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Mirrors > Home > MPE Home > Th. List > caov411 | 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 |
---|---|
caov411 | ⊢ ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caov.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | caov.2 | . . . 4 ⊢ 𝐵 ∈ V | |
3 | caov.3 | . . . 4 ⊢ 𝐶 ∈ V | |
4 | caov.com | . . . 4 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) | |
5 | caov.ass | . . . 4 ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)) | |
6 | 1, 2, 3, 4, 5 | caov31 6761 | . . 3 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴) |
7 | 6 | oveq1i 6559 | . 2 ⊢ (((𝐴𝐹𝐵)𝐹𝐶)𝐹𝐷) = (((𝐶𝐹𝐵)𝐹𝐴)𝐹𝐷) |
8 | ovex 6577 | . . 3 ⊢ (𝐴𝐹𝐵) ∈ V | |
9 | caov.4 | . . 3 ⊢ 𝐷 ∈ V | |
10 | 8, 3, 9, 5 | caovass 6732 | . 2 ⊢ (((𝐴𝐹𝐵)𝐹𝐶)𝐹𝐷) = ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) |
11 | ovex 6577 | . . 3 ⊢ (𝐶𝐹𝐵) ∈ V | |
12 | 11, 1, 9, 5 | caovass 6732 | . 2 ⊢ (((𝐶𝐹𝐵)𝐹𝐴)𝐹𝐷) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷)) |
13 | 7, 10, 12 | 3eqtr3i 2640 | 1 ⊢ ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Vcvv 3173 (class class class)co 6549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 |
This theorem is referenced by: ecopovtrn 7737 distrnq 9662 lterpq 9671 ltanq 9672 prlem936 9748 |
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