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Mirrors > Home > ILE Home > Th. List > caov411d | 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))) |
caovd.4 | ⊢ (𝜑 → 𝐷 ∈ 𝑆) |
caovd.cl | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) |
Ref | Expression |
---|---|
caov411d | ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
2 | caovd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
3 | caovd.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑆) | |
4 | caovd.com | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) | |
5 | caovd.ass | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) | |
6 | caovd.4 | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑆) | |
7 | caovd.cl | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) | |
8 | 1, 2, 3, 4, 5, 6, 7 | caov4d 5763 | . 2 ⊢ (𝜑 → ((𝐵𝐹𝐴)𝐹(𝐶𝐹𝐷)) = ((𝐵𝐹𝐶)𝐹(𝐴𝐹𝐷))) |
9 | 4, 1, 2 | caovcomd 5735 | . . 3 ⊢ (𝜑 → (𝐵𝐹𝐴) = (𝐴𝐹𝐵)) |
10 | 9 | oveq1d 5605 | . 2 ⊢ (𝜑 → ((𝐵𝐹𝐴)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷))) |
11 | 4, 1, 3 | caovcomd 5735 | . . 3 ⊢ (𝜑 → (𝐵𝐹𝐶) = (𝐶𝐹𝐵)) |
12 | 11 | oveq1d 5605 | . 2 ⊢ (𝜑 → ((𝐵𝐹𝐶)𝐹(𝐴𝐹𝐷)) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷))) |
13 | 8, 10, 12 | 3eqtr3d 2123 | 1 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∧ w3a 920 = wceq 1285 ∈ wcel 1434 (class class class)co 5590 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2614 df-un 2988 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-iota 4933 df-fv 4976 df-ov 5593 |
This theorem is referenced by: ecopovtrn 6318 ecopovtrng 6321 ltsonq 6859 ltanqg 6861 mulextsr1lem 7227 |
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