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Mirrors > Home > ILE Home > Th. List > omsuc | Unicode version |
Description: Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
omsuc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-suc 4108 | . . . . . . 7 | |
2 | iuneq1 3670 | . . . . . . 7 | |
3 | 1, 2 | ax-mp 7 | . . . . . 6 |
4 | iunxun 3735 | . . . . . 6 | |
5 | 3, 4 | eqtri 2060 | . . . . 5 |
6 | oveq2 5520 | . . . . . . . 8 | |
7 | 6 | oveq1d 5527 | . . . . . . 7 |
8 | 7 | iunxsng 3732 | . . . . . 6 |
9 | 8 | uneq2d 3097 | . . . . 5 |
10 | 5, 9 | syl5eq 2084 | . . . 4 |
11 | 10 | adantl 262 | . . 3 |
12 | suceloni 4227 | . . . 4 | |
13 | omv2 6045 | . . . 4 | |
14 | 12, 13 | sylan2 270 | . . 3 |
15 | omv2 6045 | . . . 4 | |
16 | 15 | uneq1d 3096 | . . 3 |
17 | 11, 14, 16 | 3eqtr4d 2082 | . 2 |
18 | omcl 6041 | . . 3 | |
19 | simpl 102 | . . 3 | |
20 | oaword1 6050 | . . . 4 | |
21 | ssequn1 3113 | . . . 4 | |
22 | 20, 21 | sylib 127 | . . 3 |
23 | 18, 19, 22 | syl2anc 391 | . 2 |
24 | 17, 23 | eqtrd 2072 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wcel 1393 cun 2915 wss 2917 csn 3375 ciun 3657 con0 4100 csuc 4102 (class class class)co 5512 coa 5998 comu 5999 |
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-in1 544 ax-in2 545 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-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-oadd 6005 df-omul 6006 |
This theorem is referenced by: onmsuc 6052 |
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