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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 5895 | . . 3 |
5 | 4 | oveq1d 5757 | . 2 |
6 | caovd.ass | . . 3 | |
7 | caovd.3 | . . 3 | |
8 | 6, 2, 3, 7 | caovassd 5898 | . 2 |
9 | 6, 3, 2, 7 | caovassd 5898 | . 2 |
10 | 5, 8, 9 | 3eqtr3d 2158 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 947 wceq 1316 wcel 1465 (class class class)co 5742 |
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-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-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-iota 5058 df-fv 5101 df-ov 5745 |
This theorem is referenced by: caov4d 5923 caovimo 5932 ltaddnq 7183 ltexnqq 7184 enq0tr 7210 mullocprlem 7346 1idprl 7366 1idpru 7367 cauappcvgprlemdisj 7427 mulcmpblnrlemg 7516 lttrsr 7538 ltsosr 7540 0idsr 7543 1idsr 7544 recexgt0sr 7549 mulgt0sr 7554 axmulass 7649 |
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