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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 7628 | . . 3 ⊢ (𝐵𝐹(𝐶𝐹𝐷)) = (𝐶𝐹(𝐵𝐹𝐷)) |
7 | 6 | oveq2i 7412 | . 2 ⊢ (𝐴𝐹(𝐵𝐹(𝐶𝐹𝐷))) = (𝐴𝐹(𝐶𝐹(𝐵𝐹𝐷))) |
8 | caov.1 | . . 3 ⊢ 𝐴 ∈ V | |
9 | ovex 7434 | . . 3 ⊢ (𝐶𝐹𝐷) ∈ V | |
10 | 8, 1, 9, 5 | caovass 7600 | . 2 ⊢ ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = (𝐴𝐹(𝐵𝐹(𝐶𝐹𝐷))) |
11 | ovex 7434 | . . 3 ⊢ (𝐵𝐹𝐷) ∈ V | |
12 | 8, 2, 11, 5 | caovass 7600 | . 2 ⊢ ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)) = (𝐴𝐹(𝐶𝐹(𝐵𝐹𝐷))) |
13 | 7, 10, 12 | 3eqtr4i 2762 | 1 ⊢ ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 Vcvv 3466 (class class class)co 7401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-nul 5296 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-iota 6485 df-fv 6541 df-ov 7404 |
This theorem is referenced by: caov42 7633 ecopovtrn 8810 adderpqlem 10945 mulerpqlem 10946 ltmnq 10963 reclem3pr 11040 mulcmpblnrlem 11061 distrsr 11082 ltasr 11091 mulgt0sr 11096 axdistr 11149 |
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