Theorem grpoass

Description: A group operation is associative. (Contributed by NM, 10-Oct-2006) (New usage is discouraged.)

	Ref	Expression
Hypothesis	grpfo.1	⊢ 𝑋 = ran 𝐺
Assertion	grpoass	⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝐺 𝐶 ) = ( 𝐴 𝐺 ( 𝐵 𝐺 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		grpfo.1	⊢ 𝑋 = ran 𝐺
2	1	isgrpo	⊢ ( 𝐺 ∈ GrpOp → ( 𝐺 ∈ GrpOp ↔ ( 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) ∧ ∃ 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ∃ 𝑦 ∈ 𝑋 ( 𝑦 𝐺 𝑥 ) = 𝑢 ) ) ) )
3	2	ibi	⊢ ( 𝐺 ∈ GrpOp → ( 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) ∧ ∃ 𝑢 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( ( 𝑢 𝐺 𝑥 ) = 𝑥 ∧ ∃ 𝑦 ∈ 𝑋 ( 𝑦 𝐺 𝑥 ) = 𝑢 ) ) )
4	3	simp2d	⊢ ( 𝐺 ∈ GrpOp → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) )
5		oveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝐺 𝑦 ) = ( 𝐴 𝐺 𝑦 ) )
6	5	oveq1d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( ( 𝐴 𝐺 𝑦 ) 𝐺 𝑧 ) )
7		oveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) = ( 𝐴 𝐺 ( 𝑦 𝐺 𝑧 ) ) )
8	6 7	eqeq12d	⊢ ( 𝑥 = 𝐴 → ( ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) ↔ ( ( 𝐴 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝐴 𝐺 ( 𝑦 𝐺 𝑧 ) ) ) )
9		oveq2	⊢ ( 𝑦 = 𝐵 → ( 𝐴 𝐺 𝑦 ) = ( 𝐴 𝐺 𝐵 ) )
10	9	oveq1d	⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 𝐺 𝑦 ) 𝐺 𝑧 ) = ( ( 𝐴 𝐺 𝐵 ) 𝐺 𝑧 ) )
11		oveq1	⊢ ( 𝑦 = 𝐵 → ( 𝑦 𝐺 𝑧 ) = ( 𝐵 𝐺 𝑧 ) )
12	11	oveq2d	⊢ ( 𝑦 = 𝐵 → ( 𝐴 𝐺 ( 𝑦 𝐺 𝑧 ) ) = ( 𝐴 𝐺 ( 𝐵 𝐺 𝑧 ) ) )
13	10 12	eqeq12d	⊢ ( 𝑦 = 𝐵 → ( ( ( 𝐴 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝐴 𝐺 ( 𝑦 𝐺 𝑧 ) ) ↔ ( ( 𝐴 𝐺 𝐵 ) 𝐺 𝑧 ) = ( 𝐴 𝐺 ( 𝐵 𝐺 𝑧 ) ) ) )
14		oveq2	⊢ ( 𝑧 = 𝐶 → ( ( 𝐴 𝐺 𝐵 ) 𝐺 𝑧 ) = ( ( 𝐴 𝐺 𝐵 ) 𝐺 𝐶 ) )
15		oveq2	⊢ ( 𝑧 = 𝐶 → ( 𝐵 𝐺 𝑧 ) = ( 𝐵 𝐺 𝐶 ) )
16	15	oveq2d	⊢ ( 𝑧 = 𝐶 → ( 𝐴 𝐺 ( 𝐵 𝐺 𝑧 ) ) = ( 𝐴 𝐺 ( 𝐵 𝐺 𝐶 ) ) )
17	14 16	eqeq12d	⊢ ( 𝑧 = 𝐶 → ( ( ( 𝐴 𝐺 𝐵 ) 𝐺 𝑧 ) = ( 𝐴 𝐺 ( 𝐵 𝐺 𝑧 ) ) ↔ ( ( 𝐴 𝐺 𝐵 ) 𝐺 𝐶 ) = ( 𝐴 𝐺 ( 𝐵 𝐺 𝐶 ) ) ) )
18	8 13 17	rspc3v	⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝐺 𝐶 ) = ( 𝐴 𝐺 ( 𝐵 𝐺 𝐶 ) ) ) )
19	4 18	mpan9	⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝐺 𝐶 ) = ( 𝐴 𝐺 ( 𝐵 𝐺 𝐶 ) ) )