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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 22103 | . 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 6194 | . . . 4 ⊢ (Fil‘𝐵) = (Fil‘(Base‘𝑊)) |
4 | iscusp2.2 | . . . . . . 7 ⊢ 𝑈 = (UnifSt‘𝑊) | |
5 | 4 | fveq2i 6194 | . . . . . 6 ⊢ (CauFilu‘𝑈) = (CauFilu‘(UnifSt‘𝑊)) |
6 | 5 | eleq2i 2693 | . . . . 5 ⊢ (𝑐 ∈ (CauFilu‘𝑈) ↔ 𝑐 ∈ (CauFilu‘(UnifSt‘𝑊))) |
7 | iscusp2.3 | . . . . . . 7 ⊢ 𝐽 = (TopOpen‘𝑊) | |
8 | 7 | oveq1i 6660 | . . . . . 6 ⊢ (𝐽 fLim 𝑐) = ((TopOpen‘𝑊) fLim 𝑐) |
9 | 8 | neeq1i 2858 | . . . . 5 ⊢ ((𝐽 fLim 𝑐) ≠ ∅ ↔ ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅) |
10 | 6, 9 | imbi12i 340 | . . . 4 ⊢ ((𝑐 ∈ (CauFilu‘𝑈) → (𝐽 fLim 𝑐) ≠ ∅) ↔ (𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)) |
11 | 3, 10 | raleqbii 2990 | . . 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 730 | . 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 267 | 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 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∅c0 3915 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 TopOpenctopn 16082 Filcfil 21649 fLim cflim 21738 UnifStcuss 22057 UnifSpcusp 22058 CauFiluccfilu 22090 CUnifSpccusp 22101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-cusp 22102 |
This theorem is referenced by: cmetcusp1 23149 |
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