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Mirrors > Home > MPE Home > Th. List > cfilucfil4 | 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 |
---|---|
cfilucfil4 | ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋) ∧ 𝐶 ∈ (Fil‘𝑋)) → (𝐶 ∈ (CauFilu‘(metUnif‘𝐷)) ↔ 𝐶 ∈ (CauFil‘𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cfilucfil3 25168 | . . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → ((𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (CauFilu‘(metUnif‘𝐷))) ↔ 𝐶 ∈ (CauFil‘𝐷))) | |
2 | cfilfil 25115 | . . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐶 ∈ (CauFil‘𝐷)) → 𝐶 ∈ (Fil‘𝑋)) | |
3 | 2 | ex 412 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐶 ∈ (CauFil‘𝐷) → 𝐶 ∈ (Fil‘𝑋))) |
4 | 3 | adantl 481 | . . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → (𝐶 ∈ (CauFil‘𝐷) → 𝐶 ∈ (Fil‘𝑋))) |
5 | 4 | pm4.71rd 562 | . . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → (𝐶 ∈ (CauFil‘𝐷) ↔ (𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (CauFil‘𝐷)))) |
6 | 1, 5 | bitrd 279 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → ((𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (CauFilu‘(metUnif‘𝐷))) ↔ (𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (CauFil‘𝐷)))) |
7 | pm5.32 573 | . . 3 ⊢ ((𝐶 ∈ (Fil‘𝑋) → (𝐶 ∈ (CauFilu‘(metUnif‘𝐷)) ↔ 𝐶 ∈ (CauFil‘𝐷))) ↔ ((𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (CauFilu‘(metUnif‘𝐷))) ↔ (𝐶 ∈ (Fil‘𝑋) ∧ 𝐶 ∈ (CauFil‘𝐷)))) | |
8 | 6, 7 | sylibr 233 | . 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → (𝐶 ∈ (Fil‘𝑋) → (𝐶 ∈ (CauFilu‘(metUnif‘𝐷)) ↔ 𝐶 ∈ (CauFil‘𝐷)))) |
9 | 8 | 3impia 1116 | 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 ∧ w3a 1086 ∈ wcel 2105 ≠ wne 2939 ∅c0 4322 ‘cfv 6543 ∞Metcxmet 21218 metUnifcmetu 21224 Filcfil 23669 CauFiluccfilu 24111 CauFilccfil 25100 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-2 12282 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ico 13337 df-psmet 21225 df-xmet 21226 df-fbas 21230 df-fg 21231 df-metu 21232 df-fil 23670 df-ust 24025 df-cfilu 24112 df-cfil 25103 |
This theorem is referenced by: cmetcusp1 25201 |
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