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Theorem iscmet3 22899
 Description: The property "𝐷 is a complete metric" expressed in terms of functions on ℕ (or any other upper integer set). Thus, we only have to look at functions on ℕ, and not all possible Cauchy filters, to determine completeness. (The proof uses countable choice.) (Contributed by NM, 18-Dec-2006.) (Revised by Mario Carneiro, 5-May-2014.)
Hypotheses
Ref Expression
iscmet3.1 𝑍 = (ℤ𝑀)
iscmet3.2 𝐽 = (MetOpen‘𝐷)
iscmet3.3 (𝜑𝑀 ∈ ℤ)
iscmet3.4 (𝜑𝐷 ∈ (Met‘𝑋))
Assertion
Ref Expression
iscmet3 (𝜑 → (𝐷 ∈ (CMet‘𝑋) ↔ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))))
Distinct variable groups:   𝐷,𝑓   𝑓,𝑋   𝑓,𝐽   𝑓,𝑍   𝑓,𝑀   𝜑,𝑓

Proof of Theorem iscmet3
Dummy variables 𝑔 𝑖 𝑗 𝑘 𝑛 𝑠 𝑡 𝑢 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iscmet3.2 . . . . 5 𝐽 = (MetOpen‘𝐷)
21cmetcau 22895 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑓 ∈ (Cau‘𝐷)) → 𝑓 ∈ dom (⇝𝑡𝐽))
32a1d 25 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑓 ∈ (Cau‘𝐷)) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽)))
43ralrimiva 2949 . 2 (𝐷 ∈ (CMet‘𝑋) → ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽)))
5 iscmet3.4 . . . . 5 (𝜑𝐷 ∈ (Met‘𝑋))
65adantr 480 . . . 4 ((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) → 𝐷 ∈ (Met‘𝑋))
7 simpr 476 . . . . . . . . 9 (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ 𝑔 ∈ (CauFil‘𝐷)) → 𝑔 ∈ (CauFil‘𝐷))
8 1rp 11712 . . . . . . . . . . 11 1 ∈ ℝ+
9 rphalfcl 11734 . . . . . . . . . . 11 (1 ∈ ℝ+ → (1 / 2) ∈ ℝ+)
108, 9ax-mp 5 . . . . . . . . . 10 (1 / 2) ∈ ℝ+
11 rpexpcl 12741 . . . . . . . . . 10 (((1 / 2) ∈ ℝ+𝑘 ∈ ℤ) → ((1 / 2)↑𝑘) ∈ ℝ+)
1210, 11mpan 702 . . . . . . . . 9 (𝑘 ∈ ℤ → ((1 / 2)↑𝑘) ∈ ℝ+)
13 cfili 22874 . . . . . . . . 9 ((𝑔 ∈ (CauFil‘𝐷) ∧ ((1 / 2)↑𝑘) ∈ ℝ+) → ∃𝑡𝑔𝑢𝑡𝑣𝑡 (𝑢𝐷𝑣) < ((1 / 2)↑𝑘))
147, 12, 13syl2an 493 . . . . . . . 8 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ 𝑔 ∈ (CauFil‘𝐷)) ∧ 𝑘 ∈ ℤ) → ∃𝑡𝑔𝑢𝑡𝑣𝑡 (𝑢𝐷𝑣) < ((1 / 2)↑𝑘))
1514ralrimiva 2949 . . . . . . 7 (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ 𝑔 ∈ (CauFil‘𝐷)) → ∀𝑘 ∈ ℤ ∃𝑡𝑔𝑢𝑡𝑣𝑡 (𝑢𝐷𝑣) < ((1 / 2)↑𝑘))
16 vex 3176 . . . . . . . 8 𝑔 ∈ V
17 znnen 14780 . . . . . . . . 9 ℤ ≈ ℕ
18 nnenom 12641 . . . . . . . . 9 ℕ ≈ ω
1917, 18entri 7896 . . . . . . . 8 ℤ ≈ ω
20 raleq 3115 . . . . . . . . 9 (𝑡 = (𝑠𝑘) → (∀𝑣𝑡 (𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))
2120raleqbi1dv 3123 . . . . . . . 8 (𝑡 = (𝑠𝑘) → (∀𝑢𝑡𝑣𝑡 (𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))
2216, 19, 21axcc4 9144 . . . . . . 7 (∀𝑘 ∈ ℤ ∃𝑡𝑔𝑢𝑡𝑣𝑡 (𝑢𝐷𝑣) < ((1 / 2)↑𝑘) → ∃𝑠(𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))
2315, 22syl 17 . . . . . 6 (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ 𝑔 ∈ (CauFil‘𝐷)) → ∃𝑠(𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))
24 iscmet3.3 . . . . . . . . . . . 12 (𝜑𝑀 ∈ ℤ)
2524ad2antrr 758 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) → 𝑀 ∈ ℤ)
26 iscmet3.1 . . . . . . . . . . . 12 𝑍 = (ℤ𝑀)
2726uzenom 12625 . . . . . . . . . . 11 (𝑀 ∈ ℤ → 𝑍 ≈ ω)
28 endom 7868 . . . . . . . . . . 11 (𝑍 ≈ ω → 𝑍 ≼ ω)
2925, 27, 283syl 18 . . . . . . . . . 10 (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) → 𝑍 ≼ ω)
30 dfin5 3548 . . . . . . . . . . . . . . 15 (( I ‘𝑋) ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛)) = {𝑥 ∈ ( I ‘𝑋) ∣ 𝑥 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛)}
31 fzn0 12226 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀...𝑘) ≠ ∅ ↔ 𝑘 ∈ (ℤ𝑀))
3231biimpri 217 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (ℤ𝑀) → (𝑀...𝑘) ≠ ∅)
