Theorem cmetcaulem 25408

Description: Lemma for cmetcau 25409. (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 25406	. . . . . . . . 9 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
3	1, 2	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
4		metxmet 24452	. . . . . . . 8 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
5	3, 4	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
6		cmetcau.1	. . . . . . . 8 ⊢ 𝐽 = (MetOpen‘𝐷)
7	6	mopntopon 24557	. . . . . . 7 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
8	5, 7	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
9		1z 12615	. . . . . . . 8 ⊢ 1 ∈ ℤ
10		nnuz 12892	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
11	10	uzfbas 24016	. . . . . . . 8 ⊢ (1 ∈ ℤ → (ℤ≥ “ ℕ) ∈ (fBas‘ℕ))
12	9, 11	mp1i 14	. . . . . . 7 ⊢ (𝜑 → (ℤ≥ “ ℕ) ∈ (fBas‘ℕ))
13		fgcl 23996	. . . . . . 7 ⊢ ((ℤ≥ “ ℕ) ∈ (fBas‘ℕ) → (ℕfilGen(ℤ≥ “ ℕ)) ∈ (Fil‘ℕ))
14	12, 13	syl 18	. . . . . 6 ⊢ (𝜑 → (ℕfilGen(ℤ≥ “ ℕ)) ∈ (Fil‘ℕ))
15		elfvdm 6905	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ dom CMet)
16	1, 15	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ dom CMet)
17		cnex 11169	. . . . . . . . . . . 12 ⊢ ℂ ∈ V
18	17	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ℂ ∈ V)
19		cmetcau.5	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷))
20		caufpm 25402	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝐹 ∈ (𝑋 ↑pm ℂ))
21	5, 19, 20	syl2anc 595	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (𝑋 ↑pm ℂ))
22		elpm2g 8829	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ dom CMet ∧ ℂ ∈ V) → (𝐹 ∈ (𝑋 ↑pm ℂ) ↔ (𝐹:dom 𝐹⟶𝑋 ∧ dom 𝐹 ⊆ ℂ)))
23	22	simprbda 503	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ dom CMet ∧ ℂ ∈ V) ∧ 𝐹 ∈ (𝑋 ↑pm ℂ)) → 𝐹:dom 𝐹⟶𝑋)
24	16, 18, 21, 23	syl21anc 850	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:dom 𝐹⟶𝑋)
25	24	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 𝐹:dom 𝐹⟶𝑋)
26	25	ffvelcdmda 7069	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ 𝑋)
27		cmetcau.4	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ 𝑋)
28	27	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ∈ dom 𝐹) → 𝑃 ∈ 𝑋)
29	26, 28	ifclda 4519	. . . . . . 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 7099	. . . . . 6 ⊢ (𝜑 → 𝐺:ℕ⟶𝑋)
32		flfval 24108	. . . . . 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 1394	. . . . 5 ⊢ (𝜑 → ((𝐽 fLimf (ℕfilGen(ℤ≥ “ ℕ)))‘𝐺) = (𝐽 fLim ((𝑋 FilMap 𝐺)‘(ℕfilGen(ℤ≥ “ ℕ)))))
34		eqid 2765	. . . . . . . 8 ⊢ (ℕfilGen(ℤ≥ “ ℕ)) = (ℕfilGen(ℤ≥ “ ℕ))
35	34	fmfg 24067	. . . . . . 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 1394	. . . . . 6 ⊢ (𝜑 → ((𝑋 FilMap 𝐺)‘(ℤ≥ “ ℕ)) = ((𝑋 FilMap 𝐺)‘(ℕfilGen(ℤ≥ “ ℕ))))
37	36	oveq2d 7416	. . . . 5 ⊢ (𝜑 → (𝐽 fLim ((𝑋 FilMap 𝐺)‘(ℤ≥ “ ℕ))) = (𝐽 fLim ((𝑋 FilMap 𝐺)‘(ℕfilGen(ℤ≥ “ ℕ)))))
38	33, 37	eqtr4d 2803	. . . 4 ⊢ (𝜑 → ((𝐽 fLimf (ℕfilGen(ℤ≥ “ ℕ)))‘𝐺) = (𝐽 fLim ((𝑋 FilMap 𝐺)‘(ℤ≥ “ ℕ))))
39		biidd 265	. . . . . . . 8 ⊢ (𝑧 = 1 → (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹 ↔ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹))
40		1zzd 12616	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℤ)
