Theorem cmetcaulem 25323

Description: Lemma for cmetcau 25324. (Contributed by Mario Carneiro, 14-Oct-2015.)

Hypotheses

Ref	Expression
cmetcau.1	⊢ 𝐽 = (MetOpen‘𝐷)
cmetcau.3	⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
cmetcau.4	⊢ (𝜑 → 𝑃 ∈ 𝑋)
cmetcau.5	⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷))
cmetcau.6	⊢ 𝐺 = (𝑥 ∈ ℕ ↦ if(𝑥 ∈ dom 𝐹, (𝐹‘𝑥), 𝑃))

Assertion

Ref	Expression
cmetcaulem	⊢ (𝜑 → 𝐹 ∈ dom (⇝𝑡‘𝐽))

Distinct variable groups:   𝑥,𝐷   𝑥,𝐹   𝑥,𝑃   𝑥,𝐽   𝜑,𝑥   𝑥,𝑋

Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem cmetcaulem

Dummy variables 𝑗 𝑘 𝑚 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cmetcau.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
2		cmetmet 25321	. . . . . . . . 9 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
3	1, 2	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
4		metxmet 24345	. . . . . . . 8 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
6		cmetcau.1	. . . . . . . 8 ⊢ 𝐽 = (MetOpen‘𝐷)
7	6	mopntopon 24450	. . . . . . 7 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
8	5, 7	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
9		1z 12649	. . . . . . . 8 ⊢ 1 ∈ ℤ
10		nnuz 12922	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
11	10	uzfbas 23907	. . . . . . . 8 ⊢ (1 ∈ ℤ → (ℤ≥ “ ℕ) ∈ (fBas‘ℕ))
12	9, 11	mp1i 13	. . . . . . 7 ⊢ (𝜑 → (ℤ≥ “ ℕ) ∈ (fBas‘ℕ))
13		fgcl 23887	. . . . . . 7 ⊢ ((ℤ≥ “ ℕ) ∈ (fBas‘ℕ) → (ℕfilGen(ℤ≥ “ ℕ)) ∈ (Fil‘ℕ))
14	12, 13	syl 17	. . . . . 6 ⊢ (𝜑 → (ℕfilGen(ℤ≥ “ ℕ)) ∈ (Fil‘ℕ))
15		elfvdm 6942	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ dom CMet)
16	1, 15	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ dom CMet)
17		cnex 11237	. . . . . . . . . . . 12 ⊢ ℂ ∈ V
18	17	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ℂ ∈ V)
19		cmetcau.5	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷))
20		caufpm 25317	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝐹 ∈ (𝑋 ↑pm ℂ))
21	5, 19, 20	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (𝑋 ↑pm ℂ))
22		elpm2g 8885	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ dom CMet ∧ ℂ ∈ V) → (𝐹 ∈ (𝑋 ↑pm ℂ) ↔ (𝐹:dom 𝐹⟶𝑋 ∧ dom 𝐹 ⊆ ℂ)))
23	22	simprbda 498	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ dom CMet ∧ ℂ ∈ V) ∧ 𝐹 ∈ (𝑋 ↑pm ℂ)) → 𝐹:dom 𝐹⟶𝑋)
24	16, 18, 21, 23	syl21anc 837	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:dom 𝐹⟶𝑋)
25	24	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 𝐹:dom 𝐹⟶𝑋)
26	25	ffvelcdmda 7103	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ 𝑋)
27		cmetcau.4	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ 𝑋)
28	27	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑃 ∈ 𝑋)
29	26, 28	ifclda 4560	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → if(𝑥 ∈ dom 𝐹, (𝐹‘𝑥), 𝑃) ∈ 𝑋)
30		cmetcau.6	. . . . . . 7 ⊢ 𝐺 = (𝑥 ∈ ℕ ↦ if(𝑥 ∈ dom 𝐹, (𝐹‘𝑥), 𝑃))
31	29, 30	fmptd 7133	. . . . . 6 ⊢ (𝜑 → 𝐺:ℕ⟶𝑋)
32		flfval 23999	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (ℕfilGen(ℤ≥ “ ℕ)) ∈ (Fil‘ℕ) ∧ 𝐺:ℕ⟶𝑋) → ((𝐽 fLimf (ℕfilGen(ℤ≥ “ ℕ)))‘𝐺) = (𝐽 fLim ((𝑋 FilMap 𝐺)‘(ℕfilGen(ℤ≥ “ ℕ)))))