3332, 26eleq2s 2706 . . . . . . . . . . . . . . . . . . 19 (𝑘𝑍 → (𝑀...𝑘) ≠ ∅)
34 simprr 792 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) → 𝑠:ℤ⟶𝑔)
35 elfzelz 12213 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ (𝑀...𝑘) → 𝑛 ∈ ℤ)
36 ffvelrn 6265 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠:ℤ⟶𝑔𝑛 ∈ ℤ) → (𝑠𝑛) ∈ 𝑔)
3734, 35, 36syl2an 493 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑛 ∈ (𝑀...𝑘)) → (𝑠𝑛) ∈ 𝑔)
38 metxmet 21949 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
395, 38syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐷 ∈ (∞Met‘𝑋))
4039adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) → 𝐷 ∈ (∞Met‘𝑋))
41 simpl 472 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔) → 𝑔 ∈ (CauFil‘𝐷))
42 cfilfil 22873 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑔 ∈ (CauFil‘𝐷)) → 𝑔 ∈ (Fil‘𝑋))
4340, 41, 42syl2an 493 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) → 𝑔 ∈ (Fil‘𝑋))
44 filelss 21466 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔 ∈ (Fil‘𝑋) ∧ (𝑠𝑛) ∈ 𝑔) → (𝑠𝑛) ⊆ 𝑋)
4543, 44sylan 487 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ (𝑠𝑛) ∈ 𝑔) → (𝑠𝑛) ⊆ 𝑋)
4637, 45syldan 486 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑛 ∈ (𝑀...𝑘)) → (𝑠𝑛) ⊆ 𝑋)
4746ralrimiva 2949 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) → ∀𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ⊆ 𝑋)
48 r19.2z 4012 . . . . . . . . . . . . . . . . . . 19 (((𝑀...𝑘) ≠ ∅ ∧ ∀𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ⊆ 𝑋) → ∃𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ⊆ 𝑋)
4933, 47, 48syl2anr 494 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → ∃𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ⊆ 𝑋)
50 iinss 4507 . . . . . . . . . . . . . . . . . 18 (∃𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ⊆ 𝑋 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ⊆ 𝑋)
5149, 50syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ⊆ 𝑋)
526ad2antrr 758 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → 𝐷 ∈ (Met‘𝑋))
53 elfvdm 6130 . . . . . . . . . . . . . . . . . 18 (𝐷 ∈ (Met‘𝑋) → 𝑋 ∈ dom Met)
54 fvi 6165 . . . . . . . . . . . . . . . . . 18 (𝑋 ∈ dom Met → ( I ‘𝑋) = 𝑋)
5552, 53, 543syl 18 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → ( I ‘𝑋) = 𝑋)
5651, 55sseqtr4d 3605 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ⊆ ( I ‘𝑋))
57 sseqin2 3779 . . . . . . . . . . . . . . . 16 ( 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ⊆ ( I ‘𝑋) ↔ (( I ‘𝑋) ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛)) = 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛))
5856, 57sylib 207 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → (( I ‘𝑋) ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛)) = 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛))
5930, 58syl5eqr 2658 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → {𝑥 ∈ ( I ‘𝑋) ∣ 𝑥 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛)} = 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛))
6043adantr 480 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → 𝑔 ∈ (Fil‘𝑋))
6137ralrimiva 2949 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) → ∀𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ∈ 𝑔)
6261adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → ∀𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ∈ 𝑔)
6333adantl 481 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → (𝑀...𝑘) ≠ ∅)