41	10, 5, 40	iscau3 25398	. . . . . . . . . . 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 504	. . . . . . . . . 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 699	. . . . . . . . 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 1152	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑤 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑤)) < 𝑧) → 𝑘 ∈ dom 𝐹)
45	44	ralimi 3102	. . . . . . . . . . 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 3103	. . . . . . . . . 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 3102	. . . . . . . . 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 18	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹)
49		1rp 13011	. . . . . . . . 9 ⊢ 1 ∈ ℝ+
50	49	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ+)
51	39, 48, 50	rspcdva 3585	. . . . . . 7 ⊢ (𝜑 → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹)
52		dfss3 3928	. . . . . . . . 9 ⊢ ((ℤ≥‘𝑗) ⊆ dom 𝐹 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹)
53		nnsscn 12229	. . . . . . . . . . . . . 14 ⊢ ℕ ⊆ ℂ
54	31, 53	jctir 529	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺:ℕ⟶𝑋 ∧ ℕ ⊆ ℂ))
55		elpm2r 8830	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ dom CMet ∧ ℂ ∈ V) ∧ (𝐺:ℕ⟶𝑋 ∧ ℕ ⊆ ℂ)) → 𝐺 ∈ (𝑋 ↑pm ℂ))
56	16, 18, 54, 55	syl21anc 850	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ (𝑋 ↑pm ℂ))
57	56	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) → 𝐺 ∈ (𝑋 ↑pm ℂ))
58		eqid 2765	. . . . . . . . . . . . . . 15 ⊢ (ℤ≥‘𝑗) = (ℤ≥‘𝑗)
59	5	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) → 𝐷 ∈ (∞Met‘𝑋))
60		nnz 12603	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℤ)
61	60	ad2antrl 740	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) → 𝑗 ∈ ℤ)
62		eqidd 2766	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) = (𝐹‘𝑘))
63		eqidd 2766	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑚) = (𝐹‘𝑚))
64	58, 59, 61, 62, 63	iscau4 25399	. . . . . . . . . . . . . 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 504	. . . . . . . . . . . . 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 701	. . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) → 𝑗 ∈ ℕ)
68		eluznn 12933	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ℕ ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 𝑚 ∈ ℕ)
69	67, 68	sylan 591	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 𝑚 ∈ ℕ)
70		eluznn 12933	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝑚)) → 𝑘 ∈ ℕ)
71	30, 29	dmmptd 6670	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → dom 𝐺 = ℕ)
72	71	adantr 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) → dom 𝐺 = ℕ)
73	72	eleq2d 2851	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) → (𝑘 ∈ dom 𝐺 ↔ 𝑘 ∈ ℕ))
74	73	biimpar 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ dom 𝐺)
75	74	a1d 26	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑘 ∈ ℕ) → (𝑘 ∈ dom 𝐹 → 𝑘 ∈ dom 𝐺))
76		idd 25	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) ∈ 𝑋 → (𝐹‘𝑘) ∈ 𝑋))