33	8, 14, 31, 32	syl3anc 1372	. . . . 5 ⊢ (𝜑 → ((𝐽 fLimf (ℕfilGen(ℤ≥ “ ℕ)))‘𝐺) = (𝐽 fLim ((𝑋 FilMap 𝐺)‘(ℕfilGen(ℤ≥ “ ℕ)))))
34		eqid 2736	. . . . . . . 8 ⊢ (ℕfilGen(ℤ≥ “ ℕ)) = (ℕfilGen(ℤ≥ “ ℕ))
35	34	fmfg 23958	. . . . . . 7 ⊢ ((𝑋 ∈ dom CMet ∧ (ℤ≥ “ ℕ) ∈ (fBas‘ℕ) ∧ 𝐺:ℕ⟶𝑋) → ((𝑋 FilMap 𝐺)‘(ℤ≥ “ ℕ)) = ((𝑋 FilMap 𝐺)‘(ℕfilGen(ℤ≥ “ ℕ))))
36	16, 12, 31, 35	syl3anc 1372	. . . . . 6 ⊢ (𝜑 → ((𝑋 FilMap 𝐺)‘(ℤ≥ “ ℕ)) = ((𝑋 FilMap 𝐺)‘(ℕfilGen(ℤ≥ “ ℕ))))
37	36	oveq2d 7448	. . . . 5 ⊢ (𝜑 → (𝐽 fLim ((𝑋 FilMap 𝐺)‘(ℤ≥ “ ℕ))) = (𝐽 fLim ((𝑋 FilMap 𝐺)‘(ℕfilGen(ℤ≥ “ ℕ)))))
38	33, 37	eqtr4d 2779	. . . 4 ⊢ (𝜑 → ((𝐽 fLimf (ℕfilGen(ℤ≥ “ ℕ)))‘𝐺) = (𝐽 fLim ((𝑋 FilMap 𝐺)‘(ℤ≥ “ ℕ))))
39		biidd 262	. . . . . . . 8 ⊢ (𝑧 = 1 → (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹 ↔ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹))
40		1zzd 12650	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℤ)
41	10, 5, 40	iscau3 25313	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ ∀𝑧 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑤 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑤)) < 𝑧))))
42	41	simplbda 499	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (Cau‘𝐷)) → ∀𝑧 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑤 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑤)) < 𝑧))
43	19, 42	mpdan 687	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑤 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑤)) < 𝑧))
44		simp1 1136	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑤 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑤)) < 𝑧) → 𝑘 ∈ dom 𝐹)
45	44	ralimi 3082	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑤 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑤)) < 𝑧) → ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹)
46	45	reximi 3083	. . . . . . . . . 10 ⊢ (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑤 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑤)) < 𝑧) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹)
47	46	ralimi 3082	. . . . . . . . 9 ⊢ (∀𝑧 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑤 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑤)) < 𝑧) → ∀𝑧 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹)
48	43, 47	syl 17	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹)
49		1rp 13039	. . . . . . . . 9 ⊢ 1 ∈ ℝ+
50	49	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ+)
51	39, 48, 50	rspcdva 3622	. . . . . . 7 ⊢ (𝜑 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹)
52		dfss3 3971	. . . . . . . . 9 ⊢ ((ℤ≥‘𝑗) ⊆ dom 𝐹 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹)
53		nnsscn 12272	. . . . . . . . . . . . . 14 ⊢ ℕ ⊆ ℂ
54	31, 53	jctir 520	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺:ℕ⟶𝑋 ∧ ℕ ⊆ ℂ))
55		elpm2r 8886	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ dom CMet ∧ ℂ ∈ V) ∧ (𝐺:ℕ⟶𝑋 ∧ ℕ ⊆ ℂ)) → 𝐺 ∈ (𝑋 ↑pm ℂ))
56	16, 18, 54, 55	syl21anc 837	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ (𝑋 ↑pm ℂ))
