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Theorem cmetcaulem 23086

Description: Lemma for cmetcau 23087. (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 23084	. . . . . . . . 9 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
3	1, 2	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
4		metxmet 22139	. . . . . . . 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 22244	. . . . . . 7 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
8	5, 7	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
9		1z 11407	. . . . . . . 8 ⊢ 1 ∈ ℤ
10		nnuz 11723	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
11	10	uzfbas 21702	. . . . . . . 8 ⊢ (1 ∈ ℤ → (ℤ_≥ “ ℕ) ∈ (fBas‘ℕ))
12	9, 11	mp1i 13	. . . . . . 7 ⊢ (𝜑 → (ℤ_≥ “ ℕ) ∈ (fBas‘ℕ))
13		fgcl 21682	. . . . . . 7 ⊢ ((ℤ_≥ “ ℕ) ∈ (fBas‘ℕ) → (ℕfilGen(ℤ_≥ “ ℕ)) ∈ (Fil‘ℕ))
14	12, 13	syl 17	. . . . . 6 ⊢ (𝜑 → (ℕfilGen(ℤ_≥ “ ℕ)) ∈ (Fil‘ℕ))
15		elfvdm 6220	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ dom CMet)
16	1, 15	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ dom CMet)
17		cnex 10017	. . . . . . . . . . . 12 ⊢ ℂ ∈ V
18	17	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ℂ ∈ V)
19		cmetcau.5	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷))
20		caufpm 23080	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝐹 ∈ (𝑋 ↑_pm ℂ))
21	5, 19, 20	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (𝑋 ↑_pm ℂ))
22		elpm2g 7874	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ dom CMet ∧ ℂ ∈ V) → (𝐹 ∈ (𝑋 ↑_pm ℂ) ↔ (𝐹:dom 𝐹⟶𝑋 ∧ dom 𝐹 ⊆ ℂ)))
23	22	simprbda 653	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ dom CMet ∧ ℂ ∈ V) ∧ 𝐹 ∈ (𝑋 ↑_pm ℂ)) → 𝐹:dom 𝐹⟶𝑋)
24	16, 18, 21, 23	syl21anc 1325	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:dom 𝐹⟶𝑋)
25	24	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 𝐹:dom 𝐹⟶𝑋)
26	25	ffvelrnda 6359	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ 𝑋)
27		cmetcau.4	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ 𝑋)
28	27	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑃 ∈ 𝑋)
29	26, 28	ifclda 4120	. . . . . . 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 6385	. . . . . 6 ⊢ (𝜑 → 𝐺:ℕ⟶𝑋)
32		flfval 21794	. . . . . 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 1326	. . . . 5 ⊢ (𝜑 → ((𝐽 fLimf (ℕfilGen(ℤ_≥ “ ℕ)))‘𝐺) = (𝐽 fLim ((𝑋 FilMap 𝐺)‘(ℕfilGen(ℤ_≥ “ ℕ)))))
34		eqid 2622	. . . . . . . 8 ⊢ (ℕfilGen(ℤ_≥ “ ℕ)) = (ℕfilGen(ℤ_≥ “ ℕ))
35	34	fmfg 21753	. . . . . . 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 1326	. . . . . 6 ⊢ (𝜑 → ((𝑋 FilMap 𝐺)‘(ℤ_≥ “ ℕ)) = ((𝑋 FilMap 𝐺)‘(ℕfilGen(ℤ_≥ “ ℕ))))
37	36	oveq2d 6666	. . . . 5 ⊢ (𝜑 → (𝐽 fLim ((𝑋 FilMap 𝐺)‘(ℤ_≥ “ ℕ))) = (𝐽 fLim ((𝑋 FilMap 𝐺)‘(ℕfilGen(ℤ_≥ “ ℕ)))))
38	33, 37	eqtr4d 2659	. . . 4 ⊢ (𝜑 → ((𝐽 fLimf (ℕfilGen(ℤ_≥ “ ℕ)))‘𝐺) = (𝐽 fLim ((𝑋 FilMap 𝐺)‘(ℤ_≥ “ ℕ))))
39		1rp 11836	. . . . . . . 8 ⊢ 1 ∈ ℝ⁺
40		1zzd 11408	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℤ)
41	10, 5, 40	iscau3 23076	. . . . . . . . . . 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 654	. . . . . . . . . 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 702	. . . . . . . . 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 1061	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑤 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑤)) < 𝑧) → 𝑘 ∈ dom 𝐹)
45	44	ralimi 2952	. . . . . . . . . . 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 3011	. . . . . . . . . 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 2952	. . . . . . . . 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		biidd 252	. . . . . . . . 9 ⊢ (𝑧 = 1 → (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑘 ∈ dom 𝐹 ↔ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑘 ∈ dom 𝐹))
