Theorem grpodivdiv

Description: Double group division. (Contributed by NM, 24-Feb-2008) (New usage is discouraged.)

	Ref	Expression
Hypotheses	grpdivf.1	⊢ 𝑋 = ran 𝐺
	grpdivf.3	⊢ 𝐷 = ( /𝑔 ‘ 𝐺 )
Assertion	grpodivdiv	⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 𝐷 ( 𝐵 𝐷 𝐶 ) ) = ( 𝐴 𝐺 ( 𝐶 𝐷 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		grpdivf.1	⊢ 𝑋 = ran 𝐺
2		grpdivf.3	⊢ 𝐷 = ( /𝑔 ‘ 𝐺 )
3		simpl	⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → 𝐺 ∈ GrpOp )
4		simpr1	⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → 𝐴 ∈ 𝑋 )
5	1 2	grpodivcl	⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐵 𝐷 𝐶 ) ∈ 𝑋 )
6	5	3adant3r1	⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐵 𝐷 𝐶 ) ∈ 𝑋 )
7		eqid	⊢ ( inv ‘ 𝐺 ) = ( inv ‘ 𝐺 )
8	1 7 2	grpodivval	⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝐵 𝐷 𝐶 ) ∈ 𝑋 ) → ( 𝐴 𝐷 ( 𝐵 𝐷 𝐶 ) ) = ( 𝐴 𝐺 ( ( inv ‘ 𝐺 ) ‘ ( 𝐵 𝐷 𝐶 ) ) ) )
9	3 4 6 8	syl3anc	⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 𝐷 ( 𝐵 𝐷 𝐶 ) ) = ( 𝐴 𝐺 ( ( inv ‘ 𝐺 ) ‘ ( 𝐵 𝐷 𝐶 ) ) ) )
10	1 7 2	grpoinvdiv	⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( ( inv ‘ 𝐺 ) ‘ ( 𝐵 𝐷 𝐶 ) ) = ( 𝐶 𝐷 𝐵 ) )
11	10	3adant3r1	⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( inv ‘ 𝐺 ) ‘ ( 𝐵 𝐷 𝐶 ) ) = ( 𝐶 𝐷 𝐵 ) )
12	11	oveq2d	⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 𝐺 ( ( inv ‘ 𝐺 ) ‘ ( 𝐵 𝐷 𝐶 ) ) ) = ( 𝐴 𝐺 ( 𝐶 𝐷 𝐵 ) ) )
13	9 12	eqtrd	⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 𝐷 ( 𝐵 𝐷 𝐶 ) ) = ( 𝐴 𝐺 ( 𝐶 𝐷 𝐵 ) ) )