57	56	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) → 𝐺 ∈ (𝑋 ↑pm ℂ))
58		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (ℤ≥‘𝑗) = (ℤ≥‘𝑗)
59	5	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) → 𝐷 ∈ (∞Met‘𝑋))
60		nnz 12636	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℤ)
61	60	ad2antrl 728	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) → 𝑗 ∈ ℤ)
62		eqidd 2737	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) = (𝐹‘𝑘))
63		eqidd 2737	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑚) = (𝐹‘𝑚))
64	58, 59, 61, 62, 63	iscau4 25314	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ ∀𝑧 ∈ ℝ+ ∃𝑚 ∈ (ℤ≥‘𝑗)∀𝑘 ∈ (ℤ≥‘𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑧))))
65	64	simplbda 499	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝐹 ∈ (Cau‘𝐷)) → ∀𝑧 ∈ ℝ+ ∃𝑚 ∈ (ℤ≥‘𝑗)∀𝑘 ∈ (ℤ≥‘𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑧))
66	19, 65	mpidan 689	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) → ∀𝑧 ∈ ℝ+ ∃𝑚 ∈ (ℤ≥‘𝑗)∀𝑘 ∈ (ℤ≥‘𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑧))
67		simprl 770	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) → 𝑗 ∈ ℕ)
68		eluznn 12961	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ℕ ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 𝑚 ∈ ℕ)
69	67, 68	sylan 580	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 𝑚 ∈ ℕ)
70		eluznn 12961	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝑚)) → 𝑘 ∈ ℕ)
71	30, 29	dmmptd 6712	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → dom 𝐺 = ℕ)
72	71	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) → dom 𝐺 = ℕ)
73	72	eleq2d 2826	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) → (𝑘 ∈ dom 𝐺 ↔ 𝑘 ∈ ℕ))
74	73	biimpar 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ dom 𝐺)
75	74	a1d 25	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑘 ∈ ℕ) → (𝑘 ∈ dom 𝐹 → 𝑘 ∈ dom 𝐺))
76		idd 24	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) ∈ 𝑋 → (𝐹‘𝑘) ∈ 𝑋))
77		idd 24	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑘 ∈ ℕ) → (((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑧 → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑧))
78	75, 76, 77	3anim123d 1444	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑘 ∈ ℕ) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑧) → (𝑘 ∈ dom 𝐺 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑧)))
79	70, 78	sylan2 593	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ (𝑚 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝑚))) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑧) → (𝑘 ∈ dom 𝐺 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑧)))
80	79	anassrs 467	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑚)) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑧) → (𝑘 ∈ dom 𝐺 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑧)))
81	80	ralimdva 3166	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑚 ∈ ℕ) → (∀𝑘 ∈ (ℤ≥‘𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑧) → ∀𝑘 ∈ (ℤ≥‘𝑚)(𝑘 ∈ dom 𝐺 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑧)))