50	49	rspcv 3305	. . . . . . . 8 ⊢ (1 ∈ ℝ⁺ → (∀𝑧 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑘 ∈ dom 𝐹 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑘 ∈ dom 𝐹))
51	39, 48, 50	mpsyl 68	. . . . . . 7 ⊢ (𝜑 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑘 ∈ dom 𝐹)
52		dfss3 3592	. . . . . . . . 9 ⊢ ((ℤ_≥‘𝑗) ⊆ dom 𝐹 ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑘 ∈ dom 𝐹)
53		nnsscn 11025	. . . . . . . . . . . . . 14 ⊢ ℕ ⊆ ℂ
54	31, 53	jctir 561	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺:ℕ⟶𝑋 ∧ ℕ ⊆ ℂ))
55		elpm2r 7875	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ dom CMet ∧ ℂ ∈ V) ∧ (𝐺:ℕ⟶𝑋 ∧ ℕ ⊆ ℂ)) → 𝐺 ∈ (𝑋 ↑_pm ℂ))
56	16, 18, 54, 55	syl21anc 1325	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ (𝑋 ↑_pm ℂ))
57	56	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ_≥‘𝑗) ⊆ dom 𝐹)) → 𝐺 ∈ (𝑋 ↑_pm ℂ))
58		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ (ℤ_≥‘𝑗) = (ℤ_≥‘𝑗)
59	5	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ_≥‘𝑗) ⊆ dom 𝐹)) → 𝐷 ∈ (∞Met‘𝑋))
60		nnz 11399	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℤ)
61	60	ad2antrl 764	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ_≥‘𝑗) ⊆ dom 𝐹)) → 𝑗 ∈ ℤ)
62		eqidd 2623	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ_≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) = (𝐹‘𝑘))
63		eqidd 2623	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ_≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑚) = (𝐹‘𝑚))
64	58, 59, 61, 62, 63	iscau4 23077	. . . . . . . . . . . . . 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 654	. . . . . . . . . . . . 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 704	. . . . . . . . . . . 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 794	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ_≥‘𝑗) ⊆ dom 𝐹)) → 𝑗 ∈ ℕ)
68		eluznn 11758	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → 𝑚 ∈ ℕ)
69	67, 68	sylan 488	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ_≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → 𝑚 ∈ ℕ)
70		eluznn 11758	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘𝑚)) → 𝑘 ∈ ℕ)
71	30, 29	dmmptd 6024	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → dom 𝐺 = ℕ)
72	71	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ_≥‘𝑗) ⊆ dom 𝐹)) → dom 𝐺 = ℕ)
73	72	eleq2d 2687	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ_≥‘𝑗) ⊆ dom 𝐹)) → (𝑘 ∈ dom 𝐺 ↔ 𝑘 ∈ ℕ))
74	73	biimpar 502	. . . . . . . . . . . . . . . . . . . 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 1406	. . . . . . . . . . . . . . . . . 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 491	. . . . . . . . . . . . . . . . 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 680	. . . . . . . . . . . . . . . 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 2962	. . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . 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 3017	. . . . . . . . . . . . 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 2963	. . . . . . . . . . . 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 11758	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ ℕ)
87	67, 86	sylan 488	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ_≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ ℕ)
88		simprr 796	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ_≥‘𝑗) ⊆ dom 𝐹)) → (ℤ_≥‘𝑗) ⊆ dom 𝐹)
89	88	sselda 3603	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ_≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ dom 𝐹)
90		iftrue 4092	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ dom 𝐹 → if(𝑘 ∈ dom 𝐹, (𝐹‘𝑘), 𝑃) = (𝐹‘𝑘))
91	90	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℕ ∧ 𝑘 ∈ dom 𝐹) → if(𝑘 ∈ dom 𝐹, (𝐹‘𝑘), 𝑃) = (𝐹‘𝑘))