64 fzfid 12634 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → (𝑀...𝑘) ∈ Fin)
65 iinfi 8206 . . . . . . . . . . . . . . . . 17 ((𝑔 ∈ (Fil‘𝑋) ∧ (∀𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ∈ 𝑔 ∧ (𝑀...𝑘) ≠ ∅ ∧ (𝑀...𝑘) ∈ Fin)) → 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ∈ (fi‘𝑔))
6660, 62, 63, 64, 65syl13anc 1320 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ∈ (fi‘𝑔))
67 filfi 21473 . . . . . . . . . . . . . . . . 17 (𝑔 ∈ (Fil‘𝑋) → (fi‘𝑔) = 𝑔)
6860, 67syl 17 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → (fi‘𝑔) = 𝑔)
6966, 68eleqtrd 2690 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ∈ 𝑔)
70 fileln0 21464 . . . . . . . . . . . . . . 15 ((𝑔 ∈ (Fil‘𝑋) ∧ 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ∈ 𝑔) → 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ≠ ∅)
7160, 69, 70syl2anc 691 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ≠ ∅)
7259, 71eqnetrd 2849 . . . . . . . . . . . . 13 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → {𝑥 ∈ ( I ‘𝑋) ∣ 𝑥 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛)} ≠ ∅)
73 rabn0 3912 . . . . . . . . . . . . 13 ({𝑥 ∈ ( I ‘𝑋) ∣ 𝑥 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛)} ≠ ∅ ↔ ∃𝑥 ∈ ( I ‘𝑋)𝑥 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛))
7472, 73sylib 207 . . . . . . . . . . . 12 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → ∃𝑥 ∈ ( I ‘𝑋)𝑥 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛))
7574ralrimiva 2949 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) → ∀𝑘𝑍𝑥 ∈ ( I ‘𝑋)𝑥 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛))
7675adantrrr 757 . . . . . . . . . 10 (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) → ∀𝑘𝑍𝑥 ∈ ( I ‘𝑋)𝑥 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛))
77 fvex 6113 . . . . . . . . . . 11 ( I ‘𝑋) ∈ V
78 eleq1 2676 . . . . . . . . . . . 12 (𝑥 = (𝑓𝑘) → (𝑥 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ↔ (𝑓𝑘) ∈ 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛)))
79 fvex 6113 . . . . . . . . . . . . 13 (𝑓𝑘) ∈ V
80 eliin 4461 . . . . . . . . . . . . 13 ((𝑓𝑘) ∈ V → ((𝑓𝑘) ∈ 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ↔ ∀𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)))
8179, 80ax-mp 5 . . . . . . . . . . . 12 ((𝑓𝑘) ∈ 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ↔ ∀𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛))
8278, 81syl6bb 275 . . . . . . . . . . 11 (𝑥 = (𝑓𝑘) → (𝑥 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ↔ ∀𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)))
8377, 82axcc4dom 9146 . . . . . . . . . 10 ((𝑍 ≼ ω ∧ ∀𝑘𝑍𝑥 ∈ ( I ‘𝑋)𝑥 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛)) → ∃𝑓(𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)))
8429, 76, 83syl2anc 691 . . . . . . . . 9 (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) → ∃𝑓(𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)))
85 df-ral 2901 . . . . . . . . . . . . 13 (∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽)) ↔ ∀𝑓(𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))))
86 19.29 1789 . . . . . . . . . . . . 13 ((∀𝑓(𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ ∃𝑓(𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛))) → ∃𝑓((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛))))
8785, 86sylanb 488 . . . . . . . . . . . 12 ((∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽)) ∧ ∃𝑓(𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛))) → ∃𝑓((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛))))