82	69, 81	syldan 591	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (∀𝑘 ∈ (ℤ≥‘𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑧) → ∀𝑘 ∈ (ℤ≥‘𝑚)(𝑘 ∈ dom 𝐺 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑧)))
83	82	reximdva 3167	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) → (∃𝑚 ∈ (ℤ≥‘𝑗)∀𝑘 ∈ (ℤ≥‘𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑧) → ∃𝑚 ∈ (ℤ≥‘𝑗)∀𝑘 ∈ (ℤ≥‘𝑚)(𝑘 ∈ dom 𝐺 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑧)))
84	83	ralimdv 3168	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) → (∀𝑧 ∈ ℝ+ ∃𝑚 ∈ (ℤ≥‘𝑗)∀𝑘 ∈ (ℤ≥‘𝑚)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑧) → ∀𝑧 ∈ ℝ+ ∃𝑚 ∈ (ℤ≥‘𝑗)∀𝑘 ∈ (ℤ≥‘𝑚)(𝑘 ∈ dom 𝐺 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑧)))
85	66, 84	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) → ∀𝑧 ∈ ℝ+ ∃𝑚 ∈ (ℤ≥‘𝑗)∀𝑘 ∈ (ℤ≥‘𝑚)(𝑘 ∈ dom 𝐺 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑧))
86		eluznn 12961	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ ℕ)
87	67, 86	sylan 580	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ ℕ)
88		simprr 772	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) → (ℤ≥‘𝑗) ⊆ dom 𝐹)
89	88	sselda 3982	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ dom 𝐹)
90		iftrue 4530	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ dom 𝐹 → if(𝑘 ∈ dom 𝐹, (𝐹‘𝑘), 𝑃) = (𝐹‘𝑘))
91	90	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℕ ∧ 𝑘 ∈ dom 𝐹) → if(𝑘 ∈ dom 𝐹, (𝐹‘𝑘), 𝑃) = (𝐹‘𝑘))
92		fvex 6918	. . . . . . . . . . . . . . . 16 ⊢ (𝐹‘𝑘) ∈ V
93	91, 92	eqeltrdi 2848	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℕ ∧ 𝑘 ∈ dom 𝐹) → if(𝑘 ∈ dom 𝐹, (𝐹‘𝑘), 𝑃) ∈ V)
94		eleq1w 2823	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → (𝑥 ∈ dom 𝐹 ↔ 𝑘 ∈ dom 𝐹))
95		fveq2 6905	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → (𝐹‘𝑥) = (𝐹‘𝑘))
96	94, 95	ifbieq1d 4549	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑘 → if(𝑥 ∈ dom 𝐹, (𝐹‘𝑥), 𝑃) = if(𝑘 ∈ dom 𝐹, (𝐹‘𝑘), 𝑃))
97	96, 30	fvmptg 7013	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℕ ∧ if(𝑘 ∈ dom 𝐹, (𝐹‘𝑘), 𝑃) ∈ V) → (𝐺‘𝑘) = if(𝑘 ∈ dom 𝐹, (𝐹‘𝑘), 𝑃))
98	93, 97	syldan 591	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℕ ∧ 𝑘 ∈ dom 𝐹) → (𝐺‘𝑘) = if(𝑘 ∈ dom 𝐹, (𝐹‘𝑘), 𝑃))
99	98, 91	eqtrd 2776	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ ∧ 𝑘 ∈ dom 𝐹) → (𝐺‘𝑘) = (𝐹‘𝑘))
100	87, 89, 99	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐺‘𝑘) = (𝐹‘𝑘))
101	88	sselda 3982	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 𝑚 ∈ dom 𝐹)
102	69, 101	elind 4199	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 𝑚 ∈ (ℕ ∩ dom 𝐹))
103		fveq2 6905	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → (𝐺‘𝑘) = (𝐺‘𝑚))
104		fveq2 6905	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
105	103, 104	eqeq12d 2752	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑚 → ((𝐺‘𝑘) = (𝐹‘𝑘) ↔ (𝐺‘𝑚) = (𝐹‘𝑚)))
106		elin 3966	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℕ ∩ dom 𝐹) ↔ (𝑘 ∈ ℕ ∧ 𝑘 ∈ dom 𝐹))
