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Mirrors > Home > ILE Home > Th. List > caov12d | Unicode version |
Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.) |
Ref | Expression |
---|---|
caovd.1 | |
caovd.2 | |
caovd.3 | |
caovd.com | |
caovd.ass |
Ref | Expression |
---|---|
caov12d |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovd.com | . . . 4 | |
2 | caovd.1 | . . . 4 | |
3 | caovd.2 | . . . 4 | |
4 | 1, 2, 3 | caovcomd 5657 | . . 3 |
5 | 4 | oveq1d 5527 | . 2 |
6 | caovd.ass | . . 3 | |
7 | caovd.3 | . . 3 | |
8 | 6, 2, 3, 7 | caovassd 5660 | . 2 |
9 | 6, 3, 2, 7 | caovassd 5660 | . 2 |
10 | 5, 8, 9 | 3eqtr3d 2080 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 w3a 885 wceq 1243 wcel 1393 (class class class)co 5512 |
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 |
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-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: caov4d 5685 caovimo 5694 ltaddnq 6505 ltexnqq 6506 enq0tr 6532 mullocprlem 6668 1idprl 6688 1idpru 6689 cauappcvgprlemdisj 6749 mulcmpblnrlemg 6825 lttrsr 6847 ltsosr 6849 0idsr 6852 1idsr 6853 recexgt0sr 6858 mulgt0sr 6862 axmulass 6947 |
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