MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscmet3 Structured version   Visualization version   GIF version

Theorem iscmet3 23499
Description: The property "𝐷 is a complete metric" expressed in terms of functions on (or any other upper integer set). Thus, we only have to look at functions on , 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 23495 . . . 4 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑓 ∈ (Cau‘𝐷)) → 𝑓 ∈ dom (⇝𝑡𝐽))
32a1d 25 . . 3 ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑓 ∈ (Cau‘𝐷)) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽)))
43ralrimiva 3148 . 2 (𝐷 ∈ (CMet‘𝑋) → ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽)))
5 iscmet3.4 . . . . 5 (𝜑𝐷 ∈ (Met‘𝑋))
65adantr 474 . . . 4 ((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) → 𝐷 ∈ (Met‘𝑋))
7 simpr 479 . . . . . . . . 9 (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ 𝑔 ∈ (CauFil‘𝐷)) → 𝑔 ∈ (CauFil‘𝐷))
8 1rp 12141 . . . . . . . . . . 11 1 ∈ ℝ+
9 rphalfcl 12166 . . . . . . . . . . 11 (1 ∈ ℝ+ → (1 / 2) ∈ ℝ+)
108, 9ax-mp 5 . . . . . . . . . 10 (1 / 2) ∈ ℝ+
11 rpexpcl 13197 . . . . . . . . . 10 (((1 / 2) ∈ ℝ+𝑘 ∈ ℤ) → ((1 / 2)↑𝑘) ∈ ℝ+)
1210, 11mpan 680 . . . . . . . . 9 (𝑘 ∈ ℤ → ((1 / 2)↑𝑘) ∈ ℝ+)
13 cfili 23474 . . . . . . . . 9 ((𝑔 ∈ (CauFil‘𝐷) ∧ ((1 / 2)↑𝑘) ∈ ℝ+) → ∃𝑡𝑔𝑢𝑡𝑣𝑡 (𝑢𝐷𝑣) < ((1 / 2)↑𝑘))
147, 12, 13syl2an 589 . . . . . . . 8 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ 𝑔 ∈ (CauFil‘𝐷)) ∧ 𝑘 ∈ ℤ) → ∃𝑡𝑔𝑢𝑡𝑣𝑡 (𝑢𝐷𝑣) < ((1 / 2)↑𝑘))
1514ralrimiva 3148 . . . . . . 7 (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ 𝑔 ∈ (CauFil‘𝐷)) → ∀𝑘 ∈ ℤ ∃𝑡𝑔𝑢𝑡𝑣𝑡 (𝑢𝐷𝑣) < ((1 / 2)↑𝑘))
16 vex 3401 . . . . . . . 8 𝑔 ∈ V
17 znnen 15345 . . . . . . . . 9 ℤ ≈ ℕ
18 nnenom 13098 . . . . . . . . 9 ℕ ≈ ω
1917, 18entri 8295 . . . . . . . 8 ℤ ≈ ω
20 raleq 3330 . . . . . . . . 9 (𝑡 = (𝑠𝑘) → (∀𝑣𝑡 (𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))
2120raleqbi1dv 3328 . . . . . . . 8 (𝑡 = (𝑠𝑘) → (∀𝑢𝑡𝑣𝑡 (𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))
2216, 19, 21axcc4 9596 . . . . . . 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 716 . . . . . . . . . . 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 13082 . . . . . . . . . . 11 (𝑀 ∈ ℤ → 𝑍 ≈ ω)
28 endom 8268 . . . . . . . . . . 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 3800 . . . . . . . . . . . . . . 15 (( I ‘𝑋) ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛)) = {𝑥 ∈ ( I ‘𝑋) ∣ 𝑥 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛)}
31 fzn0 12672 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀...𝑘) ≠ ∅ ↔ 𝑘 ∈ (ℤ𝑀))
3231biimpri 220 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (ℤ𝑀) → (𝑀...𝑘) ≠ ∅)
3332, 26eleq2s 2877 . . . . . . . . . . . . . . . . . . 19 (𝑘𝑍 → (𝑀...𝑘) ≠ ∅)
34 simprr 763 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) → 𝑠:ℤ⟶𝑔)
35 elfzelz 12659 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ (𝑀...𝑘) → 𝑛 ∈ ℤ)
36 ffvelrn 6621 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠:ℤ⟶𝑔𝑛 ∈ ℤ) → (𝑠𝑛) ∈ 𝑔)
3734, 35, 36syl2an 589 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑛 ∈ (𝑀...𝑘)) → (𝑠𝑛) ∈ 𝑔)
38 metxmet 22547 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
395, 38syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐷 ∈ (∞Met‘𝑋))
4039adantr 474 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) → 𝐷 ∈ (∞Met‘𝑋))
41 simpl 476 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔) → 𝑔 ∈ (CauFil‘𝐷))
42 cfilfil 23473 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑔 ∈ (CauFil‘𝐷)) → 𝑔 ∈ (Fil‘𝑋))
4340, 41, 42syl2an 589 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) → 𝑔 ∈ (Fil‘𝑋))
44 filelss 22064 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔 ∈ (Fil‘𝑋) ∧ (𝑠𝑛) ∈ 𝑔) → (𝑠𝑛) ⊆ 𝑋)
4543, 44sylan 575 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ (𝑠𝑛) ∈ 𝑔) → (𝑠𝑛) ⊆ 𝑋)
4637, 45syldan 585 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑛 ∈ (𝑀...𝑘)) → (𝑠𝑛) ⊆ 𝑋)
4746ralrimiva 3148 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) → ∀𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ⊆ 𝑋)
48 r19.2z 4283 . . . . . . . . . . . . . . . . . . 19 (((𝑀...𝑘) ≠ ∅ ∧ ∀𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ⊆ 𝑋) → ∃𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ⊆ 𝑋)
4933, 47, 48syl2anr 590 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → ∃𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ⊆ 𝑋)
50 iinss 4804 . . . . . . . . . . . . . . . . . 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 716 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → 𝐷 ∈ (Met‘𝑋))
53 elfvdm 6478 . . . . . . . . . . . . . . . . . 18 (𝐷 ∈ (Met‘𝑋) → 𝑋 ∈ dom Met)
