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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 7619 | . 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 7597 | . . 3 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝑆) |
9 | 6, 8, 2 | caovcomd 7600 | . 2 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = (𝐶𝐺(𝐴𝐹𝐵))) |
10 | 6, 3, 2 | caovcomd 7600 | . . 3 ⊢ (𝜑 → (𝐴𝐺𝐶) = (𝐶𝐺𝐴)) |
11 | 6, 4, 2 | caovcomd 7600 | . . 3 ⊢ (𝜑 → (𝐵𝐺𝐶) = (𝐶𝐺𝐵)) |
12 | 10, 11 | oveq12d 7423 | . 2 ⊢ (𝜑 → ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)) = ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵))) |
13 | 5, 9, 12 | 3eqtr4d 2776 | 1 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 (class class class)co 7405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-iota 6489 df-fv 6545 df-ov 7408 |
This theorem is referenced by: (None) |
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