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Mirrors > Home > MPE Home > Th. List > cfilucfil3 | Structured version Visualization version GIF version |
Description: Given a metric 𝐷 and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the Cauchy filters for the metric. (Contributed by Thierry Arnoux, 15-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.) |
Ref | Expression |
---|---|
cfilucfil3 | ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → ((𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (CauFilu‘(metUnif‘𝐷))) ↔ 𝐶 ∈ (CauFil‘𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xmetpsmet 23409 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋)) | |
2 | cfilucfil2 23623 | . . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐶 ∈ (CauFilu‘(metUnif‘𝐷)) ↔ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))) | |
3 | 2 | anbi2d 628 | . . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (CauFilu‘(metUnif‘𝐷))) ↔ (𝐶 ∈ (Fil‘𝑋) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))) |
4 | filfbas 22907 | . . . . . . 7 ⊢ (𝐶 ∈ (Fil‘𝑋) → 𝐶 ∈ (fBas‘𝑋)) | |
5 | 4 | pm4.71i 559 | . . . . . 6 ⊢ (𝐶 ∈ (Fil‘𝑋) ↔ (𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (fBas‘𝑋))) |
6 | 5 | anbi1i 623 | . . . . 5 ⊢ ((𝐶 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)) ↔ ((𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (fBas‘𝑋)) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) |
7 | anass 468 | . . . . 5 ⊢ (((𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (fBas‘𝑋)) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)) ↔ (𝐶 ∈ (Fil‘𝑋) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))) | |
8 | 6, 7 | bitr2i 275 | . . . 4 ⊢ ((𝐶 ∈ (Fil‘𝑋) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ↔ (𝐶 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) |
9 | 3, 8 | bitrdi 286 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (CauFilu‘(metUnif‘𝐷))) ↔ (𝐶 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))) |
10 | 1, 9 | sylan2 592 | . 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → ((𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (CauFilu‘(metUnif‘𝐷))) ↔ (𝐶 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))) |
11 | iscfil 24334 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐶 ∈ (CauFil‘𝐷) ↔ (𝐶 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))) | |
12 | 11 | adantl 481 | . 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → (𝐶 ∈ (CauFil‘𝐷) ↔ (𝐶 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))) |
13 | 10, 12 | bitr4d 281 | 1 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → ((𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (CauFilu‘(metUnif‘𝐷))) ↔ 𝐶 ∈ (CauFil‘𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 ⊆ wss 3883 ∅c0 4253 × cxp 5578 “ cima 5583 ‘cfv 6418 (class class class)co 7255 0cc0 10802 ℝ+crp 12659 [,)cico 13010 PsMetcpsmet 20494 ∞Metcxmet 20495 fBascfbas 20498 metUnifcmetu 20501 Filcfil 22904 CauFiluccfilu 23346 CauFilccfil 24321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-po 5494 df-so 5495 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-1st 7804 df-2nd 7805 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-2 11966 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ico 13014 df-psmet 20502 df-xmet 20503 df-fbas 20507 df-fg 20508 df-metu 20509 df-fil 22905 df-ust 23260 df-cfilu 23347 df-cfil 24324 |
This theorem is referenced by: cfilucfil4 24390 |
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