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Mirrors > Home > ILE Home > Th. List > caov12d | 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 |
---|---|
caov12d | ⊢ (𝜑 → (𝐴𝐹(𝐵𝐹𝐶)) = (𝐵𝐹(𝐴𝐹𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovd.com | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) | |
2 | caovd.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
3 | caovd.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
4 | 1, 2, 3 | caovcomd 5793 | . . 3 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)) |
5 | 4 | oveq1d 5659 | . 2 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐵𝐹𝐴)𝐹𝐶)) |
6 | caovd.ass | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) | |
7 | caovd.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑆) | |
8 | 6, 2, 3, 7 | caovassd 5796 | . 2 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) |
9 | 6, 3, 2, 7 | caovassd 5796 | . 2 ⊢ (𝜑 → ((𝐵𝐹𝐴)𝐹𝐶) = (𝐵𝐹(𝐴𝐹𝐶))) |
10 | 5, 8, 9 | 3eqtr3d 2128 | 1 ⊢ (𝜑 → (𝐴𝐹(𝐵𝐹𝐶)) = (𝐵𝐹(𝐴𝐹𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∧ w3a 924 = wceq 1289 ∈ wcel 1438 (class class class)co 5644 |
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 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-un 3003 df-sn 3450 df-pr 3451 df-op 3453 df-uni 3652 df-br 3844 df-iota 4975 df-fv 5018 df-ov 5647 |
This theorem is referenced by: caov4d 5821 caovimo 5830 ltaddnq 6956 ltexnqq 6957 enq0tr 6983 mullocprlem 7119 1idprl 7139 1idpru 7140 cauappcvgprlemdisj 7200 mulcmpblnrlemg 7276 lttrsr 7298 ltsosr 7300 0idsr 7303 1idsr 7304 recexgt0sr 7309 mulgt0sr 7313 axmulass 7398 |
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