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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 7023 | . 2 | |
2 | recn 7014 | . . . . . 6 | |
3 | ax-icn 6979 | . . . . . . 7 | |
4 | recn 7014 | . . . . . . 7 | |
5 | mulcl 7008 | . . . . . . 7 | |
6 | 3, 4, 5 | sylancr 393 | . . . . . 6 |
7 | ax-1cn 6977 | . . . . . . 7 | |
8 | adddir 7018 | . . . . . . 7 | |
9 | 7, 8 | mp3an3 1221 | . . . . . 6 |
10 | 2, 6, 9 | syl2an 273 | . . . . 5 |
11 | ax-1rid 6991 | . . . . . 6 | |
12 | mulass 7012 | . . . . . . . . 9 | |
13 | 3, 7, 12 | mp3an13 1223 | . . . . . . . 8 |
14 | 4, 13 | syl 14 | . . . . . . 7 |
15 | ax-1rid 6991 | . . . . . . . 8 | |
16 | 15 | oveq2d 5528 | . . . . . . 7 |
17 | 14, 16 | eqtrd 2072 | . . . . . 6 |
18 | 11, 17 | oveqan12d 5531 | . . . . 5 |
19 | 10, 18 | eqtrd 2072 | . . . 4 |
20 | oveq1 5519 | . . . . 5 | |
21 | id 19 | . . . . 5 | |
22 | 20, 21 | eqeq12d 2054 | . . . 4 |
23 | 19, 22 | syl5ibrcom 146 | . . 3 |
24 | 23 | rexlimivv 2438 | . 2 |
25 | 1, 24 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wcel 1393 wrex 2307 (class class class)co 5512 cc 6887 cr 6888 c1 6890 ci 6891 caddc 6892 cmul 6894 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-resscn 6976 ax-1cn 6977 ax-icn 6979 ax-addcl 6980 ax-mulcl 6982 ax-mulcom 6985 ax-mulass 6987 ax-distr 6988 ax-1rid 6991 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-iota 4867 df-fv 4910 df-ov 5515 |
This theorem is referenced by: mulid2 7025 mulid1i 7029 mulid1d 7044 muleqadd 7649 divdivap1 7699 conjmulap 7705 nnmulcl 7935 expmul 9300 binom21 9363 bernneq 9369 |
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