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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 5952 | . . 3 ⊢ (𝜑 → (𝐵𝐹(𝐶𝐹𝐷)) = (𝐶𝐹(𝐵𝐹𝐷))) |
7 | 6 | oveq2d 5790 | . 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 5924 | . . 3 ⊢ (𝜑 → (𝐶𝐹𝐷) ∈ 𝑆) |
11 | 5, 8, 1, 10 | caovassd 5930 | . 2 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = (𝐴𝐹(𝐵𝐹(𝐶𝐹𝐷)))) |
12 | 9, 1, 3 | caovcld 5924 | . . 3 ⊢ (𝜑 → (𝐵𝐹𝐷) ∈ 𝑆) |
13 | 5, 8, 2, 12 | caovassd 5930 | . 2 ⊢ (𝜑 → ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)) = (𝐴𝐹(𝐶𝐹(𝐵𝐹𝐷)))) |
14 | 7, 11, 13 | 3eqtr4d 2182 | 1 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷))) |
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: caov411d 5956 caov42d 5957 ecopovtrn 6526 ecopovtrng 6529 addcmpblnq 7175 mulcmpblnq 7176 ordpipqqs 7182 distrnqg 7195 ltsonq 7206 ltanqg 7208 ltmnqg 7209 addcmpblnq0 7251 mulcmpblnq0 7252 distrnq0 7267 prarloclemlo 7302 addlocprlemeqgt 7340 addcanprleml 7422 recexprlem1ssl 7441 recexprlem1ssu 7442 mulcmpblnrlemg 7548 distrsrg 7567 ltasrg 7578 mulgt0sr 7586 prsradd 7594 axdistr 7682 |
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