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Mirrors > Home > ILE Home > Th. List > mulid1 | Unicode version |
Description: is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) |
Ref | Expression |
---|---|
mulid1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnre 7762 | . 2 | |
2 | recn 7753 | . . . . . 6 | |
3 | ax-icn 7715 | . . . . . . 7 | |
4 | recn 7753 | . . . . . . 7 | |
5 | mulcl 7747 | . . . . . . 7 | |
6 | 3, 4, 5 | sylancr 410 | . . . . . 6 |
7 | ax-1cn 7713 | . . . . . . 7 | |
8 | adddir 7757 | . . . . . . 7 | |
9 | 7, 8 | mp3an3 1304 | . . . . . 6 |
10 | 2, 6, 9 | syl2an 287 | . . . . 5 |
11 | ax-1rid 7727 | . . . . . 6 | |
12 | mulass 7751 | . . . . . . . . 9 | |
13 | 3, 7, 12 | mp3an13 1306 | . . . . . . . 8 |
14 | 4, 13 | syl 14 | . . . . . . 7 |
15 | ax-1rid 7727 | . . . . . . . 8 | |
16 | 15 | oveq2d 5790 | . . . . . . 7 |
17 | 14, 16 | eqtrd 2172 | . . . . . 6 |
18 | 11, 17 | oveqan12d 5793 | . . . . 5 |
19 | 10, 18 | eqtrd 2172 | . . . 4 |
20 | oveq1 5781 | . . . . 5 | |
21 | id 19 | . . . . 5 | |
22 | 20, 21 | eqeq12d 2154 | . . . 4 |
23 | 19, 22 | syl5ibrcom 156 | . . 3 |
24 | 23 | rexlimivv 2555 | . 2 |
25 | 1, 24 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wrex 2417 (class class class)co 5774 cc 7618 cr 7619 c1 7621 ci 7622 caddc 7623 cmul 7625 |
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 7712 ax-1cn 7713 ax-icn 7715 ax-addcl 7716 ax-mulcl 7718 ax-mulcom 7721 ax-mulass 7723 ax-distr 7724 ax-1rid 7727 ax-cnre 7731 |
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: mulid2 7764 mulid1i 7768 mulid1d 7783 muleqadd 8429 divdivap1 8483 conjmulap 8489 nnmulcl 8741 expmul 10338 binom21 10404 binom2sub1 10406 bernneq 10412 hashiun 11247 fproddccvg 11341 prodmodclem2a 11345 efexp 11388 |
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