54 fvi 6515 . . . . . . . . . . . . . . . . . 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 3861 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ⊆ ( I ‘𝑋))
57 sseqin2 4040 . . . . . . . . . . . . . . . 16 ( 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ⊆ ( I ‘𝑋) ↔ (( I ‘𝑋) ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛)) = 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛))
5856, 57sylib 210 . . . . . . . . . . . . . . 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 2828 . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → 𝑔 ∈ (Fil‘𝑋))
6137ralrimiva 3148 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) → ∀𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ∈ 𝑔)
6261adantr 474 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → ∀𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ∈ 𝑔)
6333adantl 475 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → (𝑀...𝑘) ≠ ∅)
64 fzfid 13091 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → (𝑀...𝑘) ∈ Fin)
65 iinfi 8611 . . . . . . . . . . . . . . . . 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 1440 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ∈ (fi‘𝑔))
67 filfi 22071 . . . . . . . . . . . . . . . . 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 2861 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ∈ 𝑔)
70 fileln0 22062 . . . . . . . . . . . . . . 15 ((𝑔 ∈ (Fil‘𝑋) ∧ 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ∈ 𝑔) → 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ≠ ∅)
7160, 69, 70syl2anc 579 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘𝑍) → 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ≠ ∅)
7259, 71eqnetrd 3036 . . . . . . . . . . . . 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 4188 . . . . . . . . . . . . 13 ({𝑥 ∈ ( I ‘𝑋) ∣ 𝑥 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛)} ≠ ∅ ↔ ∃𝑥 ∈ ( I ‘𝑋)𝑥 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛))
7472, 73sylib 210 . . . . . . . . . . . 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 3148 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) → ∀𝑘𝑍𝑥 ∈ ( I ‘𝑋)𝑥 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛))
7675adantrrr 715 . . . . . . . . . 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 6459 . . . . . . . . . . 11 ( I ‘𝑋) ∈ V
78 eleq1 2847 . . . . . . . . . . . 12 (𝑥 = (𝑓𝑘) → (𝑥 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ↔ (𝑓𝑘) ∈ 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛)))
79 fvex 6459 . . . . . . . . . . . . 13 (𝑓𝑘) ∈ V
80 eliin 4758 . . . . . . . . . . . . 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 279 . . . . . . . . . . 11 (𝑥 = (𝑓𝑘) → (𝑥 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛) ↔ ∀𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)))
8377, 82axcc4dom 9598 . . . . . . . . . 10 ((𝑍 ≼ ω ∧ ∀𝑘𝑍𝑥 ∈ ( I ‘𝑋)𝑥 𝑛 ∈ (𝑀...𝑘)(𝑠𝑛)) → ∃𝑓(𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛)))
8429, 76, 83syl2anc 579 . . . . . . . . 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 3095 . . . . . . . . . . . . 13 (∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽)) ↔ ∀𝑓(𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))))
86 19.29 1920 . . . . . . . . . . . . 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 576 . . . . . . . . . . . 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 716 . . . . . . . . . . . . . . 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 716 . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . 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 6274 . . . . . . . . . . . . . . . . 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 224 . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . . 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 491 . . . . . . . . . . . . . . . 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 6446 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑖 → (𝑠𝑘) = (𝑠𝑖))
97 oveq2 6930 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑖 → ((1 / 2)↑𝑘) = ((1 / 2)↑𝑖))
9897breq2d 4898 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑖 → ((𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ (𝑢𝐷𝑣) < ((1 / 2)↑𝑖)))
9996, 98raleqbidv 3326 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑖 → (∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ∀𝑣 ∈ (𝑠𝑖)(𝑢𝐷𝑣) < ((1 / 2)↑𝑖)))
10096, 99raleqbidv 3326 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑖 → (∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ∀𝑢 ∈ (𝑠𝑖)∀𝑣 ∈ (𝑠𝑖)(𝑢𝐷𝑣) < ((1 / 2)↑𝑖)))
101100cbvralv 3367 . . . . . . . . . . . . . . . 16 (∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠𝑘)∀𝑣 ∈ (𝑠𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ∀𝑖 ∈ ℤ ∀𝑢 ∈ (𝑠𝑖)∀𝑣 ∈ (𝑠𝑖)(𝑢𝐷𝑣) < ((1 / 2)↑𝑖))
