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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 22515 | . 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 6451 | . . . 4 ⊢ (Fil‘𝐵) = (Fil‘(Base‘𝑊)) |
4 | iscusp2.2 | . . . . . . 7 ⊢ 𝑈 = (UnifSt‘𝑊) | |
5 | 4 | fveq2i 6451 | . . . . . 6 ⊢ (CauFilu‘𝑈) = (CauFilu‘(UnifSt‘𝑊)) |
6 | 5 | eleq2i 2851 | . . . . 5 ⊢ (𝑐 ∈ (CauFilu‘𝑈) ↔ 𝑐 ∈ (CauFilu‘(UnifSt‘𝑊))) |
7 | iscusp2.3 | . . . . . . 7 ⊢ 𝐽 = (TopOpen‘𝑊) | |
8 | 7 | oveq1i 6934 | . . . . . 6 ⊢ (𝐽 fLim 𝑐) = ((TopOpen‘𝑊) fLim 𝑐) |
9 | 8 | neeq1i 3033 | . . . . 5 ⊢ ((𝐽 fLim 𝑐) ≠ ∅ ↔ ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅) |
10 | 6, 9 | imbi12i 342 | . . . 4 ⊢ ((𝑐 ∈ (CauFilu‘𝑈) → (𝐽 fLim 𝑐) ≠ ∅) ↔ (𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)) |
11 | 3, 10 | raleqbii 3172 | . . 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 616 | . 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 270 | 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 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∀wral 3090 ∅c0 4141 ‘cfv 6137 (class class class)co 6924 Basecbs 16259 TopOpenctopn 16472 Filcfil 22061 fLim cflim 22150 UnifStcuss 22469 UnifSpcusp 22470 CauFiluccfilu 22502 CUnifSpccusp 22513 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-iota 6101 df-fv 6145 df-ov 6927 df-cusp 22514 |
This theorem is referenced by: cmetcusp1 23563 |
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