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| Mirrors > Home > MPE Home > Th. List > caovdir2d | Structured version Visualization version GIF version | ||
| Description: Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.) | 
| Ref | Expression | 
|---|---|
| caovdir2d.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧))) | 
| caovdir2d.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑆) | 
| caovdir2d.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑆) | 
| caovdir2d.4 | ⊢ (𝜑 → 𝐶 ∈ 𝑆) | 
| caovdir2d.cl | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) | 
| caovdir2d.com | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥)) | 
| Ref | Expression | 
|---|---|
| caovdir2d | ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | caovdir2d.1 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧))) | |
| 2 | caovdir2d.4 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑆) | |
| 3 | caovdir2d.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
| 4 | caovdir2d.3 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
| 5 | 1, 2, 3, 4 | caovdid 7649 | . 2 ⊢ (𝜑 → (𝐶𝐺(𝐴𝐹𝐵)) = ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵))) | 
| 6 | caovdir2d.com | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥)) | |
| 7 | caovdir2d.cl | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) | |
| 8 | 7, 3, 4 | caovcld 7627 | . . 3 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝑆) | 
| 9 | 6, 8, 2 | caovcomd 7630 | . 2 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = (𝐶𝐺(𝐴𝐹𝐵))) | 
| 10 | 6, 3, 2 | caovcomd 7630 | . . 3 ⊢ (𝜑 → (𝐴𝐺𝐶) = (𝐶𝐺𝐴)) | 
| 11 | 6, 4, 2 | caovcomd 7630 | . . 3 ⊢ (𝜑 → (𝐵𝐺𝐶) = (𝐶𝐺𝐵)) | 
| 12 | 10, 11 | oveq12d 7450 | . 2 ⊢ (𝜑 → ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)) = ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵))) | 
| 13 | 5, 9, 12 | 3eqtr4d 2786 | 1 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 (class class class)co 7432 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 | 
| This theorem is referenced by: (None) | 
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