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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 7916 | . 2 | |
2 | recn 7907 | . . . . . 6 | |
3 | ax-icn 7869 | . . . . . . 7 | |
4 | recn 7907 | . . . . . . 7 | |
5 | mulcl 7901 | . . . . . . 7 | |
6 | 3, 4, 5 | sylancr 412 | . . . . . 6 |
7 | ax-1cn 7867 | . . . . . . 7 | |
8 | adddir 7911 | . . . . . . 7 | |
9 | 7, 8 | mp3an3 1321 | . . . . . 6 |
10 | 2, 6, 9 | syl2an 287 | . . . . 5 |
11 | ax-1rid 7881 | . . . . . 6 | |
12 | mulass 7905 | . . . . . . . . 9 | |
13 | 3, 7, 12 | mp3an13 1323 | . . . . . . . 8 |
14 | 4, 13 | syl 14 | . . . . . . 7 |
15 | ax-1rid 7881 | . . . . . . . 8 | |
16 | 15 | oveq2d 5869 | . . . . . . 7 |
17 | 14, 16 | eqtrd 2203 | . . . . . 6 |
18 | 11, 17 | oveqan12d 5872 | . . . . 5 |
19 | 10, 18 | eqtrd 2203 | . . . 4 |
20 | oveq1 5860 | . . . . 5 | |
21 | id 19 | . . . . 5 | |
22 | 20, 21 | eqeq12d 2185 | . . . 4 |
23 | 19, 22 | syl5ibrcom 156 | . . 3 |
24 | 23 | rexlimivv 2593 | . 2 |
25 | 1, 24 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 wrex 2449 (class class class)co 5853 cc 7772 cr 7773 c1 7775 ci 7776 caddc 7777 cmul 7779 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-resscn 7866 ax-1cn 7867 ax-icn 7869 ax-addcl 7870 ax-mulcl 7872 ax-mulcom 7875 ax-mulass 7877 ax-distr 7878 ax-1rid 7881 ax-cnre 7885 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-iota 5160 df-fv 5206 df-ov 5856 |
This theorem is referenced by: mulid2 7918 mulid1i 7922 mulid1d 7937 muleqadd 8586 divdivap1 8640 conjmulap 8646 nnmulcl 8899 expmul 10521 binom21 10588 binom2sub1 10590 bernneq 10596 hashiun 11441 fproddccvg 11535 prodmodclem2a 11539 efexp 11645 ecxp 13616 lgsdilem2 13731 |
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