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Mirrors > Home > MPE Home > Th. List > caov12 | Structured version Visualization version Unicode version |
Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) |
Ref | Expression |
---|---|
caov.1 | |
caov.2 | |
caov.3 | |
caov.com | |
caov.ass |
Ref | Expression |
---|---|
caov12 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caov.1 | . . . 4 | |
2 | caov.2 | . . . 4 | |
3 | caov.com | . . . 4 | |
4 | 1, 2, 3 | caovcom 6831 | . . 3 |
5 | 4 | oveq1i 6660 | . 2 |
6 | caov.3 | . . 3 | |
7 | caov.ass | . . 3 | |
8 | 1, 2, 6, 7 | caovass 6834 | . 2 |
9 | 2, 1, 6, 7 | caovass 6834 | . 2 |
10 | 5, 8, 9 | 3eqtr3i 2652 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wceq 1483 wcel 1990 cvv 3200 (class class class)co 6650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 |
This theorem is referenced by: caov31 6863 caov4 6865 caovmo 6871 distrnq 9783 ltaddnq 9796 ltexnq 9797 1idpr 9851 prlem934 9855 prlem936 9869 mulcmpblnrlem 9891 ltsosr 9915 0idsr 9918 1idsr 9919 recexsrlem 9924 mulgt0sr 9926 axmulass 9978 |
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