77		idd 25	. . . . . . . . . . . . . . . . . . 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 1467	. . . . . . . . . . . . . . . . . 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 604	. . . . . . . . . . . . . . . . 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 472	. . . . . . . . . . . . . . . 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 3177	. . . . . . . . . . . . . . 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 602	. . . . . . . . . . . . . 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 3178	. . . . . . . . . . . . 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 3179	. . . . . . . . . . . 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 16	. . . . . . . . . . 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 12933	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ ℕ)
87	67, 86	sylan 591	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ ℕ)
88		simprr 784	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) → (ℤ≥‘𝑗) ⊆ dom 𝐹)
89	88	sselda 3939	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ dom 𝐹)
90		iftrue 4489	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ dom 𝐹 → if(𝑘 ∈ dom 𝐹, (𝐹‘𝑘), 𝑃) = (𝐹‘𝑘))
91	90	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℕ ∧ 𝑘 ∈ dom 𝐹) → if(𝑘 ∈ dom 𝐹, (𝐹‘𝑘), 𝑃) = (𝐹‘𝑘))
92		fvex 6884	. . . . . . . . . . . . . . . 16 ⊢ (𝐹‘𝑘) ∈ V
93	91, 92	eqeltrdi 2873	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℕ ∧ 𝑘 ∈ dom 𝐹) → if(𝑘 ∈ dom 𝐹, (𝐹‘𝑘), 𝑃) ∈ V)
94		eleq1w 2848	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → (𝑥 ∈ dom 𝐹 ↔ 𝑘 ∈ dom 𝐹))
95		fveq2 6871	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → (𝐹‘𝑥) = (𝐹‘𝑘))
96	94, 95	ifbieq1d 4508	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑘 → if(𝑥 ∈ dom 𝐹, (𝐹‘𝑥), 𝑃) = if(𝑘 ∈ dom 𝐹, (𝐹‘𝑘), 𝑃))
97	96, 30	fvmptg 6977	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℕ ∧ if(𝑘 ∈ dom 𝐹, (𝐹‘𝑘), 𝑃) ∈ V) → (𝐺‘𝑘) = if(𝑘 ∈ dom 𝐹, (𝐹‘𝑘), 𝑃))
98	93, 97	syldan 602	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℕ ∧ 𝑘 ∈ dom 𝐹) → (𝐺‘𝑘) = if(𝑘 ∈ dom 𝐹, (𝐹‘𝑘), 𝑃))
99	98, 91	eqtrd 2800	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ ∧ 𝑘 ∈ dom 𝐹) → (𝐺‘𝑘) = (𝐹‘𝑘))
100	87, 89, 99	syl2anc 595	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐺‘𝑘) = (𝐹‘𝑘))
101	88	sselda 3939	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 𝑚 ∈ dom 𝐹)
102	69, 101	elind 4155	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → 𝑚 ∈ (ℕ ∩ dom 𝐹))
103		fveq2 6871	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → (𝐺‘𝑘) = (𝐺‘𝑚))
104		fveq2 6871	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
105	103, 104	eqeq12d 2781	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑚 → ((𝐺‘𝑘) = (𝐹‘𝑘) ↔ (𝐺‘𝑚) = (𝐹‘𝑚)))
106		elin 3923	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℕ ∩ dom 𝐹) ↔ (𝑘 ∈ ℕ ∧ 𝑘 ∈ dom 𝐹))
107	106, 99	sylbi 220	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (ℕ ∩ dom 𝐹) → (𝐺‘𝑘) = (𝐹‘𝑘))
108	105, 107	vtoclga 3544	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (ℕ ∩ dom 𝐹) → (𝐺‘𝑚) = (𝐹‘𝑚))
109	102, 108	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) ∧ 𝑚 ∈ (ℤ≥‘𝑗)) → (𝐺‘𝑚) = (𝐹‘𝑚))
