![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cmetcvg | Structured version Visualization version GIF version |
Description: The convergence of a Cauchy filter in a complete metric space. (Contributed by Mario Carneiro, 14-Oct-2015.) |
Ref | Expression |
---|---|
iscmet.1 | ⊢ 𝐽 = (MetOpen‘𝐷) |
Ref | Expression |
---|---|
cmetcvg | ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → (𝐽 fLim 𝐹) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscmet.1 | . . . 4 ⊢ 𝐽 = (MetOpen‘𝐷) | |
2 | 1 | iscmet 25332 | . . 3 ⊢ (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅)) |
3 | 2 | simprbi 496 | . 2 ⊢ (𝐷 ∈ (CMet‘𝑋) → ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅) |
4 | oveq2 7439 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝐽 fLim 𝑓) = (𝐽 fLim 𝐹)) | |
5 | 4 | neeq1d 2998 | . . 3 ⊢ (𝑓 = 𝐹 → ((𝐽 fLim 𝑓) ≠ ∅ ↔ (𝐽 fLim 𝐹) ≠ ∅)) |
6 | 5 | rspccva 3621 | . 2 ⊢ ((∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅ ∧ 𝐹 ∈ (CauFil‘𝐷)) → (𝐽 fLim 𝐹) ≠ ∅) |
7 | 3, 6 | sylan 580 | 1 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → (𝐽 fLim 𝐹) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∅c0 4339 ‘cfv 6563 (class class class)co 7431 Metcmet 21368 MetOpencmopn 21372 fLim cflim 23958 CauFilccfil 25300 CMetccmet 25302 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-cmet 25305 |
This theorem is referenced by: cmetcaulem 25336 metsscmetcld 25363 cmetss 25364 cmetcusp 25402 minveclem4a 25478 fmcncfil 33892 |
Copyright terms: Public domain | W3C validator |