Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 22900 | . 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 6666 | . . . 4 ⊢ (Fil‘𝐵) = (Fil‘(Base‘𝑊)) |
4 | iscusp2.2 | . . . . . . 7 ⊢ 𝑈 = (UnifSt‘𝑊) | |
5 | 4 | fveq2i 6666 | . . . . . 6 ⊢ (CauFilu‘𝑈) = (CauFilu‘(UnifSt‘𝑊)) |
6 | 5 | eleq2i 2902 | . . . . 5 ⊢ (𝑐 ∈ (CauFilu‘𝑈) ↔ 𝑐 ∈ (CauFilu‘(UnifSt‘𝑊))) |
7 | iscusp2.3 | . . . . . . 7 ⊢ 𝐽 = (TopOpen‘𝑊) | |
8 | 7 | oveq1i 7158 | . . . . . 6 ⊢ (𝐽 fLim 𝑐) = ((TopOpen‘𝑊) fLim 𝑐) |
9 | 8 | neeq1i 3078 | . . . . 5 ⊢ ((𝐽 fLim 𝑐) ≠ ∅ ↔ ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅) |
10 | 6, 9 | imbi12i 353 | . . . 4 ⊢ ((𝑐 ∈ (CauFilu‘𝑈) → (𝐽 fLim 𝑐) ≠ ∅) ↔ (𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)) |
11 | 3, 10 | raleqbii 3232 | . . 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 624 | . 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 280 | 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 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 ∀wral 3136 ∅c0 4289 ‘cfv 6348 (class class class)co 7148 Basecbs 16475 TopOpenctopn 16687 Filcfil 22445 fLim cflim 22534 UnifStcuss 22854 UnifSpcusp 22855 CauFiluccfilu 22887 CUnifSpccusp 22898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7151 df-cusp 22899 |
This theorem is referenced by: cmetcusp1 23948 |
Copyright terms: Public domain | W3C validator |