Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > caov32d | Structured version Visualization version GIF version |
Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.) |
Ref | Expression |
---|---|
caovd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
caovd.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑆) |
caovd.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑆) |
caovd.com | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) |
caovd.ass | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) |
Ref | Expression |
---|---|
caov32d | ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovd.com | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) | |
2 | caovd.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
3 | caovd.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑆) | |
4 | 1, 2, 3 | caovcomd 6830 | . . 3 ⊢ (𝜑 → (𝐵𝐹𝐶) = (𝐶𝐹𝐵)) |
5 | 4 | oveq2d 6666 | . 2 ⊢ (𝜑 → (𝐴𝐹(𝐵𝐹𝐶)) = (𝐴𝐹(𝐶𝐹𝐵))) |
6 | caovd.ass | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) | |
7 | caovd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
8 | 6, 7, 2, 3 | caovassd 6833 | . 2 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) |
9 | 6, 7, 3, 2 | caovassd 6833 | . 2 ⊢ (𝜑 → ((𝐴𝐹𝐶)𝐹𝐵) = (𝐴𝐹(𝐶𝐹𝐵))) |
10 | 5, 8, 9 | 3eqtr4d 2666 | 1 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 (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 |
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-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 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: caov31d 6856 |
Copyright terms: Public domain | W3C validator |