Theorem cmetcaulem 25245

Description: Lemma for cmetcau 25246. (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 25243	. . . . . . . . 9 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
3	1, 2	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
4		metxmet 24278	. . . . . . . 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 24383	. . . . . . 7 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
8	5, 7	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
9		1z 12627	. . . . . . . 8 ⊢ 1 ∈ ℤ
10		nnuz 12900	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
11	10	uzfbas 23841	. . . . . . . 8 ⊢ (1 ∈ ℤ → (ℤ≥ “ ℕ) ∈ (fBas‘ℕ))
12	9, 11	mp1i 13	. . . . . . 7 ⊢ (𝜑 → (ℤ≥ “ ℕ) ∈ (fBas‘ℕ))
13		fgcl 23821	. . . . . . 7 ⊢ ((ℤ≥ “ ℕ) ∈ (fBas‘ℕ) → (ℕfilGen(ℤ≥ “ ℕ)) ∈ (Fil‘ℕ))
14	12, 13	syl 17	. . . . . 6 ⊢ (𝜑 → (ℕfilGen(ℤ≥ “ ℕ)) ∈ (Fil‘ℕ))
15		elfvdm 6918	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ dom CMet)
16	1, 15	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ dom CMet)
17		cnex 11215	. . . . . . . . . . . 12 ⊢ ℂ ∈ V
18	17	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ℂ ∈ V)
19		cmetcau.5	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷))
20		caufpm 25239	. . . . . . . . . . . 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 8863	. . . . . . . . . . . 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 7079	. . . . . . . 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 4541	. . . . . . 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 7109	. . . . . 6 ⊢ (𝜑 → 𝐺:ℕ⟶𝑋)
32		flfval 23933	. . . . . 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 1373	. . . . 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 23892	. . . . . . 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 1373	. . . . . 6 ⊢ (𝜑 → ((𝑋 FilMap 𝐺)‘(ℤ≥ “ ℕ)) = ((𝑋 FilMap 𝐺)‘(ℕfilGen(ℤ≥ “ ℕ))))
37	36	oveq2d 7426	. . . . 5 ⊢ (𝜑 → (𝐽 fLim ((𝑋 FilMap 𝐺)‘(ℤ≥ “ ℕ))) = (𝐽 fLim ((𝑋 FilMap 𝐺)‘(ℕfilGen(ℤ≥ “ ℕ)))))
38	33, 37	eqtr4d 2774	. . . 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 12628	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℤ)
41	10, 5, 40	iscau3 25235	. . . . . . . . . . 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 3074	. . . . . . . . . . 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 3075	. . . . . . . . . 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 3074	. . . . . . . . 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 13017	. . . . . . . . 9 ⊢ 1 ∈ ℝ+
50	49	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ+)
51	39, 48, 50	rspcdva 3607	. . . . . . 7 ⊢ (𝜑 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹)
52		dfss3 3952	. . . . . . . . 9 ⊢ ((ℤ≥‘𝑗) ⊆ dom 𝐹 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹)
53		nnsscn 12250	. . . . . . . . . . . . . 14 ⊢ ℕ ⊆ ℂ
54	31, 53	jctir 520	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺:ℕ⟶𝑋 ∧ ℕ ⊆ ℂ))
55		elpm2r 8864	. . . . . . . . . . . . 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 12614	. . . . . . . . . . . . . . . 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 25236	. . . . . . . . . . . . . 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 12939	. . . . . . . . . . . . . . . 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 12939	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝑚)) → 𝑘 ∈ ℕ)
71	30, 29	dmmptd 6688	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → dom 𝐺 = ℕ)
72	71	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) → dom 𝐺 = ℕ)
73	72	eleq2d 2821	. . . . . . . . . . . . . . . . . . . . 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 1445	. . . . . . . . . . . . . . . . . 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 3153	. . . . . . . . . . . . . . 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 3154	. . . . . . . . . . . . 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 3155	. . . . . . . . . . . 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 12939	. . . . . . . . . . . . . 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 3963	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ dom 𝐹)
90		iftrue 4511	. . . . . . . . . . . . . . . . 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 6894	. . . . . . . . . . . . . . . 16 ⊢ (𝐹‘𝑘) ∈ V
93	91, 92	eqeltrdi 2843	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℕ ∧ 𝑘 ∈ dom 𝐹) → if(𝑘 ∈ dom 𝐹, (𝐹‘𝑘), 𝑃) ∈ V)
94		eleq1w 2818	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → (𝑥 ∈ dom 𝐹 ↔ 𝑘 ∈ dom 𝐹))
