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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 6853 | . 2 ⊢ (𝐶𝐺(𝐴𝐹𝐵)) = ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵)) |
6 | ovex 6678 | . . 3 ⊢ (𝐴𝐹𝐵) ∈ V | |
7 | caovdir.com | . . 3 ⊢ (𝑥𝐺𝑦) = (𝑦𝐺𝑥) | |
8 | 1, 6, 7 | caovcom 6831 | . 2 ⊢ (𝐶𝐺(𝐴𝐹𝐵)) = ((𝐴𝐹𝐵)𝐺𝐶) |
9 | 1, 2, 7 | caovcom 6831 | . . 3 ⊢ (𝐶𝐺𝐴) = (𝐴𝐺𝐶) |
10 | 1, 3, 7 | caovcom 6831 | . . 3 ⊢ (𝐶𝐺𝐵) = (𝐵𝐺𝐶) |
11 | 9, 10 | oveq12i 6662 | . 2 ⊢ ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵)) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)) |
12 | 5, 8, 11 | 3eqtr3i 2652 | 1 ⊢ ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 Vcvv 3200 (class class class)co 6650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 |
This theorem is referenced by: caovdilem 6869 adderpqlem 9776 addassnq 9780 prlem934 9855 prlem936 9869 recexsrlem 9924 mulgt0sr 9926 |
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