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Mirrors > Home > ILE Home > Th. List > caov31 | 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 6013 | . . 3 ⊢ ((𝐴𝐹𝐶)𝐹𝐵) = (𝐴𝐹(𝐶𝐹𝐵)) |
6 | caov.com | . . . 4 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) | |
7 | 1, 2, 3, 6, 4 | caov12 6041 | . . 3 ⊢ (𝐴𝐹(𝐶𝐹𝐵)) = (𝐶𝐹(𝐴𝐹𝐵)) |
8 | 5, 7 | eqtri 2191 | . 2 ⊢ ((𝐴𝐹𝐶)𝐹𝐵) = (𝐶𝐹(𝐴𝐹𝐵)) |
9 | 1, 3, 2, 6, 4 | caov32 6040 | . 2 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵) |
10 | 2, 1, 3, 6, 4 | caov32 6040 | . . 3 ⊢ ((𝐶𝐹𝐴)𝐹𝐵) = ((𝐶𝐹𝐵)𝐹𝐴) |
11 | 2, 1, 3, 4 | caovass 6013 | . . 3 ⊢ ((𝐶𝐹𝐴)𝐹𝐵) = (𝐶𝐹(𝐴𝐹𝐵)) |
12 | 10, 11 | eqtr3i 2193 | . 2 ⊢ ((𝐶𝐹𝐵)𝐹𝐴) = (𝐶𝐹(𝐴𝐹𝐵)) |
13 | 8, 9, 12 | 3eqtr4i 2201 | 1 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴) |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∈ wcel 2141 Vcvv 2730 (class class class)co 5853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-iota 5160 df-fv 5206 df-ov 5856 |
This theorem is referenced by: caov13 6043 |
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