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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 7874 | . 2 | |
2 | recn 7865 | . . . . . 6 | |
3 | ax-icn 7827 | . . . . . . 7 | |
4 | recn 7865 | . . . . . . 7 | |
5 | mulcl 7859 | . . . . . . 7 | |
6 | 3, 4, 5 | sylancr 411 | . . . . . 6 |
7 | ax-1cn 7825 | . . . . . . 7 | |
8 | adddir 7869 | . . . . . . 7 | |
9 | 7, 8 | mp3an3 1308 | . . . . . 6 |
10 | 2, 6, 9 | syl2an 287 | . . . . 5 |
11 | ax-1rid 7839 | . . . . . 6 | |
12 | mulass 7863 | . . . . . . . . 9 | |
13 | 3, 7, 12 | mp3an13 1310 | . . . . . . . 8 |
14 | 4, 13 | syl 14 | . . . . . . 7 |
15 | ax-1rid 7839 | . . . . . . . 8 | |
16 | 15 | oveq2d 5840 | . . . . . . 7 |
17 | 14, 16 | eqtrd 2190 | . . . . . 6 |
18 | 11, 17 | oveqan12d 5843 | . . . . 5 |
19 | 10, 18 | eqtrd 2190 | . . . 4 |
20 | oveq1 5831 | . . . . 5 | |
21 | id 19 | . . . . 5 | |
22 | 20, 21 | eqeq12d 2172 | . . . 4 |
23 | 19, 22 | syl5ibrcom 156 | . . 3 |
24 | 23 | rexlimivv 2580 | . 2 |
25 | 1, 24 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 wrex 2436 (class class class)co 5824 cc 7730 cr 7731 c1 7733 ci 7734 caddc 7735 cmul 7737 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 ax-resscn 7824 ax-1cn 7825 ax-icn 7827 ax-addcl 7828 ax-mulcl 7830 ax-mulcom 7833 ax-mulass 7835 ax-distr 7836 ax-1rid 7839 ax-cnre 7843 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-iota 5135 df-fv 5178 df-ov 5827 |
This theorem is referenced by: mulid2 7876 mulid1i 7880 mulid1d 7895 muleqadd 8542 divdivap1 8596 conjmulap 8602 nnmulcl 8854 expmul 10464 binom21 10530 binom2sub1 10532 bernneq 10538 hashiun 11375 fproddccvg 11469 prodmodclem2a 11473 efexp 11579 ecxp 13233 |
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