133	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝐹:𝑍⟶ℝ^*)
48	47, 30	ffvelcdmd 7119	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ^*)
49	48	pnfged 45389	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ≤ +∞)
50	49	ad5ant134 1367	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 < (𝐹‘𝑘)) → (𝐹‘𝑘) ≤ +∞)
51	40, 41, 45, 46, 50	eliocd 45425	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 < (𝐹‘𝑘)) → (𝐹‘𝑘) ∈ (𝑥(,]+∞))
52	51	adantl3r 749	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 < (𝐹‘𝑘)) → (𝐹‘𝑘) ∈ (𝑥(,]+∞))
53	37, 52	sseldd 4009	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 < (𝐹‘𝑘)) → (𝐹‘𝑘) ∈ 𝑢)
54	35, 53	jca 511	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 < (𝐹‘𝑘)) → (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))
55	54	ex 412	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝑥 < (𝐹‘𝑘) → (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
56	28, 55	ralimdaa 3266	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)𝑥 < (𝐹‘𝑘) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
57	56	adantrr 716	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑢) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑥 < (𝐹‘𝑘))) → (∀𝑘 ∈ (ℤ_≥‘𝑗)𝑥 < (𝐹‘𝑘) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
58	21, 57	mpd 15	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑢) ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑥 < (𝐹‘𝑘))) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))
59	58	3impb 1115	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑥 < (𝐹‘𝑘)) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))
60		xlimpnfvlem2.g	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑥 < (𝐹‘𝑘))
61	60	r19.21bi 3257	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑥 < (𝐹‘𝑘))
62	61	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑢) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑥 < (𝐹‘𝑘))
63	20, 59, 62	reximdd 45053	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑢) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))
64		xlimpnfvlem2.m	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
65	7	rexuz3 15397	. . . . . . . . . . . 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)))
66	64, 65	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)))
67	66	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)))
68	63, 67	mpbid 232	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑢) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))
69	68	rexlimdva2 3163	. . . . . . . 8 ⊢ (𝜑 → (∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
70	69	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ (ordTop‘ ≤ )) ∧ +∞ ∈ 𝑢) → (∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
71	15, 70	mpd 15	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ (ordTop‘ ≤ )) ∧ +∞ ∈ 𝑢) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))
72	71	ex 412	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (ordTop‘ ≤ )) → (+∞ ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
73	72	ralrimiva 3152	. . . 4 ⊢ (𝜑 → ∀𝑢 ∈ (ordTop‘ ≤ )(+∞ ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
74	11, 13, 73	3jca 1128	. . 3 ⊢ (𝜑 → (𝐹 ∈ (ℝ^* ↑_pm ℂ) ∧ +∞ ∈ ℝ^* ∧ ∀𝑢 ∈ (ordTop‘ ≤ )(+∞ ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))))
75		nfcv 2908	. . . 4 ⊢ Ⅎ𝑘𝐹
76	75, 2	lmbr3 45668	. . 3 ⊢ (𝜑 → (𝐹(⇝_𝑡‘(ordTop‘ ≤ ))+∞ ↔ (𝐹 ∈ (ℝ^* ↑_pm ℂ) ∧ +∞ ∈ ℝ^* ∧ ∀𝑢 ∈ (ordTop‘ ≤ )(+∞ ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))))
77	74, 76	mpbird 257	. 2 ⊢ (𝜑 → 𝐹(⇝_𝑡‘(ordTop‘ ≤ ))+∞)
78		df-xlim 45740	. . . 4 ⊢ ~~>* = (⇝_𝑡‘(ordTop‘ ≤ ))
79	78	breqi 5172	. . 3 ⊢ (𝐹~~>*+∞ ↔ 𝐹(⇝_𝑡‘(ordTop‘ ≤ ))+∞)
80	79	a1i 11	. 2 ⊢ (𝜑 → (𝐹~~>*+∞ ↔ 𝐹(⇝_𝑡‘(ordTop‘ ≤ ))+∞))
81	77, 80	mpbird 257	1 ⊢ (𝜑 → 𝐹~~>*+∞)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 Ⅎwnf 1781 ∈ wcel 2108 ∀wral 3067 ∃wrex 3076 Vcvv 3488 ⊆ wss 3976 class class class wbr 5166 dom cdm 5700 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ↑_pm cpm 8885 ℂcc 11182 ℝcr 11183 +∞cpnf 11321 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 ℤcz 12639 ℤ_≥cuz 12903 (,]cioc 13408 ordTopcordt 17559 TopOnctopon 22937 ⇝_𝑡clm 23255 ~~>*clsxlim 45739

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-1o 8522 df-2o 8523 df-er 8763 df-pm 8887 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fi 9480 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-z 12640 df-uz 12904 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-topgen 17503 df-ordt 17561 df-ps 18636 df-tsr 18637 df-top 22921 df-topon 22938 df-bases 22974 df-lm 23258 df-xlim 45740

This theorem is referenced by: xlimpnfv 45759

W3C validator