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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 7369 | . 2 ⊢ (𝐶𝐺(𝐴𝐹𝐵)) = ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵)) |
6 | ovex 7191 | . . 3 ⊢ (𝐴𝐹𝐵) ∈ V | |
7 | caovdir.com | . . 3 ⊢ (𝑥𝐺𝑦) = (𝑦𝐺𝑥) | |
8 | 1, 6, 7 | caovcom 7347 | . 2 ⊢ (𝐶𝐺(𝐴𝐹𝐵)) = ((𝐴𝐹𝐵)𝐺𝐶) |
9 | 1, 2, 7 | caovcom 7347 | . . 3 ⊢ (𝐶𝐺𝐴) = (𝐴𝐺𝐶) |
10 | 1, 3, 7 | caovcom 7347 | . . 3 ⊢ (𝐶𝐺𝐵) = (𝐵𝐺𝐶) |
11 | 9, 10 | oveq12i 7170 | . 2 ⊢ ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵)) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)) |
12 | 5, 8, 11 | 3eqtr3i 2854 | 1 ⊢ ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3496 (class class class)co 7158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-nul 5212 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 |
This theorem is referenced by: caovdilem 7385 adderpqlem 10378 addassnq 10382 prlem934 10457 prlem936 10471 recexsrlem 10527 mulgt0sr 10529 |
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