caov411d - Metamath Proof Explorer

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Theorem caov411d 6859

Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)

**Hypotheses**
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)))
caovd.4	⊢ (𝜑 → 𝐷 ∈ 𝑆)
caovd.cl	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)

**Assertion**
Ref	Expression
caov411d	⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷)))

Distinct variable groups: 𝑥,𝑦,𝑧,𝐴 𝑥,𝐵,𝑦,𝑧 𝑥,𝐶,𝑦,𝑧 𝑥,𝐷,𝑦,𝑧 𝜑,𝑥,𝑦,𝑧 𝑥,𝐹,𝑦,𝑧 𝑥,𝑆,𝑦,𝑧

**Proof of Theorem caov411d**
Step	Hyp	Ref	Expression
1		caovd.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
2		caovd.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
3		caovd.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑆)
4		caovd.com	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
5		caovd.ass	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
6		caovd.4	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑆)
7		caovd.cl	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
8	1, 2, 3, 4, 5, 6, 7	caov4d 6858	. 2 ⊢ (𝜑 → ((𝐵𝐹𝐴)𝐹(𝐶𝐹𝐷)) = ((𝐵𝐹𝐶)𝐹(𝐴𝐹𝐷)))
9	4, 1, 2	caovcomd 6830	. . 3 ⊢ (𝜑 → (𝐵𝐹𝐴) = (𝐴𝐹𝐵))
10	9	oveq1d 6665	. 2 ⊢ (𝜑 → ((𝐵𝐹𝐴)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)))
11	4, 1, 3	caovcomd 6830	. . 3 ⊢ (𝜑 → (𝐵𝐹𝐶) = (𝐶𝐹𝐵))
12	11	oveq1d 6665	. 2 ⊢ (𝜑 → ((𝐵𝐹𝐶)𝐹(𝐴𝐹𝐷)) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷)))
13	8, 10, 12	3eqtr3d 2664	1 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷)))

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: (None)

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