8824ad2antrr 758 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)))) → 𝑀 ∈ ℤ)
895ad2antrr 758 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)))) → 𝐷 ∈ (Met‘𝑋))
90 simprrl 800 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)))) → 𝑓:𝑍⟶( I ‘𝑋))
91 feq3 5941 . . . . . . . . . . . . . . . . 17 (( I ‘𝑋) = 𝑋 → (𝑓:𝑍⟶( I ‘𝑋) ↔ 𝑓:𝑍𝑋))
9289, 53, 54, 914syl 19 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)))) → (𝑓:𝑍⟶( I ‘𝑋) ↔ 𝑓:𝑍𝑋))
9390, 92mpbid 221 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)))) → 𝑓:𝑍𝑋)
94 simplrr 797 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)))) → (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))
9594simprd 478 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)))) → ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘))
96 fveq2 6103 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑖 → (𝑠𝑘) = (𝑠𝑖))
97 oveq2 6557 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑖 → ((1 / 2)↑𝑘) = ((1 / 2)↑𝑖))
9897breq2d 4595 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑖 → ((𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ (𝑢𝐷𝑣) < ((1 / 2)↑𝑖)))
9996, 98raleqbidv 3129 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑖 → (∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ∀𝑣 ∈ (𝑠𝑖)(𝑢𝐷𝑣) < ((1 / 2)↑𝑖)))
10096, 99raleqbidv 3129 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑖 → (∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ∀𝑢 ∈ (𝑠𝑖)∀𝑣 ∈ (𝑠𝑖)(𝑢𝐷𝑣) < ((1 / 2)↑𝑖)))
101100cbvralv 3147 . . . . . . . . . . . . . . . 16 (∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ∀𝑖 ∈ ℤ ∀𝑢 ∈ (𝑠𝑖)∀𝑣 ∈ (𝑠𝑖)(𝑢𝐷𝑣) < ((1 / 2)↑𝑖))
10295, 101sylib 207 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)))) → ∀𝑖 ∈ ℤ ∀𝑢 ∈ (𝑠𝑖)∀𝑣 ∈ (𝑠𝑖)(𝑢𝐷𝑣) < ((1 / 2)↑𝑖))
103 simprrr 801 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)))) → ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛))
104 fveq2 6103 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑗 → (𝑠𝑛) = (𝑠𝑗))
105104eleq2d 2673 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑗 → ((𝑓𝑘) ∈ (𝑠𝑛) ↔ (𝑓𝑘) ∈ (𝑠𝑗)))
106105cbvralv 3147 . . . . . . . . . . . . . . . . . 18 (∀𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛) ↔ ∀𝑗 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑗))
107 oveq2 6557 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑖 → (𝑀...𝑘) = (𝑀...𝑖))
108 fveq2 6103 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑖 → (𝑓𝑘) = (𝑓𝑖))
109108eleq1d 2672 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑖 → ((𝑓𝑘) ∈ (𝑠𝑗) ↔ (𝑓𝑖) ∈ (𝑠𝑗)))
110107, 109raleqbidv 3129 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑖 → (∀𝑗 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑗) ↔ ∀𝑗 ∈ (𝑀...𝑖)(𝑓𝑖) ∈ (𝑠𝑗)))
111106, 110syl5bb 271 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑖 → (∀𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛) ↔ ∀𝑗 ∈ (𝑀...𝑖)(𝑓𝑖) ∈ (𝑠𝑗)))
112111cbvralv 3147 . . . . . . . . . . . . . . . 16 (∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛) ↔ ∀𝑖𝑍𝑗 ∈ (𝑀...𝑖)(𝑓𝑖) ∈ (𝑠𝑗))
113103, 112sylib 207 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)))) → ∀𝑖𝑍𝑗 ∈ (𝑀...𝑖)(𝑓𝑖) ∈ (𝑠𝑗))
11489, 38syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)))) → 𝐷 ∈ (∞Met‘𝑋))
115 simplrl 796 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)))) → 𝑔 ∈ (CauFil‘𝐷))
116114, 115, 42syl2anc 691 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)))) → 𝑔 ∈ (Fil‘𝑋))
11794simpld 474 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)))) → 𝑠:ℤ⟶𝑔)
