xlimpnfvlem2 - Mathbox for Glauco Siliprandi

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Theorem xlimpnfvlem2 45866

Description: Lemma for xlimpnfv 45867: 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 23143	. . . . . . 7 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*)
2	1	a1i 11	. . . . . 6 ⊢ (𝜑 → (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*))
3	2	elfvexd 6915	. . . . 5 ⊢ (𝜑 → ℝ^* ∈ V)
4		cnex 11210	. . . . . 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 45504	. . . . . 6 ⊢ 𝑍 ⊆ ℂ
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑍 ⊆ ℂ)
10		elpm2r 8859	. . . . 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 11289	. . . . 5 ⊢ +∞ ∈ ℝ^*
13	12	a1i 11	. . . 4 ⊢ (𝜑 → +∞ ∈ ℝ^*)
14		pnfnei 23158	. . . . . . . 8 ⊢ ((𝑢 ∈ (ordTop‘ ≤ ) ∧ +∞ ∈ 𝑢) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝑢)
15	14	adantll 714	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ (ordTop‘ ≤ )) ∧ +∞ ∈ 𝑢) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝑢)
16		xlimpnfvlem2.j	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝜑
17		nfv 1914	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 𝑥 ∈ ℝ
18	16, 17	nfan 1899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑥 ∈ ℝ)
19		nfv 1914	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝑥(,]+∞) ⊆ 𝑢
20	18, 19	nfan 1899	. . . . . . . . . . 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 1914	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘 𝑥 ∈ ℝ
24	22, 23	nfan 1899	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ ℝ)
25		nfv 1914	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝑥(,]+∞) ⊆ 𝑢
26	24, 25	nfan 1899	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑢)
27		nfv 1914	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
28	26, 27	nfan 1899	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍)
29	7	uztrn2 12871	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
30	29	3adant1 1130	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
31	6	fdmd 6716	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → dom 𝐹 = 𝑍)
32	31	3ad2ant1 1133	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → dom 𝐹 = 𝑍)
33	30, 32	eleqtrrd 2837	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ dom 𝐹)
34	33	ad5ant134 1369	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (𝑥(,]+∞) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 < (𝐹‘𝑘)) → 𝑘 ∈ dom 𝐹)
35	34	adantl4r 755	. . . . . . . . . . . . . . . . 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 755	. . . . . . . . . . . . . . . . . 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 11281	. . . . . . . . . . . . . . . . . . . . 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 753	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 < (𝐹‘𝑘)) → 𝑘 ∈ 𝑍)
44	6	ffvelcdmda 7074	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ^*)
45	42, 43, 44	syl2anc 584	. . . . . . . . . . . . . . . . . . . 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 7075	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ^*)
49	48	pnfged 13147	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ≤ +∞)
50	49	ad5ant134 1369	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 < (𝐹‘𝑘)) → (𝐹‘𝑘) ≤ +∞)
51	40, 41, 45, 46, 50	eliocd 45536	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 < (𝐹‘𝑘)) → (𝐹‘𝑘) ∈ (𝑥(,]+∞))
52	51	adantl3r 750	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥(,]+∞) ⊆ 𝑢) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑥 < (𝐹‘𝑘)) → (𝐹‘𝑘) ∈ (𝑥(,]+∞))
53	37, 52	sseldd 3959	. . . . . . . . . . . . . . . . 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 3243	. . . . . . . . . . . . . 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 717	. . . . . . . . . . . . 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 1114	. . . . . . . . . . 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 3234	. . . . . . . . . . . 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 45172	. . . . . . . . . 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 15367	. . . . . . . . . . . 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 726	. . . . . . . . . 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 3143	. . . . . . . 8 ⊢ (𝜑 → (∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
70	69	ad2antrr 726	. . . . . . 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 3132	. . . 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 2898	. . . 4 ⊢ Ⅎ𝑘𝐹
76	75, 2	lmbr3 45776	. . 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 45848	. . . 4 ⊢ ~~>* = (⇝_𝑡‘(ordTop‘ ≤ ))
79	78	breqi 5125	. . 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 1086 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 ∀wral 3051 ∃wrex 3060 Vcvv 3459 ⊆ wss 3926 class class class wbr 5119 dom cdm 5654 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 ↑_pm cpm 8841 ℂcc 11127 ℝcr 11128 +∞cpnf 11266 ℝ^*cxr 11268 < clt 11269 ≤ cle 11270 ℤcz 12588 ℤ_≥cuz 12852 (,]cioc 13363 ordTopcordt 17513 TopOnctopon 22848 ⇝_𝑡clm 23164 ~~>*clsxlim 45847

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-1o 8480 df-2o 8481 df-er 8719 df-pm 8843 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fi 9423 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-z 12589 df-uz 12853 df-ioo 13366 df-ioc 13367 df-ico 13368 df-icc 13369 df-topgen 17457 df-ordt 17515 df-ps 18576 df-tsr 18577 df-top 22832 df-topon 22849 df-bases 22884 df-lm 23167 df-xlim 45848

This theorem is referenced by: xlimpnfv 45867

W3C validator