92		fvex 6201	. . . . . . . . . . . . . . . 16 ⊢ (𝐹‘𝑘) ∈ V
93	91, 92	syl6eqel 2709	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℕ ∧ 𝑘 ∈ dom 𝐹) → if(𝑘 ∈ dom 𝐹, (𝐹‘𝑘), 𝑃) ∈ V)
94		eleq1 2689	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → (𝑥 ∈ dom 𝐹 ↔ 𝑘 ∈ dom 𝐹))
95		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → (𝐹‘𝑥) = (𝐹‘𝑘))
96	94, 95	ifbieq1d 4109	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑘 → if(𝑥 ∈ dom 𝐹, (𝐹‘𝑥), 𝑃) = if(𝑘 ∈ dom 𝐹, (𝐹‘𝑘), 𝑃))
97	96, 30	fvmptg 6280	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℕ ∧ if(𝑘 ∈ dom 𝐹, (𝐹‘𝑘), 𝑃) ∈ V) → (𝐺‘𝑘) = if(𝑘 ∈ dom 𝐹, (𝐹‘𝑘), 𝑃))
98	93, 97	syldan 487	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℕ ∧ 𝑘 ∈ dom 𝐹) → (𝐺‘𝑘) = if(𝑘 ∈ dom 𝐹, (𝐹‘𝑘), 𝑃))
99	98, 91	eqtrd 2656	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ ∧ 𝑘 ∈ dom 𝐹) → (𝐺‘𝑘) = (𝐹‘𝑘))
100	87, 89, 99	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ_≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐺‘𝑘) = (𝐹‘𝑘))
101	88	sselda 3603	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ_≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → 𝑚 ∈ dom 𝐹)
102	69, 101	elind 3798	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ_≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → 𝑚 ∈ (ℕ ∩ dom 𝐹))
103		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → (𝐺‘𝑘) = (𝐺‘𝑚))
104		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
105	103, 104	eqeq12d 2637	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑚 → ((𝐺‘𝑘) = (𝐹‘𝑘) ↔ (𝐺‘𝑚) = (𝐹‘𝑚)))
106		elin 3796	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℕ ∩ dom 𝐹) ↔ (𝑘 ∈ ℕ ∧ 𝑘 ∈ dom 𝐹))
107	106, 99	sylbi 207	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (ℕ ∩ dom 𝐹) → (𝐺‘𝑘) = (𝐹‘𝑘))
108	105, 107	vtoclga 3272	. . . . . . . . . . . . 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 23077	. . . . . . . . . . 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 957	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ_≥‘𝑗) ⊆ dom 𝐹)) → 𝐺 ∈ (Cau‘𝐷))
112	111	expr 643	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((ℤ_≥‘𝑗) ⊆ dom 𝐹 → 𝐺 ∈ (Cau‘𝐷)))
113	52, 112	syl5bir 233	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (∀𝑘 ∈ (ℤ_≥‘𝑗)𝑘 ∈ dom 𝐹 → 𝐺 ∈ (Cau‘𝐷)))
114	113	rexlimdva 3031	. . . . . . 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 2622	. . . . . . . 8 ⊢ ((𝑋 FilMap 𝐺)‘(ℤ_≥ “ ℕ)) = ((𝑋 FilMap 𝐺)‘(ℤ_≥ “ ℕ))
117	10, 116	caucfil 23081	. . . . . . 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 1326	. . . . . 6 ⊢ (𝜑 → (𝐺 ∈ (Cau‘𝐷) ↔ ((𝑋 FilMap 𝐺)‘(ℤ_≥ “ ℕ)) ∈ (CauFil‘𝐷)))
119	115, 118	mpbid 222	. . . . 5 ⊢ (𝜑 → ((𝑋 FilMap 𝐺)‘(ℤ_≥ “ ℕ)) ∈ (CauFil‘𝐷))
120	6	cmetcvg 23083	. . . . 5 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ ((𝑋 FilMap 𝐺)‘(ℤ_≥ “ ℕ)) ∈ (CauFil‘𝐷)) → (𝐽 fLim ((𝑋 FilMap 𝐺)‘(ℤ_≥ “ ℕ))) ≠ ∅)
121	1, 119, 120	syl2anc 693	. . . 4 ⊢ (𝜑 → (𝐽 fLim ((𝑋 FilMap 𝐺)‘(ℤ_≥ “ ℕ))) ≠ ∅)
122	38, 121	eqnetrd 2861	. . 3 ⊢ (𝜑 → ((𝐽 fLimf (ℕfilGen(ℤ_≥ “ ℕ)))‘𝐺) ≠ ∅)
123		n0 3931	. . 3 ⊢ (((𝐽 fLimf (ℕfilGen(ℤ_≥ “ ℕ)))‘𝐺) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ((𝐽 fLimf (ℕfilGen(ℤ_≥ “ ℕ)))‘𝐺))
124	122, 123	sylib 208	. 2 ⊢ (𝜑 → ∃𝑦 𝑦 ∈ ((𝐽 fLimf (ℕfilGen(ℤ_≥ “ ℕ)))‘𝐺))
125	10, 34	lmflf 21809	. . . . 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 1326	. . . 4 ⊢ (𝜑 → (𝐺(⇝_𝑡‘𝐽)𝑦 ↔ 𝑦 ∈ ((𝐽 fLimf (ℕfilGen(ℤ_≥ “ ℕ)))‘𝐺)))
127	21	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐺(⇝_𝑡‘𝐽)𝑦) → 𝐹 ∈ (𝑋 ↑_pm ℂ))
128		lmcl 21101	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐺(⇝_𝑡‘𝐽)𝑦) → 𝑦 ∈ 𝑋)
129	8, 128	sylan 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐺(⇝_𝑡‘𝐽)𝑦) → 𝑦 ∈ 𝑋)
