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Mirrors > Home > MPE Home > Th. List > caovdir | Structured version Visualization version GIF version |
Description: Reverse distributive law. (Contributed by NM, 26-Aug-1995.) |
Ref | Expression |
---|---|
caovdir.1 | ⊢ 𝐴 ∈ V |
caovdir.2 | ⊢ 𝐵 ∈ V |
caovdir.3 | ⊢ 𝐶 ∈ V |
caovdir.com | ⊢ (𝑥𝐺𝑦) = (𝑦𝐺𝑥) |
caovdir.distr | ⊢ (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧)) |
Ref | Expression |
---|---|
caovdir | ⊢ ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovdir.3 | . . 3 ⊢ 𝐶 ∈ V | |
2 | caovdir.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | caovdir.2 | . . 3 ⊢ 𝐵 ∈ V | |
4 | caovdir.distr | . . 3 ⊢ (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧)) | |
5 | 1, 2, 3, 4 | caovdi 7577 | . 2 ⊢ (𝐶𝐺(𝐴𝐹𝐵)) = ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵)) |
6 | ovex 7394 | . . 3 ⊢ (𝐴𝐹𝐵) ∈ V | |
7 | caovdir.com | . . 3 ⊢ (𝑥𝐺𝑦) = (𝑦𝐺𝑥) | |
8 | 1, 6, 7 | caovcom 7555 | . 2 ⊢ (𝐶𝐺(𝐴𝐹𝐵)) = ((𝐴𝐹𝐵)𝐺𝐶) |
9 | 1, 2, 7 | caovcom 7555 | . . 3 ⊢ (𝐶𝐺𝐴) = (𝐴𝐺𝐶) |
10 | 1, 3, 7 | caovcom 7555 | . . 3 ⊢ (𝐶𝐺𝐵) = (𝐵𝐺𝐶) |
11 | 9, 10 | oveq12i 7373 | . 2 ⊢ ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵)) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)) |
12 | 5, 8, 11 | 3eqtr3i 2769 | 1 ⊢ ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 Vcvv 3447 (class class class)co 7361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-nul 5267 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-iota 6452 df-fv 6508 df-ov 7364 |
This theorem is referenced by: caovdilem 7593 adderpqlem 10898 addassnq 10902 prlem934 10977 prlem936 10991 recexsrlem 11047 mulgt0sr 11049 |
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