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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 4263 | . . . . . . 7 | |
2 | iuneq1 3796 | . . . . . . 7 | |
3 | 1, 2 | ax-mp 5 | . . . . . 6 |
4 | iunxun 3862 | . . . . . 6 | |
5 | 3, 4 | eqtri 2138 | . . . . 5 |
6 | oveq2 5750 | . . . . . . . 8 | |
7 | 6 | oveq1d 5757 | . . . . . . 7 |
8 | 7 | iunxsng 3858 | . . . . . 6 |
9 | 8 | uneq2d 3200 | . . . . 5 |
10 | 5, 9 | syl5eq 2162 | . . . 4 |
11 | 10 | adantl 275 | . . 3 |
12 | suceloni 4387 | . . . 4 | |
13 | omv2 6329 | . . . 4 | |
14 | 12, 13 | sylan2 284 | . . 3 |
15 | omv2 6329 | . . . 4 | |
16 | 15 | uneq1d 3199 | . . 3 |
17 | 11, 14, 16 | 3eqtr4d 2160 | . 2 |
18 | omcl 6325 | . . 3 | |
19 | simpl 108 | . . 3 | |
20 | oaword1 6335 | . . . 4 | |
21 | ssequn1 3216 | . . . 4 | |
22 | 20, 21 | sylib 121 | . . 3 |
23 | 18, 19, 22 | syl2anc 408 | . 2 |
24 | 17, 23 | eqtrd 2150 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 cun 3039 wss 3041 csn 3497 ciun 3783 con0 4255 csuc 4257 (class class class)co 5742 coa 6278 comu 6279 |
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-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-oadd 6285 df-omul 6286 |
This theorem is referenced by: onmsuc 6337 |
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