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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 7115 | . 2 | |
2 | recn 7106 | . . . . . 6 | |
3 | ax-icn 7071 | . . . . . . 7 | |
4 | recn 7106 | . . . . . . 7 | |
5 | mulcl 7100 | . . . . . . 7 | |
6 | 3, 4, 5 | sylancr 405 | . . . . . 6 |
7 | ax-1cn 7069 | . . . . . . 7 | |
8 | adddir 7110 | . . . . . . 7 | |
9 | 7, 8 | mp3an3 1257 | . . . . . 6 |
10 | 2, 6, 9 | syl2an 283 | . . . . 5 |
11 | ax-1rid 7083 | . . . . . 6 | |
12 | mulass 7104 | . . . . . . . . 9 | |
13 | 3, 7, 12 | mp3an13 1259 | . . . . . . . 8 |
14 | 4, 13 | syl 14 | . . . . . . 7 |
15 | ax-1rid 7083 | . . . . . . . 8 | |
16 | 15 | oveq2d 5548 | . . . . . . 7 |
17 | 14, 16 | eqtrd 2113 | . . . . . 6 |
18 | 11, 17 | oveqan12d 5551 | . . . . 5 |
19 | 10, 18 | eqtrd 2113 | . . . 4 |
20 | oveq1 5539 | . . . . 5 | |
21 | id 19 | . . . . 5 | |
22 | 20, 21 | eqeq12d 2095 | . . . 4 |
23 | 19, 22 | syl5ibrcom 155 | . . 3 |
24 | 23 | rexlimivv 2482 | . 2 |
25 | 1, 24 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 wrex 2349 (class class class)co 5532 cc 6979 cr 6980 c1 6982 ci 6983 caddc 6984 cmul 6986 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-resscn 7068 ax-1cn 7069 ax-icn 7071 ax-addcl 7072 ax-mulcl 7074 ax-mulcom 7077 ax-mulass 7079 ax-distr 7080 ax-1rid 7083 ax-cnre 7087 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-iota 4887 df-fv 4930 df-ov 5535 |
This theorem is referenced by: mulid2 7117 mulid1i 7121 mulid1d 7136 muleqadd 7758 divdivap1 7811 conjmulap 7817 nnmulcl 8060 expmul 9521 binom21 9586 bernneq 9593 |
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