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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 24308 | . 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 6909 | . . . 4 ⊢ (Fil‘𝐵) = (Fil‘(Base‘𝑊)) |
| 4 | iscusp2.2 | . . . . . . 7 ⊢ 𝑈 = (UnifSt‘𝑊) | |
| 5 | 4 | fveq2i 6909 | . . . . . 6 ⊢ (CauFilu‘𝑈) = (CauFilu‘(UnifSt‘𝑊)) |
| 6 | 5 | eleq2i 2833 | . . . . 5 ⊢ (𝑐 ∈ (CauFilu‘𝑈) ↔ 𝑐 ∈ (CauFilu‘(UnifSt‘𝑊))) |
| 7 | iscusp2.3 | . . . . . . 7 ⊢ 𝐽 = (TopOpen‘𝑊) | |
| 8 | 7 | oveq1i 7441 | . . . . . 6 ⊢ (𝐽 fLim 𝑐) = ((TopOpen‘𝑊) fLim 𝑐) |
| 9 | 8 | neeq1i 3005 | . . . . 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 3344 | . . 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 278 | 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∅c0 4333 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 TopOpenctopn 17466 Filcfil 23853 fLim cflim 23942 UnifStcuss 24262 UnifSpcusp 24263 CauFiluccfilu 24295 CUnifSpccusp 24306 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-cusp 24307 |
| This theorem is referenced by: cmetcusp1 25387 |
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