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Mirrors > Home > ILE Home > Th. List > caov4d | 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 |
---|---|
caov4d | ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovd.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
2 | caovd.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑆) | |
3 | caovd.4 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑆) | |
4 | caovd.com | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) | |
5 | caovd.ass | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) | |
6 | 1, 2, 3, 4, 5 | caov12d 5864 | . . 3 ⊢ (𝜑 → (𝐵𝐹(𝐶𝐹𝐷)) = (𝐶𝐹(𝐵𝐹𝐷))) |
7 | 6 | oveq2d 5706 | . 2 ⊢ (𝜑 → (𝐴𝐹(𝐵𝐹(𝐶𝐹𝐷))) = (𝐴𝐹(𝐶𝐹(𝐵𝐹𝐷)))) |
8 | caovd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
9 | caovd.cl | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) | |
10 | 9, 2, 3 | caovcld 5836 | . . 3 ⊢ (𝜑 → (𝐶𝐹𝐷) ∈ 𝑆) |
11 | 5, 8, 1, 10 | caovassd 5842 | . 2 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = (𝐴𝐹(𝐵𝐹(𝐶𝐹𝐷)))) |
12 | 9, 1, 3 | caovcld 5836 | . . 3 ⊢ (𝜑 → (𝐵𝐹𝐷) ∈ 𝑆) |
13 | 5, 8, 2, 12 | caovassd 5842 | . 2 ⊢ (𝜑 → ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)) = (𝐴𝐹(𝐶𝐹(𝐵𝐹𝐷)))) |
14 | 7, 11, 13 | 3eqtr4d 2137 | 1 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 927 = wceq 1296 ∈ wcel 1445 (class class class)co 5690 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-un 3017 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-iota 5014 df-fv 5057 df-ov 5693 |
This theorem is referenced by: caov411d 5868 caov42d 5869 ecopovtrn 6429 ecopovtrng 6432 addcmpblnq 7023 mulcmpblnq 7024 ordpipqqs 7030 distrnqg 7043 ltsonq 7054 ltanqg 7056 ltmnqg 7057 addcmpblnq0 7099 mulcmpblnq0 7100 distrnq0 7115 prarloclemlo 7150 addlocprlemeqgt 7188 addcanprleml 7270 recexprlem1ssl 7289 recexprlem1ssu 7290 mulcmpblnrlemg 7383 distrsrg 7402 ltasrg 7413 mulgt0sr 7420 prsradd 7428 axdistr 7506 |
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