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Mirrors > Home > MPE Home > Th. List > iscusp2 | Structured version Visualization version Unicode version |
Description: The predicate " is a complete uniform space." (Contributed by Thierry Arnoux, 15-Dec-2017.) |
Ref | Expression |
---|---|
iscusp2.1 | |
iscusp2.2 | UnifSt |
iscusp2.3 |
Ref | Expression |
---|---|
iscusp2 | CUnifSp UnifSp CauFilu |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscusp 22103 | . 2 CUnifSp UnifSp CauFiluUnifSt | |
2 | iscusp2.1 | . . . . 5 | |
3 | 2 | fveq2i 6194 | . . . 4 |
4 | iscusp2.2 | . . . . . . 7 UnifSt | |
5 | 4 | fveq2i 6194 | . . . . . 6 CauFilu CauFiluUnifSt |
6 | 5 | eleq2i 2693 | . . . . 5 CauFilu CauFiluUnifSt |
7 | iscusp2.3 | . . . . . . 7 | |
8 | 7 | oveq1i 6660 | . . . . . 6 |
9 | 8 | neeq1i 2858 | . . . . 5 |
10 | 6, 9 | imbi12i 340 | . . . 4 CauFilu CauFiluUnifSt |
11 | 3, 10 | raleqbii 2990 | . . 3 CauFilu CauFiluUnifSt |
12 | 11 | anbi2i 730 | . 2 UnifSp CauFilu UnifSp CauFiluUnifSt |
13 | 1, 12 | bitr4i 267 | 1 CUnifSp UnifSp CauFilu |
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 cbs 15857 ctopn 16082 cfil 21649 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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