ablocom - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ablocom		Structured version Visualization version GIF version

Theorem ablocom 30529

Description: An Abelian group operation is commutative. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)

**Hypothesis**
Ref	Expression
ablcom.1	⊢ 𝑋 = ran 𝐺

**Assertion**
Ref	Expression
ablocom	⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))

Proof of Theorem ablocom Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ablcom.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
2	1	isablo 30527	. . . 4 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
3	2	simprbi 496	. . 3 ⊢ (𝐺 ∈ AbelOp → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
4		oveq1 7412	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
5		oveq2 7413	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦𝐺𝑥) = (𝑦𝐺𝐴))
6	4, 5	eqeq12d 2751	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑦) = (𝑦𝐺𝑥) ↔ (𝐴𝐺𝑦) = (𝑦𝐺𝐴)))
7		oveq2 7413	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
8		oveq1 7412	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐺𝐴) = (𝐵𝐺𝐴))
9	7, 8	eqeq12d 2751	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐺𝑦) = (𝑦𝐺𝐴) ↔ (𝐴𝐺𝐵) = (𝐵𝐺𝐴)))
10	6, 9	rspc2v 3612	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴)))
11	3, 10	syl5com 31	. 2 ⊢ (𝐺 ∈ AbelOp → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴)))
12	11	3impib 1116	1 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ∀wral 3051 ran crn 5655 (class class class)co 7405 GrpOpcgr 30470 AbelOpcablo 30525

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2707

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2727 df-clel 2809 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-dif 3929 df-un 3931 df-ss 3943 df-nul 4309 df-if 4501 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-br 5120 df-opab 5182 df-cnv 5662 df-dm 5664 df-rn 5665 df-iota 6484 df-fv 6539 df-ov 7408 df-ablo 30526

This theorem is referenced by: ablo32 30530 ablomuldiv 30533 ablodiv32 30536 nvcom 30602 rngocom 37937 iscringd 38022

W3C validator