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Mirrors > Home > MPE Home > Th. List > caov42 | Structured version Visualization version GIF version |
Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) |
Ref | Expression |
---|---|
caov.1 | ⊢ 𝐴 ∈ V |
caov.2 | ⊢ 𝐵 ∈ V |
caov.3 | ⊢ 𝐶 ∈ V |
caov.com | ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) |
caov.ass | ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)) |
caov.4 | ⊢ 𝐷 ∈ V |
Ref | Expression |
---|---|
caov42 | ⊢ ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐷𝐹𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caov.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | caov.2 | . . 3 ⊢ 𝐵 ∈ V | |
3 | caov.3 | . . 3 ⊢ 𝐶 ∈ V | |
4 | caov.com | . . 3 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) | |
5 | caov.ass | . . 3 ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)) | |
6 | caov.4 | . . 3 ⊢ 𝐷 ∈ V | |
7 | 1, 2, 3, 4, 5, 6 | caov4 7379 | . 2 ⊢ ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)) |
8 | 2, 6, 4 | caovcom 7345 | . . 3 ⊢ (𝐵𝐹𝐷) = (𝐷𝐹𝐵) |
9 | 8 | oveq2i 7167 | . 2 ⊢ ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐷𝐹𝐵)) |
10 | 7, 9 | eqtri 2844 | 1 ⊢ ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐷𝐹𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3494 (class class class)co 7156 |
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 2793 ax-nul 5210 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 |
This theorem is referenced by: caovlem2 7384 mulcmpblnrlem 10492 ltasr 10522 axmulass 10579 |
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