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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 7563 | . 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 7541 | . . 3 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝑆) |
9 | 6, 8, 2 | caovcomd 7544 | . 2 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = (𝐶𝐺(𝐴𝐹𝐵))) |
10 | 6, 3, 2 | caovcomd 7544 | . . 3 ⊢ (𝜑 → (𝐴𝐺𝐶) = (𝐶𝐺𝐴)) |
11 | 6, 4, 2 | caovcomd 7544 | . . 3 ⊢ (𝜑 → (𝐵𝐺𝐶) = (𝐶𝐺𝐵)) |
12 | 10, 11 | oveq12d 7369 | . 2 ⊢ (𝜑 → ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)) = ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵))) |
13 | 5, 9, 12 | 3eqtr4d 2787 | 1 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 (class class class)co 7351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3063 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-iota 6445 df-fv 6501 df-ov 7354 |
This theorem is referenced by: (None) |
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