110	58, 59, 61, 100, 109	iscau4 25399	. . . . . . . . . . 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 725	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ ℕ ∧ (ℤ≥‘𝑗) ⊆ dom 𝐹)) → 𝐺 ∈ (Cau‘𝐷))
112	111	expr 461	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((ℤ≥‘𝑗) ⊆ dom 𝐹 → 𝐺 ∈ (Cau‘𝐷)))
113	52, 112	biimtrrid 246	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹 → 𝐺 ∈ (Cau‘𝐷)))
114	113	rexlimdva 3166	. . . . . . 7 ⊢ (𝜑 → (∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑘 ∈ dom 𝐹 → 𝐺 ∈ (Cau‘𝐷)))
115	51, 114	mpd 16	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (Cau‘𝐷))
116		eqid 2765	. . . . . . . 8 ⊢ ((𝑋 FilMap 𝐺)‘(ℤ≥ “ ℕ)) = ((𝑋 FilMap 𝐺)‘(ℤ≥ “ ℕ))
117	10, 116	caucfil 25403	. . . . . . 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 1394	. . . . . 6 ⊢ (𝜑 → (𝐺 ∈ (Cau‘𝐷) ↔ ((𝑋 FilMap 𝐺)‘(ℤ≥ “ ℕ)) ∈ (CauFil‘𝐷)))
119	115, 118	mpbid 235	. . . . 5 ⊢ (𝜑 → ((𝑋 FilMap 𝐺)‘(ℤ≥ “ ℕ)) ∈ (CauFil‘𝐷))
120	6	cmetcvg 25405	. . . . 5 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ ((𝑋 FilMap 𝐺)‘(ℤ≥ “ ℕ)) ∈ (CauFil‘𝐷)) → (𝐽 fLim ((𝑋 FilMap 𝐺)‘(ℤ≥ “ ℕ))) ≠ ∅)
121	1, 119, 120	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝐽 fLim ((𝑋 FilMap 𝐺)‘(ℤ≥ “ ℕ))) ≠ ∅)
122	38, 121	eqnetrd 3027	. . 3 ⊢ (𝜑 → ((𝐽 fLimf (ℕfilGen(ℤ≥ “ ℕ)))‘𝐺) ≠ ∅)
123		n0 4308	. . 3 ⊢ (((𝐽 fLimf (ℕfilGen(ℤ≥ “ ℕ)))‘𝐺) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ((𝐽 fLimf (ℕfilGen(ℤ≥ “ ℕ)))‘𝐺))
124	122, 123	sylib 221	. 2 ⊢ (𝜑 → ∃𝑦 𝑦 ∈ ((𝐽 fLimf (ℕfilGen(ℤ≥ “ ℕ)))‘𝐺))
125	10, 34	lmflf 24123	. . . . 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 1394	. . . 4 ⊢ (𝜑 → (𝐺(⇝𝑡‘𝐽)𝑦 ↔ 𝑦 ∈ ((𝐽 fLimf (ℕfilGen(ℤ≥ “ ℕ)))‘𝐺)))
127	21	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐺(⇝𝑡‘𝐽)𝑦) → 𝐹 ∈ (𝑋 ↑pm ℂ))
128		lmcl 23415	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐺(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ 𝑋)
129	8, 128	sylan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐺(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ 𝑋)
130	6, 5, 10, 40	lmmbr3 25380	. . . . . . . . . 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 481	. . . . . . . . 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 1160	. . . . . . . 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 3125	. . . . . . . . . . 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 15391	. . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧))) → 𝑘 ∈ dom 𝐹)
136	99	ad2ant2lr 760	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧))) → (𝐺‘𝑘) = (𝐹‘𝑘))
137		simprr2 1239	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧))) → (𝐺‘𝑘) ∈ 𝑋)
138	136, 137	eqeltrrd 2866	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ ((𝐺‘𝑘)𝐷𝑦) < 𝑧))) → (𝐹‘𝑘) ∈ 𝑋)