130	6, 5, 10, 40	lmmbr3 23058	. . . . . . . . . 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 501	. . . . . . . . 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 1075	. . . . . . . 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 3064	. . . . . . . . . . 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 14089	. . . . . . . . . . . . 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 794	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧))) → 𝑘 ∈ dom 𝐹)
136	99	ad2ant2lr 784	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧))) → (𝐺‘𝑘) = (𝐹‘𝑘))
137		simprr2 1110	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧))) → (𝐺‘𝑘) ∈ 𝑋)
138	136, 137	eqeltrrd 2702	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧))) → (𝐹‘𝑘) ∈ 𝑋)
139	136	oveq1d 6665	. . . . . . . . . . . . . . . . . . . 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 1111	. . . . . . . . . . . . . . . . . . . 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 4677	. . . . . . . . . . . . . . . . . . 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 1242	. . . . . . . . . . . . . . . . . 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 450	. . . . . . . . . . . . . . . . 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 491	. . . . . . . . . . . . . . . 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 680	. . . . . . . . . . . . . . 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 2962	. . . . . . . . . . . . . 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 3017	. . . . . . . . . . . . 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	syl5bir 233	. . . . . . . . . . . 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 2963	. . . . . . . . . . 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	syl5bir 233	. . . . . . . . . 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 711	. . . . . . . . 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 481	. . . . . . . 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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐺(⇝_𝑡‘𝐽)𝑦) → 𝐷 ∈ (∞Met‘𝑋))
155		1zzd 11408	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐺(⇝_𝑡‘𝐽)𝑦) → 1 ∈ ℤ)
156	6, 154, 10, 155	lmmbr3 23058	. . . . . . 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 1245	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺(⇝_𝑡‘𝐽)𝑦) → 𝐹(⇝_𝑡‘𝐽)𝑦)
158		lmrel 21034	. . . . . . 7 ⊢ Rel (⇝_𝑡‘𝐽)
159	158	releldmi 5362	. . . . . 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 450	. . . 4 ⊢ (𝜑 → (𝐺(⇝_𝑡‘𝐽)𝑦 → 𝐹 ∈ dom (⇝_𝑡‘𝐽)))
162	126, 161	sylbird 250	. . 3 ⊢ (𝜑 → (𝑦 ∈ ((𝐽 fLimf (ℕfilGen(ℤ_≥ “ ℕ)))‘𝐺) → 𝐹 ∈ dom (⇝_𝑡‘𝐽)))
163	162	exlimdv 1861	. 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))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 ifcif 4086 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 “ cima 5117 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑_pm cpm 7858 ℂcc 9934 1c1 9937 < clt 10074 ℕcn 11020 ℤcz 11377 ℤ_≥cuz 11687 ℝ⁺crp 11832 ∞Metcxmt 19731 Metcme 19732 fBascfbas 19734 filGencfg 19735 MetOpencmopn 19736 TopOnctopon 20715 ⇝_𝑡clm 21030 Filcfil 21649 FilMap cfm 21737 fLim cflim 21738 fLimf cflf 21739 CauFilccfil 23050 Caucca 23051 CMetcms 23052

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-rep 4771 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 ax-pre-sup 10014

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-rmo 2920 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-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-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ico 12181 df-rest 16083 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-top 20699 df-topon 20716 df-bases 20750 df-ntr 20824 df-nei 20902 df-lm 21033 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-cfil 23053 df-cau 23054 df-cmet 23055

This theorem is referenced by: cmetcau 23087

W3C validator