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Mirrors > Home > ILE Home > Th. List > caov31d | GIF version |
Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.) |
Ref | Expression |
---|---|
caovd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
caovd.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑆) |
caovd.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑆) |
caovd.com | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) |
caovd.ass | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) |
Ref | Expression |
---|---|
caov31d | ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovd.com | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) | |
2 | caovd.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
3 | caovd.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑆) | |
4 | 1, 2, 3 | caovcomd 5927 | . . 3 ⊢ (𝜑 → (𝐴𝐹𝐶) = (𝐶𝐹𝐴)) |
5 | 4 | oveq1d 5789 | . 2 ⊢ (𝜑 → ((𝐴𝐹𝐶)𝐹𝐵) = ((𝐶𝐹𝐴)𝐹𝐵)) |
6 | caovd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
7 | caovd.ass | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) | |
8 | 2, 6, 3, 1, 7 | caov32d 5951 | . 2 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵)) |
9 | 3, 6, 2, 1, 7 | caov32d 5951 | . 2 ⊢ (𝜑 → ((𝐶𝐹𝐵)𝐹𝐴) = ((𝐶𝐹𝐴)𝐹𝐵)) |
10 | 5, 8, 9 | 3eqtr4d 2182 | 1 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 (class class class)co 5774 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-iota 5088 df-fv 5131 df-ov 5777 |
This theorem is referenced by: caov13d 5954 |
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