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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 6082 | . . 3 ⊢ (𝜑 → (𝐵𝐹(𝐶𝐹𝐷)) = (𝐶𝐹(𝐵𝐹𝐷))) |
7 | 6 | oveq2d 5916 | . 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 6054 | . . 3 ⊢ (𝜑 → (𝐶𝐹𝐷) ∈ 𝑆) |
11 | 5, 8, 1, 10 | caovassd 6060 | . 2 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = (𝐴𝐹(𝐵𝐹(𝐶𝐹𝐷)))) |
12 | 9, 1, 3 | caovcld 6054 | . . 3 ⊢ (𝜑 → (𝐵𝐹𝐷) ∈ 𝑆) |
13 | 5, 8, 2, 12 | caovassd 6060 | . 2 ⊢ (𝜑 → ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)) = (𝐴𝐹(𝐶𝐹(𝐵𝐹𝐷)))) |
14 | 7, 11, 13 | 3eqtr4d 2232 | 1 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 (class class class)co 5900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-br 4022 df-iota 5199 df-fv 5246 df-ov 5903 |
This theorem is referenced by: caov411d 6086 caov42d 6087 ecopovtrn 6662 ecopovtrng 6665 addcmpblnq 7401 mulcmpblnq 7402 ordpipqqs 7408 distrnqg 7421 ltsonq 7432 ltanqg 7434 ltmnqg 7435 addcmpblnq0 7477 mulcmpblnq0 7478 distrnq0 7493 prarloclemlo 7528 addlocprlemeqgt 7566 addcanprleml 7648 recexprlem1ssl 7667 recexprlem1ssu 7668 mulcmpblnrlemg 7774 distrsrg 7793 ltasrg 7804 mulgt0sr 7812 prsradd 7820 axdistr 7908 |
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