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Mirrors > Home > MPE Home > Th. List > caov32 | Structured version Visualization version GIF version |
Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) |
Ref | Expression |
---|---|
caov.1 | ⊢ 𝐴 ∈ V |
caov.2 | ⊢ 𝐵 ∈ V |
caov.3 | ⊢ 𝐶 ∈ V |
caov.com | ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) |
caov.ass | ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)) |
Ref | Expression |
---|---|
caov32 | ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caov.2 | . . . 4 ⊢ 𝐵 ∈ V | |
2 | caov.3 | . . . 4 ⊢ 𝐶 ∈ V | |
3 | caov.com | . . . 4 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) | |
4 | 1, 2, 3 | caovcom 7630 | . . 3 ⊢ (𝐵𝐹𝐶) = (𝐶𝐹𝐵) |
5 | 4 | oveq2i 7442 | . 2 ⊢ (𝐴𝐹(𝐵𝐹𝐶)) = (𝐴𝐹(𝐶𝐹𝐵)) |
6 | caov.1 | . . 3 ⊢ 𝐴 ∈ V | |
7 | caov.ass | . . 3 ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)) | |
8 | 6, 1, 2, 7 | caovass 7633 | . 2 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)) |
9 | 6, 2, 1, 7 | caovass 7633 | . 2 ⊢ ((𝐴𝐹𝐶)𝐹𝐵) = (𝐴𝐹(𝐶𝐹𝐵)) |
10 | 5, 8, 9 | 3eqtr4i 2773 | 1 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 Vcvv 3478 (class class class)co 7431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 |
This theorem is referenced by: caov31 7662 addassnq 10996 ltexprlem7 11080 mulcmpblnrlem 11108 recexsrlem 11141 mulgt0sr 11143 |
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