139	136	oveq1d 7415	. . . . . . . . . . . . . . . . . . . 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 1240	. . . . . . . . . . . . . . . . . . . 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 5129	. . . . . . . . . . . . . . . . . . 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 1144	. . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . 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 604	. . . . . . . . . . . . . . . 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 472	. . . . . . . . . . . . . . 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 3177	. . . . . . . . . . . . . 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 3178	. . . . . . . . . . . . 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 246	. . . . . . . . . . . 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 3179	. . . . . . . . . . 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 246	. . . . . . . . . 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 707	. . . . . . . . 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 485	. . . . . . . 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 16	. . . . . . 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 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐺(⇝𝑡‘𝐽)𝑦) → 𝐷 ∈ (∞Met‘𝑋))
155		1zzd 12616	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐺(⇝𝑡‘𝐽)𝑦) → 1 ∈ ℤ)
156	6, 154, 10, 155	lmmbr3 25380	. . . . . . 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 1359	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺(⇝𝑡‘𝐽)𝑦) → 𝐹(⇝𝑡‘𝐽)𝑦)
158		lmrel 23348	. . . . . . 7 ⊢ Rel (⇝𝑡‘𝐽)
159	158	releldmi 5929	. . . . . 6 ⊢ (𝐹(⇝𝑡‘𝐽)𝑦 → 𝐹 ∈ dom (⇝𝑡‘𝐽))
160	157, 159	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝐺(⇝𝑡‘𝐽)𝑦) → 𝐹 ∈ dom (⇝𝑡‘𝐽))
161	160	ex 417	. . . 4 ⊢ (𝜑 → (𝐺(⇝𝑡‘𝐽)𝑦 → 𝐹 ∈ dom (⇝𝑡‘𝐽)))
162	126, 161	sylbird 263	. . 3 ⊢ (𝜑 → (𝑦 ∈ ((𝐽 fLimf (ℕfilGen(ℤ≥ “ ℕ)))‘𝐺) → 𝐹 ∈ dom (⇝𝑡‘𝐽)))
163	162	exlimdv 1956	. 2 ⊢ (𝜑 → (∃𝑦 𝑦 ∈ ((𝐽 fLimf (ℕfilGen(ℤ≥ “ ℕ)))‘𝐺) → 𝐹 ∈ dom (⇝𝑡‘𝐽)))
164	124, 163	mpd 16	1 ⊢ (𝜑 → 𝐹 ∈ dom (⇝𝑡‘𝐽))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563  ∃wex 1802   ∈ wcel 2145   ≠ wne 2960  ∀wral 3079  ∃wrex 3089  Vcvv 3457   ∩ cin 3906   ⊆ wss 3907  ∅c0 4288  ifcif 4483   class class class wbr 5105   ↦ cmpt 5186  dom cdm 5652   “ cima 5655  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400   ↑_pm cpm 8813  ℂcc 11086  1c1 11089   < clt 11231  ℕcn 12224  ℤcz 12582  ℤ_≥cuz 12853  ℝ⁺crp 13007  ∞Metcxmet 21467  Metcmet 21468  fBascfbas 21470  filGencfg 21471  MetOpencmopn 21472  TopOnctopon 23028  ⇝_𝑡clm 23344  Filcfil 23963   FilMap cfm 24051   fLim cflim 24052   fLimf cflf 24053  CauFilccfil 25372  Cauccau 25373  CMetccmet 25374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-pre-sup 11166

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-er 8682  df-map 8814  df-pm 8815  df-en 8932  df-dom 8933  df-sdom 8934  df-sup 9390  df-inf 9391  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-div 11860  df-nn 12225  df-2 12294  df-n0 12496  df-z 12583  df-uz 12854  df-q 12964  df-rp 13008  df-xneg 13128  df-xadd 13129  df-xmul 13130  df-ico 13369  df-rest 17465  df-topgen 17486  df-psmet 21474  df-xmet 21475  df-met 21476  df-bl 21477  df-mopn 21478  df-fbas 21479  df-fg 21480  df-top 23012  df-topon 23029  df-bases 23064  df-ntr 23138  df-nei 23216  df-lm 23347  df-fil 23964  df-fm 24056  df-flim 24057  df-flf 24058  df-cfil 25375  df-cau 25376  df-cmet 25377

This theorem is referenced by:  cmetcau  25409