95		fveq2 6881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → (𝐹‘𝑥) = (𝐹‘𝑘))
96	94, 95	ifbieq1d 4530	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑘 → if(𝑥 ∈ dom 𝐹, (𝐹‘𝑥), 𝑃) = if(𝑘 ∈ dom 𝐹, (𝐹‘𝑘), 𝑃))
97	96, 30	fvmptg 6989	. . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . 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 3963	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 𝑚 ∈ dom 𝐹)
102	69, 101	elind 4180	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 𝑚 ∈ (ℕ ∩ dom 𝐹))
103		fveq2 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → (𝐺‘𝑘) = (𝐺‘𝑚))
104		fveq2 6881	. . . . . . . . . . . . . . 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 3947	. . . . . . . . . . . . . . 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 3561	. . . . . . . . . . . . 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 25236	. . . . . . . . . . 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 3142	. . . . . . 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 25240	. . . . . . 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 1373	. . . . . 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 25242	. . . . 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 3000	. . 3 ⊢ (𝜑 → ((𝐽 fLimf (ℕfilGen(ℤ≥ “ ℕ)))‘𝐺) ≠ ∅)
123		n0 4333	. . 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 23948	. . . . 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 1373	. . . 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 23240	. . . . . . . 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 25217	. . . . . . . . . 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 3099	. . . . . . . . . . 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 15373	. . . . . . . . . . . . 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 1223	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧))) → (𝐺‘𝑘) ∈ 𝑋)
138	136, 137	eqeltrrd 2836	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧))) → (𝐹‘𝑘) ∈ 𝑋)
139	136	oveq1d 7425	. . . . . . . . . . . . . . . . . . . 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 1224	. . . . . . . . . . . . . . . . . . . 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 5148	. . . . . . . . . . . . . . . . . . 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 3153	. . . . . . . . . . . . . 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 3154	. . . . . . . . . . . . 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 3155	. . . . . . . . . . 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 12628	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐺(⇝𝑡‘𝐽)𝑦) → 1 ∈ ℤ)
156	6, 154, 10, 155	lmmbr3 25217	. . . . . . 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 1343	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺(⇝𝑡‘𝐽)𝑦) → 𝐹(⇝𝑡‘𝐽)𝑦)
158		lmrel 23173	. . . . . . 7 ⊢ Rel (⇝𝑡‘𝐽)
159	158	releldmi 5933	. . . . . 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 1933	. 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 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  Vcvv 3464   ∩ cin 3930   ⊆ wss 3931  ∅c0 4313  ifcif 4505   class class class wbr 5124   ↦ cmpt 5206  dom cdm 5659   “ cima 5662  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ↑_pm cpm 8846  ℂcc 11132  1c1 11135   < clt 11274  ℕcn 12245  ℤcz 12593  ℤ_≥cuz 12857  ℝ⁺crp 13013  ∞Metcxmet 21305  Metcmet 21306  fBascfbas 21308  filGencfg 21309  MetOpencmopn 21310  TopOnctopon 22853  ⇝_𝑡clm 23169  Filcfil 23788   FilMap cfm 23876   fLim cflim 23877   fLimf cflf 23878  CauFilccfil 25209  Cauccau 25210  CMetccmet 25211

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211  ax-pre-sup 11212

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 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8724  df-map 8847  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-sup 9459  df-inf 9460  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-div 11900  df-nn 12246  df-2 12308  df-n0 12507  df-z 12594  df-uz 12858  df-q 12970  df-rp 13014  df-xneg 13133  df-xadd 13134  df-xmul 13135  df-ico 13373  df-rest 17441  df-topgen 17462  df-psmet 21312  df-xmet 21313  df-met 21314  df-bl 21315  df-mopn 21316  df-fbas 21317  df-fg 21318  df-top 22837  df-topon 22854  df-bases 22889  df-ntr 22963  df-nei 23041  df-lm 23172  df-fil 23789  df-fm 23881  df-flim 23882  df-flf 23883  df-cfil 25212  df-cau 25213  df-cmet 25214

This theorem is referenced by:  cmetcau  25246