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Mirrors > Home > ILE Home > Th. List > mulid2 | Unicode version |
Description: Identity law for multiplication. Note: see mulid1 7770 for commuted version. (Contributed by NM, 8-Oct-1999.) |
Ref | Expression |
---|---|
mulid2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 7720 | . . 3 | |
2 | mulcom 7756 | . . 3 | |
3 | 1, 2 | mpan 420 | . 2 |
4 | mulid1 7770 | . 2 | |
5 | 3, 4 | eqtrd 2172 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1331 wcel 1480 (class class class)co 5774 cc 7625 c1 7628 cmul 7632 |
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 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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-resscn 7719 ax-1cn 7720 ax-icn 7722 ax-addcl 7723 ax-mulcl 7725 ax-mulcom 7728 ax-mulass 7730 ax-distr 7731 ax-1rid 7734 ax-cnre 7738 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-iota 5088 df-fv 5131 df-ov 5777 |
This theorem is referenced by: mulid2i 7776 mulid2d 7791 muladd11 7902 1p1times 7903 mulm1 8169 div1 8470 recdivap 8485 divdivap2 8491 conjmulap 8496 expp1 10307 recan 10888 arisum 11274 geo2sum 11290 prodrbdclem 11347 prodmodclem2a 11352 demoivreALT 11487 gcdadd 11680 gcdid 11681 |
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