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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 4293 | . . . . . . 7 | |
2 | iuneq1 3826 | . . . . . . 7 | |
3 | 1, 2 | ax-mp 5 | . . . . . 6 |
4 | iunxun 3892 | . . . . . 6 | |
5 | 3, 4 | eqtri 2160 | . . . . 5 |
6 | oveq2 5782 | . . . . . . . 8 | |
7 | 6 | oveq1d 5789 | . . . . . . 7 |
8 | 7 | iunxsng 3888 | . . . . . 6 |
9 | 8 | uneq2d 3230 | . . . . 5 |
10 | 5, 9 | syl5eq 2184 | . . . 4 |
11 | 10 | adantl 275 | . . 3 |
12 | suceloni 4417 | . . . 4 | |
13 | omv2 6361 | . . . 4 | |
14 | 12, 13 | sylan2 284 | . . 3 |
15 | omv2 6361 | . . . 4 | |
16 | 15 | uneq1d 3229 | . . 3 |
17 | 11, 14, 16 | 3eqtr4d 2182 | . 2 |
18 | omcl 6357 | . . 3 | |
19 | simpl 108 | . . 3 | |
20 | oaword1 6367 | . . . 4 | |
21 | ssequn1 3246 | . . . 4 | |
22 | 20, 21 | sylib 121 | . . 3 |
23 | 18, 19, 22 | syl2anc 408 | . 2 |
24 | 17, 23 | eqtrd 2172 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cun 3069 wss 3071 csn 3527 ciun 3813 con0 4285 csuc 4287 (class class class)co 5774 coa 6310 comu 6311 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-oadd 6317 df-omul 6318 |
This theorem is referenced by: onmsuc 6369 |
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