11826, 1, 88, 89, 93, 102, 113iscmet3lem1 22897 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)))) → 𝑓 ∈ (Cau‘𝐷))
119 simprl 790 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)))) → (𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))))
120118, 93, 119mp2d 47 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)))) → 𝑓 ∈ dom (⇝𝑡𝐽))
12126, 1, 88, 89, 93, 102, 113, 116, 117, 120iscmet3lem2 22898 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)))) → (𝐽 fLim 𝑔) ≠ ∅)
122121ex 449 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) → (((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛))) → (𝐽 fLim 𝑔) ≠ ∅))
123122exlimdv 1848 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) → (∃𝑓((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛))) → (𝐽 fLim 𝑔) ≠ ∅))
12487, 123syl5 33 . . . . . . . . . . 11 ((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) → ((∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽)) ∧ ∃𝑓(𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛))) → (𝐽 fLim 𝑔) ≠ ∅))
125124expdimp 452 . . . . . . . . . 10 (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) → (∃𝑓(𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)) → (𝐽 fLim 𝑔) ≠ ∅))
126125an32s 842 . . . . . . . . 9 (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) → (∃𝑓(𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)) → (𝐽 fLim 𝑔) ≠ ∅))
12784, 126mpd 15 . . . . . . . 8 (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) → (𝐽 fLim 𝑔) ≠ ∅)
128127expr 641 . . . . . . 7 (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ 𝑔 ∈ (CauFil‘𝐷)) → ((𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)) → (𝐽 fLim 𝑔) ≠ ∅))
129128exlimdv 1848 . . . . . 6 (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ 𝑔 ∈ (CauFil‘𝐷)) → (∃𝑠(𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)) → (𝐽 fLim 𝑔) ≠ ∅))
13023, 129mpd 15 . . . . 5 (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ 𝑔 ∈ (CauFil‘𝐷)) → (𝐽 fLim 𝑔) ≠ ∅)
131130ralrimiva 2949 . . . 4 ((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) → ∀𝑔 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑔) ≠ ∅)
1321iscmet 22890 . . . 4 (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑔 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑔) ≠ ∅))
1336, 131, 132sylanbrc 695 . . 3 ((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) → 𝐷 ∈ (CMet‘𝑋))
134133ex 449 . 2 (𝜑 → (∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽)) → 𝐷 ∈ (CMet‘𝑋)))
1354, 134impbid2 215 1 (𝜑 → (𝐷 ∈ (CMet‘𝑋) ↔ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ∩ ciin 4456   class class class wbr 4583   I cid 4948  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ωcom 6957   ≈ cen 7838   ≼ cdom 7839  Fincfn 7841  ficfi 8199  1c1 9816   < clt 9953   / cdiv 10563  ℕcn 10897  2c2 10947  ℤcz 11254  ℤ≥cuz 11563  ℝ+crp 11708  ...cfz 12197  ↑cexp 12722  ∞Metcxmt 19552  Metcme 19553  MetOpencmopn 19557  ⇝𝑡clm 20840  Filcfil 21459   fLim cflim 21548  CauFilccfil 22858  Caucca 22859  CMetcms 22860 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cc 9140  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ico 12052  df-fz 12198  df-fl 12455  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-top 20521  df-bases 20522  df-topon 20523  df-ntr 20634  df-nei 20712  df-lm 20843  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-cfil 22861  df-cau 22862  df-cmet 22863 This theorem is referenced by:  iscmet2  22900  iscmet3i  22918  heibor1  32779  rrncms  32802
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