107	106, 99	sylbi 217	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (ℕ ∩ dom 𝐹) → (𝐺‘𝑘) = (𝐹‘𝑘))
108	105, 107	vtoclga 3576	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (ℕ ∩ dom 𝐹) → (𝐺‘𝑚) = (𝐹‘𝑚))
109	102, 108	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (𝐺‘𝑚) = (𝐹‘𝑚))
110	58, 59, 61, 100, 109	iscau4 25314	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) → (𝐺 ∈ (Cau‘𝐷) ↔ (𝐺 ∈ (𝑋 ↑pm ℂ) ∧ ∀𝑧 ∈ ℝ+ ∃𝑚 ∈ (ℤ≥‘𝑗)∀𝑘 ∈ (ℤ≥‘𝑚)(𝑘 ∈ dom 𝐺 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑧))))
111	57, 85, 110	mpbir2and 713	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) → 𝐺 ∈ (Cau‘𝐷))
112	111	expr 456	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((ℤ≥‘𝑗) ⊆ dom 𝐹 → 𝐺 ∈ (Cau‘𝐷)))
113	52, 112	biimtrrid 243	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹 → 𝐺 ∈ (Cau‘𝐷)))
114	113	rexlimdva 3154	. . . . . . 7 ⊢ (𝜑 → (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹 → 𝐺 ∈ (Cau‘𝐷)))
115	51, 114	mpd 15	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (Cau‘𝐷))
116		eqid 2736	. . . . . . . 8 ⊢ ((𝑋 FilMap 𝐺)‘(ℤ≥ “ ℕ)) = ((𝑋 FilMap 𝐺)‘(ℤ≥ “ ℕ))
117	10, 116	caucfil 25318	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 1 ∈ ℤ ∧ 𝐺:ℕ⟶𝑋) → (𝐺 ∈ (Cau‘𝐷) ↔ ((𝑋 FilMap 𝐺)‘(ℤ≥ “ ℕ)) ∈ (CauFil‘𝐷)))
118	5, 40, 31, 117	syl3anc 1372	. . . . . 6 ⊢ (𝜑 → (𝐺 ∈ (Cau‘𝐷) ↔ ((𝑋 FilMap 𝐺)‘(ℤ≥ “ ℕ)) ∈ (CauFil‘𝐷)))
119	115, 118	mpbid 232	. . . . 5 ⊢ (𝜑 → ((𝑋 FilMap 𝐺)‘(ℤ≥ “ ℕ)) ∈ (CauFil‘𝐷))
120	6	cmetcvg 25320	. . . . 5 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ ((𝑋 FilMap 𝐺)‘(ℤ≥ “ ℕ)) ∈ (CauFil‘𝐷)) → (𝐽 fLim ((𝑋 FilMap 𝐺)‘(ℤ≥ “ ℕ))) ≠ ∅)
121	1, 119, 120	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐽 fLim ((𝑋 FilMap 𝐺)‘(ℤ≥ “ ℕ))) ≠ ∅)
122	38, 121	eqnetrd 3007	. . 3 ⊢ (𝜑 → ((𝐽 fLimf (ℕfilGen(ℤ≥ “ ℕ)))‘𝐺) ≠ ∅)
123		n0 4352	. . 3 ⊢ (((𝐽 fLimf (ℕfilGen(ℤ≥ “ ℕ)))‘𝐺) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ((𝐽 fLimf (ℕfilGen(ℤ≥ “ ℕ)))‘𝐺))
124	122, 123	sylib 218	. 2 ⊢ (𝜑 → ∃𝑦 𝑦 ∈ ((𝐽 fLimf (ℕfilGen(ℤ≥ “ ℕ)))‘𝐺))
125	10, 34	lmflf 24014	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 1 ∈ ℤ ∧ 𝐺:ℕ⟶𝑋) → (𝐺(⇝𝑡‘𝐽)𝑦 ↔ 𝑦 ∈ ((𝐽 fLimf (ℕfilGen(ℤ≥ “ ℕ)))‘𝐺)))
126	8, 40, 31, 125	syl3anc 1372	. . . 4 ⊢ (𝜑 → (𝐺(⇝𝑡‘𝐽)𝑦 ↔ 𝑦 ∈ ((𝐽 fLimf (ℕfilGen(ℤ≥ “ ℕ)))‘𝐺)))
127	21	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐺(⇝𝑡‘𝐽)𝑦) → 𝐹 ∈ (𝑋 ↑pm ℂ))
128		lmcl 23306	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐺(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ 𝑋)
129	8, 128	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐺(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ 𝑋)
130	6, 5, 10, 40	lmmbr3 25295	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺(⇝𝑡‘𝐽)𝑦 ↔ (𝐺 ∈ (𝑋 ↑pm ℂ) ∧ 𝑦 ∈ 𝑋 ∧ ∀𝑧 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧))))
131	130	biimpa 476	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐺(⇝𝑡‘𝐽)𝑦) → (𝐺 ∈ (𝑋 ↑pm ℂ) ∧ 𝑦 ∈ 𝑋 ∧ ∀𝑧 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧)))