10295, 101sylib 210 . . . . . . . . . . . . . . 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 772 . . . . . . . . . . . . . . . 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 6446 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑗 → (𝑠𝑛) = (𝑠𝑗))
105104eleq2d 2845 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑗 → ((𝑓𝑘) ∈ (𝑠𝑛) ↔ (𝑓𝑘) ∈ (𝑠𝑗)))
106105cbvralv 3367 . . . . . . . . . . . . . . . . . 18 (∀𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛) ↔ ∀𝑗 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑗))
107 oveq2 6930 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑖 → (𝑀...𝑘) = (𝑀...𝑖))
108 fveq2 6446 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑖 → (𝑓𝑘) = (𝑓𝑖))
109108eleq1d 2844 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑖 → ((𝑓𝑘) ∈ (𝑠𝑗) ↔ (𝑓𝑖) ∈ (𝑠𝑗)))
110107, 109raleqbidv 3326 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑖 → (∀𝑗 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑗) ↔ ∀𝑗 ∈ (𝑀...𝑖)(𝑓𝑖) ∈ (𝑠𝑗)))
111106, 110syl5bb 275 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑖 → (∀𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛) ↔ ∀𝑗 ∈ (𝑀...𝑖)(𝑓𝑖) ∈ (𝑠𝑗)))
112111cbvralv 3367 . . . . . . . . . . . . . . . 16 (∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝑓𝑘) ∈ (𝑠𝑛) ↔ ∀𝑖𝑍𝑗 ∈ (𝑀...𝑖)(𝑓𝑖) ∈ (𝑠𝑗))
113103, 112sylib 210 . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . 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 579 . . . . . . . . . . . . . . 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 490 . . . . . . . . . . . . . . 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 23497 . . . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . . 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 49 . . . . . . . . . . . . . . 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 23498 . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . 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 1976 . . . . . . . . . . . 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 34 . . . . . . . . . . 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 446 . . . . . . . . . 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 642 . . . . . . . . 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 450 . . . . . . 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 1976 . . . . . 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 3148 . . . 4 ((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) → ∀𝑔 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑔) ≠ ∅)
1321iscmet 23490 . . . 4 (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑔 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑔) ≠ ∅))
1336, 131, 132sylanbrc 578 . . 3 ((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))) → 𝐷 ∈ (CMet‘𝑋))
134133ex 403 . 2 (𝜑 → (∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽)) → 𝐷 ∈ (CMet‘𝑋)))
1354, 134impbid2 218 1 (𝜑 → (𝐷 ∈ (CMet‘𝑋) ↔ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wal 1599   = wceq 1601  wex 1823  wcel 2107  wne 2969  wral 3090  wrex 3091  {crab 3094  Vcvv 3398  cin 3791  wss 3792  c0 4141   ciin 4754   class class class wbr 4886   I cid 5260  dom cdm 5355  wf 6131  cfv 6135  (class class class)co 6922  ωcom 7343  cen 8238  cdom 8239  Fincfn 8241  ficfi 8604  1c1 10273   < clt 10411   / cdiv 11032  cn 11374  2c2 11430  cz 11728  cuz 11992  +crp 12137  ...cfz 12643  cexp 13178  ∞Metcxmet 20127  Metcmet 20128  MetOpencmopn 20132  𝑡clm 21438  Filcfil 22057   fLim cflim 22146  CauFilccfil 23458  Cauccau 23459  CMetccmet 23460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cc 9592  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-iin 4756  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-omul 7848  df-er 8026  df-map 8142  df-pm 8143  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-fi 8605  df-sup 8636  df-inf 8637  df-oi 8704  df-card 9098  df-acn 9101  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-n0 11643  df-z 11729  df-uz 11993  df-q 12096  df-rp 12138  df-xneg 12257  df-xadd 12258  df-xmul 12259  df-ico 12493  df-fz 12644  df-fl 12912  df-seq 13120  df-exp 13179  df-cj 14246  df-re 14247  df-im 14248  df-sqrt 14382  df-abs 14383  df-clim 14627  df-rlim 14628  df-rest 16469  df-topgen 16490  df-psmet 20134  df-xmet 20135  df-met 20136  df-bl 20137  df-mopn 20138  df-fbas 20139  df-fg 20140  df-top 21106  df-topon 21123  df-bases 21158  df-ntr 21232  df-nei 21310  df-lm 21441  df-fil 22058  df-fm 22150  df-flim 22151  df-flf 22152  df-cfil 23461  df-cau 23462  df-cmet 23463
This theorem is referenced by:  iscmet2  23500  iscmet3i  23518  heibor1  34233  rrncms  34256
  Copyright terms: Public domain W3C validator