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Mirrors > Home > MPE Home > Th. List > caov31 | 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 |
---|---|
caov31 | ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caov.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | caov.3 | . . . 4 ⊢ 𝐶 ∈ V | |
3 | caov.2 | . . . 4 ⊢ 𝐵 ∈ V | |
4 | caov.ass | . . . 4 ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)) | |
5 | 1, 2, 3, 4 | caovass 7386 | . . 3 ⊢ ((𝐴𝐹𝐶)𝐹𝐵) = (𝐴𝐹(𝐶𝐹𝐵)) |
6 | caov.com | . . . 4 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) | |
7 | 1, 2, 3, 6, 4 | caov12 7414 | . . 3 ⊢ (𝐴𝐹(𝐶𝐹𝐵)) = (𝐶𝐹(𝐴𝐹𝐵)) |
8 | 5, 7 | eqtri 2759 | . 2 ⊢ ((𝐴𝐹𝐶)𝐹𝐵) = (𝐶𝐹(𝐴𝐹𝐵)) |
9 | 1, 3, 2, 6, 4 | caov32 7413 | . 2 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵) |
10 | 2, 1, 3, 6, 4 | caov32 7413 | . . 3 ⊢ ((𝐶𝐹𝐴)𝐹𝐵) = ((𝐶𝐹𝐵)𝐹𝐴) |
11 | 2, 1, 3, 4 | caovass 7386 | . . 3 ⊢ ((𝐶𝐹𝐴)𝐹𝐵) = (𝐶𝐹(𝐴𝐹𝐵)) |
12 | 10, 11 | eqtr3i 2761 | . 2 ⊢ ((𝐶𝐹𝐵)𝐹𝐴) = (𝐶𝐹(𝐴𝐹𝐵)) |
13 | 8, 9, 12 | 3eqtr4i 2769 | 1 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2112 Vcvv 3398 (class class class)co 7191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-iota 6316 df-fv 6366 df-ov 7194 |
This theorem is referenced by: caov13 7416 caov411 7418 |
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