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Mirrors > Home > ILE Home > Th. List > isms | Unicode version |
Description: Express the predicate " is a metric space" with underlying set and distance function . (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.) |
Ref | Expression |
---|---|
isms.j | |
isms.x | |
isms.d |
Ref | Expression |
---|---|
isms |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 5480 | . . . . 5 | |
2 | fveq2 5480 | . . . . . . 7 | |
3 | isms.x | . . . . . . 7 | |
4 | 2, 3 | eqtr4di 2215 | . . . . . 6 |
5 | 4 | sqxpeqd 4624 | . . . . 5 |
6 | 1, 5 | reseq12d 4879 | . . . 4 |
7 | isms.d | . . . 4 | |
8 | 6, 7 | eqtr4di 2215 | . . 3 |
9 | 4 | fveq2d 5484 | . . 3 |
10 | 8, 9 | eleq12d 2235 | . 2 |
11 | df-ms 12887 | . 2 | |
12 | 10, 11 | elrab2 2880 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1342 wcel 2135 cxp 4596 cres 4600 cfv 5182 cbs 12337 cds 12408 ctopn 12499 cmet 12528 cxms 12883 cms 12884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-rab 2451 df-v 2723 df-un 3115 df-in 3117 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-xp 4604 df-res 4610 df-iota 5147 df-fv 5190 df-ms 12887 |
This theorem is referenced by: isms2 13001 msxms 13005 mspropd 13025 |
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