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| Mirrors > Home > MPE Home > Th. List > caov4 | 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)) |
| caov.4 | ⊢ 𝐷 ∈ V |
| Ref | Expression |
|---|---|
| caov4 | ⊢ ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | caov.2 | . . . 4 ⊢ 𝐵 ∈ V | |
| 2 | caov.3 | . . . 4 ⊢ 𝐶 ∈ V | |
| 3 | caov.4 | . . . 4 ⊢ 𝐷 ∈ V | |
| 4 | caov.com | . . . 4 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) | |
| 5 | caov.ass | . . . 4 ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)) | |
| 6 | 1, 2, 3, 4, 5 | caov12 7661 | . . 3 ⊢ (𝐵𝐹(𝐶𝐹𝐷)) = (𝐶𝐹(𝐵𝐹𝐷)) |
| 7 | 6 | oveq2i 7442 | . 2 ⊢ (𝐴𝐹(𝐵𝐹(𝐶𝐹𝐷))) = (𝐴𝐹(𝐶𝐹(𝐵𝐹𝐷))) |
| 8 | caov.1 | . . 3 ⊢ 𝐴 ∈ V | |
| 9 | ovex 7464 | . . 3 ⊢ (𝐶𝐹𝐷) ∈ V | |
| 10 | 8, 1, 9, 5 | caovass 7633 | . 2 ⊢ ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = (𝐴𝐹(𝐵𝐹(𝐶𝐹𝐷))) |
| 11 | ovex 7464 | . . 3 ⊢ (𝐵𝐹𝐷) ∈ V | |
| 12 | 8, 2, 11, 5 | caovass 7633 | . 2 ⊢ ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)) = (𝐴𝐹(𝐶𝐹(𝐵𝐹𝐷))) |
| 13 | 7, 10, 12 | 3eqtr4i 2775 | 1 ⊢ ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 (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 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: caov42 7666 ecopovtrn 8860 adderpqlem 10994 mulerpqlem 10995 ltmnq 11012 reclem3pr 11089 mulcmpblnrlem 11110 distrsr 11131 ltasr 11140 mulgt0sr 11145 axdistr 11198 |
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