132	131	simp3d 1144	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐺(⇝𝑡‘𝐽)𝑦) → ∀𝑧 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧))
133		r19.26 3110	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ ℝ+ (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹 ∧ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧)) ↔ (∀𝑧 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹 ∧ ∀𝑧 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧)))
134	10	rexanuz2 15389	. . . . . . . . . . . . 13 ⊢ (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧)) ↔ (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹 ∧ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧)))
135		simprl 770	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧))) → 𝑘 ∈ dom 𝐹)
136	99	ad2ant2lr 748	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧))) → (𝐺‘𝑘) = (𝐹‘𝑘))
137		simprr2 1222	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧))) → (𝐺‘𝑘) ∈ 𝑋)
138	136, 137	eqeltrrd 2841	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧))) → (𝐹‘𝑘) ∈ 𝑋)
139	136	oveq1d 7447	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧))) → ((𝐺‘𝑘)𝐷𝑦) = ((𝐹‘𝑘)𝐷𝑦))
140		simprr3 1223	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧))) → ((𝐺‘𝑘)𝐷𝑦) < 𝑧)
141	139, 140	eqbrtrrd 5166	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧))) → ((𝐹‘𝑘)𝐷𝑦) < 𝑧)
142	135, 138, 141	3jca 1128	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧))) → (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷𝑦) < 𝑧))
143	142	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 ∈ dom 𝐹 ∧ (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧)) → (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷𝑦) < 𝑧)))
144	86, 143	sylan2 593	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝑗))) → ((𝑘 ∈ dom 𝐹 ∧ (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧)) → (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷𝑦) < 𝑧)))
145	144	anassrs 467	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ((𝑘 ∈ dom 𝐹 ∧ (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧)) → (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷𝑦) < 𝑧)))
146	145	ralimdva 3166	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧)) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷𝑦) < 𝑧)))
147	146	reximdva 3167	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧)) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷𝑦) < 𝑧)))
148	134, 147	biimtrrid 243	. . . . . . . . . . . 12 ⊢ (𝜑 → ((∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹 ∧ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧)) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷𝑦) < 𝑧)))
149	148	ralimdv 3168	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑧 ∈ ℝ+ (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹 ∧ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧)) → ∀𝑧 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷𝑦) < 𝑧)))
150	133, 149	biimtrrid 243	. . . . . . . . . 10 ⊢ (𝜑 → ((∀𝑧 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹 ∧ ∀𝑧 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧)) → ∀𝑧 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷𝑦) < 𝑧)))
