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Mirrors > Home > ILE Home > Th. List > ispsmet | Unicode version |
Description: Express the predicate " is a pseudometric." (Contributed by Thierry Arnoux, 7-Feb-2018.) |
Ref | Expression |
---|---|
ispsmet | PsMet |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-psmet 12156 | . . . . 5 PsMet | |
2 | id 19 | . . . . . . . 8 | |
3 | 2 | sqxpeqd 4565 | . . . . . . 7 |
4 | 3 | oveq2d 5790 | . . . . . 6 |
5 | raleq 2626 | . . . . . . . . 9 | |
6 | 5 | raleqbi1dv 2634 | . . . . . . . 8 |
7 | 6 | anbi2d 459 | . . . . . . 7 |
8 | 7 | raleqbi1dv 2634 | . . . . . 6 |
9 | 4, 8 | rabeqbidv 2681 | . . . . 5 |
10 | elex 2697 | . . . . 5 | |
11 | xrex 9639 | . . . . . . . 8 | |
12 | sqxpexg 4655 | . . . . . . . 8 | |
13 | mapvalg 6552 | . . . . . . . 8 | |
14 | 11, 12, 13 | sylancr 410 | . . . . . . 7 |
15 | mapex 6548 | . . . . . . . 8 | |
16 | 12, 11, 15 | sylancl 409 | . . . . . . 7 |
17 | 14, 16 | eqeltrd 2216 | . . . . . 6 |
18 | rabexg 4071 | . . . . . 6 | |
19 | 17, 18 | syl 14 | . . . . 5 |
20 | 1, 9, 10, 19 | fvmptd3 5514 | . . . 4 PsMet |
21 | 20 | eleq2d 2209 | . . 3 PsMet |
22 | oveq 5780 | . . . . . . 7 | |
23 | 22 | eqeq1d 2148 | . . . . . 6 |
24 | oveq 5780 | . . . . . . . 8 | |
25 | oveq 5780 | . . . . . . . . 9 | |
26 | oveq 5780 | . . . . . . . . 9 | |
27 | 25, 26 | oveq12d 5792 | . . . . . . . 8 |
28 | 24, 27 | breq12d 3942 | . . . . . . 7 |
29 | 28 | 2ralbidv 2459 | . . . . . 6 |
30 | 23, 29 | anbi12d 464 | . . . . 5 |
31 | 30 | ralbidv 2437 | . . . 4 |
32 | 31 | elrab 2840 | . . 3 |
33 | 21, 32 | syl6bb 195 | . 2 PsMet |
34 | elmapg 6555 | . . . 4 | |
35 | 11, 12, 34 | sylancr 410 | . . 3 |
36 | 35 | anbi1d 460 | . 2 |
37 | 33, 36 | bitrd 187 | 1 PsMet |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 cab 2125 wral 2416 crab 2420 cvv 2686 class class class wbr 3929 cxp 4537 wf 5119 cfv 5123 (class class class)co 5774 cmap 6542 cc0 7620 cxr 7799 cle 7801 cxad 9557 PsMetcpsmet 12148 |
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-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-map 6544 df-pnf 7802 df-mnf 7803 df-xr 7804 df-psmet 12156 |
This theorem is referenced by: psmetdmdm 12493 psmetf 12494 psmet0 12496 psmettri2 12497 psmetres2 12502 xmetpsmet 12538 |
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