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Theorem ablocom 30484

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 30482	. . . 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 7397	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
5		oveq2 7398	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦𝐺𝑥) = (𝑦𝐺𝐴))
6	4, 5	eqeq12d 2746	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑦) = (𝑦𝐺𝑥) ↔ (𝐴𝐺𝑦) = (𝑦𝐺𝐴)))
7		oveq2 7398	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
8		oveq1 7397	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐺𝐴) = (𝐵𝐺𝐴))
9	7, 8	eqeq12d 2746	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐺𝑦) = (𝑦𝐺𝐴) ↔ (𝐴𝐺𝐵) = (𝐵𝐺𝐴)))
10	6, 9	rspc2v 3602	. . 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 2109 ∀wral 3045 ran crn 5642 (class class class)co 7390 GrpOpcgr 30425 AbelOpcablo 30480

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 2008 ax-8 2111 ax-9 2119 ax-ext 2702

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 2066 df-clab 2709 df-cleq 2722 df-clel 2804 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-dif 3920 df-un 3922 df-ss 3934 df-nul 4300 df-if 4492 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-cnv 5649 df-dm 5651 df-rn 5652 df-iota 6467 df-fv 6522 df-ov 7393 df-ablo 30481

This theorem is referenced by: ablo32 30485 ablomuldiv 30488 ablodiv32 30491 nvcom 30557 rngocom 37914 iscringd 37999

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