151	48, 150	mpand 695	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑧 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧) → ∀𝑧 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷𝑦) < 𝑧)))
152	151	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐺(⇝𝑡‘𝐽)𝑦) → (∀𝑧 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧) → ∀𝑧 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷𝑦) < 𝑧)))
153	132, 152	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐺(⇝𝑡‘𝐽)𝑦) → ∀𝑧 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷𝑦) < 𝑧))
154	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐺(⇝𝑡‘𝐽)𝑦) → 𝐷 ∈ (∞Met‘𝑋))
155		1zzd 12650	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐺(⇝𝑡‘𝐽)𝑦) → 1 ∈ ℤ)
156	6, 154, 10, 155	lmmbr3 25295	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐺(⇝𝑡‘𝐽)𝑦) → (𝐹(⇝𝑡‘𝐽)𝑦 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑦 ∈ 𝑋 ∧ ∀𝑧 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷𝑦) < 𝑧))))
157	127, 129, 153, 156	mpbir3and 1342	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺(⇝𝑡‘𝐽)𝑦) → 𝐹(⇝𝑡‘𝐽)𝑦)
158		lmrel 23239	. . . . . . 7 ⊢ Rel (⇝𝑡‘𝐽)
159	158	releldmi 5958	. . . . . 6 ⊢ (𝐹(⇝𝑡‘𝐽)𝑦 → 𝐹 ∈ dom (⇝𝑡‘𝐽))
160	157, 159	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐺(⇝𝑡‘𝐽)𝑦) → 𝐹 ∈ dom (⇝𝑡‘𝐽))
161	160	ex 412	. . . 4 ⊢ (𝜑 → (𝐺(⇝𝑡‘𝐽)𝑦 → 𝐹 ∈ dom (⇝𝑡‘𝐽)))
162	126, 161	sylbird 260	. . 3 ⊢ (𝜑 → (𝑦 ∈ ((𝐽 fLimf (ℕfilGen(ℤ≥ “ ℕ)))‘𝐺) → 𝐹 ∈ dom (⇝𝑡‘𝐽)))
163	162	exlimdv 1932	. 2 ⊢ (𝜑 → (∃𝑦 𝑦 ∈ ((𝐽 fLimf (ℕfilGen(ℤ≥ “ ℕ)))‘𝐺) → 𝐹 ∈ dom (⇝𝑡‘𝐽)))
164	124, 163	mpd 15	1 ⊢ (𝜑 → 𝐹 ∈ dom (⇝𝑡‘𝐽))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539  ∃wex 1778   ∈ wcel 2107   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  Vcvv 3479   ∩ cin 3949   ⊆ wss 3950  ∅c0 4332  ifcif 4524   class class class wbr 5142   ↦ cmpt 5224  dom cdm 5684   “ cima 5687  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432   ↑_pm cpm 8868  ℂcc 11154  1c1 11157   < clt 11296  ℕcn 12267  ℤcz 12615  ℤ_≥cuz 12879  ℝ⁺crp 13035  ∞Metcxmet 21350  Metcmet 21351  fBascfbas 21353  filGencfg 21354  MetOpencmopn 21355  TopOnctopon 22917  ⇝_𝑡clm 23235  Filcfil 23854   FilMap cfm 23942   fLim cflim 23943   fLimf cflf 23944  CauFilccfil 25287  Cauccau 25288  CMetccmet 25289

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-map 8869  df-pm 8870  df-en 8987  df-dom 8988  df-sdom 8989  df-sup 9483  df-inf 9484  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-n0 12529  df-z 12616  df-uz 12880  df-q 12992  df-rp 13036  df-xneg 13155  df-xadd 13156  df-xmul 13157  df-ico 13394  df-rest 17468  df-topgen 17489  df-psmet 21357  df-xmet 21358  df-met 21359  df-bl 21360  df-mopn 21361  df-fbas 21362  df-fg 21363  df-top 22901  df-topon 22918  df-bases 22954  df-ntr 23029  df-nei 23107  df-lm 23238  df-fil 23855  df-fm 23947  df-flim 23948  df-flf 23949  df-cfil 25290  df-cau 25291  df-cmet 25292

This theorem is referenced by:  cmetcau  25324