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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 |
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caovd.2 |
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caovd.3 |
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caovd.com |
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caovd.ass |
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Ref | Expression |
---|---|
caov12d |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovd.com |
. . . 4
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2 | caovd.1 |
. . . 4
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3 | caovd.2 |
. . . 4
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4 | 1, 2, 3 | caovcomd 5839 |
. . 3
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5 | 4 | oveq1d 5705 |
. 2
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6 | caovd.ass |
. . 3
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7 | caovd.3 |
. . 3
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8 | 6, 2, 3, 7 | caovassd 5842 |
. 2
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9 | 6, 3, 2, 7 | caovassd 5842 |
. 2
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10 | 5, 8, 9 | 3eqtr3d 2135 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-un 3017 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-iota 5014 df-fv 5057 df-ov 5693 |
This theorem is referenced by: caov4d 5867 caovimo 5876 ltaddnq 7063 ltexnqq 7064 enq0tr 7090 mullocprlem 7226 1idprl 7246 1idpru 7247 cauappcvgprlemdisj 7307 mulcmpblnrlemg 7383 lttrsr 7405 ltsosr 7407 0idsr 7410 1idsr 7411 recexgt0sr 7416 mulgt0sr 7420 axmulass 7505 |
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