Theorem ablocom 30305

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 30303	. . . 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 7411	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
5		oveq2 7412	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦𝐺𝑥) = (𝑦𝐺𝐴))
6	4, 5	eqeq12d 2742	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑦) = (𝑦𝐺𝑥) ↔ (𝐴𝐺𝑦) = (𝑦𝐺𝐴)))
7		oveq2 7412	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
8		oveq1 7411	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐺𝐴) = (𝐵𝐺𝐴))
9	7, 8	eqeq12d 2742	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐺𝑦) = (𝑦𝐺𝐴) ↔ (𝐴𝐺𝐵) = (𝐵𝐺𝐴)))
10	6, 9	rspc2v 3617	. . 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 1113	1 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3055  ran crn 5670  (class class class)co 7404  GrpOpcgr 30246  AbelOpcablo 30301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-cnv 5677  df-dm 5679  df-rn 5680  df-iota 6488  df-fv 6544  df-ov 7407  df-ablo 30302

This theorem is referenced by:  ablo32  30306  ablomuldiv  30309  ablodiv32  30312  nvcom  30378  rngocom  37293  iscringd  37378