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Mirrors > Home > MPE Home > Th. List > iscusp2 | Structured version Visualization version GIF version |
Description: The predicate "𝑊 is a complete uniform space." (Contributed by Thierry Arnoux, 15-Dec-2017.) |
Ref | Expression |
---|---|
iscusp2.1 | ⊢ 𝐵 = (Base‘𝑊) |
iscusp2.2 | ⊢ 𝑈 = (UnifSt‘𝑊) |
iscusp2.3 | ⊢ 𝐽 = (TopOpen‘𝑊) |
Ref | Expression |
---|---|
iscusp2 | ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘𝐵)(𝑐 ∈ (CauFilu‘𝑈) → (𝐽 fLim 𝑐) ≠ ∅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscusp 23635 | . 2 ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))) | |
2 | iscusp2.1 | . . . . 5 ⊢ 𝐵 = (Base‘𝑊) | |
3 | 2 | fveq2i 6842 | . . . 4 ⊢ (Fil‘𝐵) = (Fil‘(Base‘𝑊)) |
4 | iscusp2.2 | . . . . . . 7 ⊢ 𝑈 = (UnifSt‘𝑊) | |
5 | 4 | fveq2i 6842 | . . . . . 6 ⊢ (CauFilu‘𝑈) = (CauFilu‘(UnifSt‘𝑊)) |
6 | 5 | eleq2i 2829 | . . . . 5 ⊢ (𝑐 ∈ (CauFilu‘𝑈) ↔ 𝑐 ∈ (CauFilu‘(UnifSt‘𝑊))) |
7 | iscusp2.3 | . . . . . . 7 ⊢ 𝐽 = (TopOpen‘𝑊) | |
8 | 7 | oveq1i 7363 | . . . . . 6 ⊢ (𝐽 fLim 𝑐) = ((TopOpen‘𝑊) fLim 𝑐) |
9 | 8 | neeq1i 3006 | . . . . 5 ⊢ ((𝐽 fLim 𝑐) ≠ ∅ ↔ ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅) |
10 | 6, 9 | imbi12i 350 | . . . 4 ⊢ ((𝑐 ∈ (CauFilu‘𝑈) → (𝐽 fLim 𝑐) ≠ ∅) ↔ (𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)) |
11 | 3, 10 | raleqbii 3313 | . . 3 ⊢ (∀𝑐 ∈ (Fil‘𝐵)(𝑐 ∈ (CauFilu‘𝑈) → (𝐽 fLim 𝑐) ≠ ∅) ↔ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)) |
12 | 11 | anbi2i 623 | . 2 ⊢ ((𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘𝐵)(𝑐 ∈ (CauFilu‘𝑈) → (𝐽 fLim 𝑐) ≠ ∅)) ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))) |
13 | 1, 12 | bitr4i 277 | 1 ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘𝐵)(𝑐 ∈ (CauFilu‘𝑈) → (𝐽 fLim 𝑐) ≠ ∅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ∀wral 3062 ∅c0 4280 ‘cfv 6493 (class class class)co 7353 Basecbs 17075 TopOpenctopn 17295 Filcfil 23180 fLim cflim 23269 UnifStcuss 23589 UnifSpcusp 23590 CauFiluccfilu 23622 CUnifSpccusp 23633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-ral 3063 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-iota 6445 df-fv 6501 df-ov 7356 df-cusp 23634 |
This theorem is referenced by: cmetcusp1 24701 |
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