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Metamath Proof Explorer

Theorem grpoinvop

Description: The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of Herstein p. 55. (Contributed by NM, 27-Oct-2006) (New usage is discouraged.)

Ref Expression
Hypotheses grpasscan1.1 ⊢ 𝑋 = ran 𝐺
grpasscan1.2 ⊢ 𝑁 = ( inv ‘ 𝐺 )
Assertion grpoinvop ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝑁 ‘ ( 𝐴 𝐺 𝐵 ) ) = ( ( 𝑁 ‘ 𝐵 ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 grpasscan1.1 ⊢ 𝑋 = ran 𝐺
2 grpasscan1.2 ⊢ 𝑁 = ( inv ‘ 𝐺 )
3 simp1 ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → 𝐺 ∈ GrpOp )
4 simp2 ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → 𝐴 ∈ 𝑋 )
5 simp3 ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → 𝐵 ∈ 𝑋 )
6 1 2 grpoinvcl ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐵 ) ∈ 𝑋 )
7 6 3adant2 ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐵 ) ∈ 𝑋 )
8 1 2 grpoinvcl ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐴 ) ∈ 𝑋 )
9 8 3adant3 ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐴 ) ∈ 𝑋 )
10 1 grpocl ⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝑁 ‘ 𝐵 ) ∈ 𝑋 ∧ ( 𝑁 ‘ 𝐴 ) ∈ 𝑋 ) → ( ( 𝑁 ‘ 𝐵 ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) ∈ 𝑋 )
11 3 7 9 10 syl3anc ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝑁 ‘ 𝐵 ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) ∈ 𝑋 )
12 1 grpoass ⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ( ( 𝑁 ‘ 𝐵 ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝐺 ( ( 𝑁 ‘ 𝐵 ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) ) = ( 𝐴 𝐺 ( 𝐵 𝐺 ( ( 𝑁 ‘ 𝐵 ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) ) ) )
13 3 4 5 11 12 syl13anc ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 𝐺 𝐵 ) 𝐺 ( ( 𝑁 ‘ 𝐵 ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) ) = ( 𝐴 𝐺 ( 𝐵 𝐺 ( ( 𝑁 ‘ 𝐵 ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) ) ) )
14 eqid ⊢ ( GId ‘ 𝐺 ) = ( GId ‘ 𝐺 )
15 1 14 2 grporinv ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ) → ( 𝐵 𝐺 ( 𝑁 ‘ 𝐵 ) ) = ( GId ‘ 𝐺 ) )
16 15 3adant2 ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐵 𝐺 ( 𝑁 ‘ 𝐵 ) ) = ( GId ‘ 𝐺 ) )
17 16 oveq1d ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐵 𝐺 ( 𝑁 ‘ 𝐵 ) ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) = ( ( GId ‘ 𝐺 ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) )
18 1 grpoass ⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝐵 ∈ 𝑋 ∧ ( 𝑁 ‘ 𝐵 ) ∈ 𝑋 ∧ ( 𝑁 ‘ 𝐴 ) ∈ 𝑋 ) ) → ( ( 𝐵 𝐺 ( 𝑁 ‘ 𝐵 ) ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) = ( 𝐵 𝐺 ( ( 𝑁 ‘ 𝐵 ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) ) )
19 3 5 7 9 18 syl13anc ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐵 𝐺 ( 𝑁 ‘ 𝐵 ) ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) = ( 𝐵 𝐺 ( ( 𝑁 ‘ 𝐵 ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) ) )
20 1 14 grpolid ⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝑁 ‘ 𝐴 ) ∈ 𝑋 ) → ( ( GId ‘ 𝐺 ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) = ( 𝑁 ‘ 𝐴 ) )
21 8 20 syldan ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ) → ( ( GId ‘ 𝐺 ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) = ( 𝑁 ‘ 𝐴 ) )
22 21 3adant3 ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( GId ‘ 𝐺 ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) = ( 𝑁 ‘ 𝐴 ) )
23 17 19 22 3eqtr3d ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐵 𝐺 ( ( 𝑁 ‘ 𝐵 ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) ) = ( 𝑁 ‘ 𝐴 ) )
24 23 oveq2d ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐺 ( 𝐵 𝐺 ( ( 𝑁 ‘ 𝐵 ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) ) ) = ( 𝐴 𝐺 ( 𝑁 ‘ 𝐴 ) ) )
25 1 14 2 grporinv ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐺 ( 𝑁 ‘ 𝐴 ) ) = ( GId ‘ 𝐺 ) )
26 25 3adant3 ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐺 ( 𝑁 ‘ 𝐴 ) ) = ( GId ‘ 𝐺 ) )
27 13 24 26 3eqtrd ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 𝐺 𝐵 ) 𝐺 ( ( 𝑁 ‘ 𝐵 ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) ) = ( GId ‘ 𝐺 ) )
28 1 grpocl ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐺 𝐵 ) ∈ 𝑋 )
29 1 14 2 grpoinvid1 ⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝐴 𝐺 𝐵 ) ∈ 𝑋 ∧ ( ( 𝑁 ‘ 𝐵 ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) ∈ 𝑋 ) → ( ( 𝑁 ‘ ( 𝐴 𝐺 𝐵 ) ) = ( ( 𝑁 ‘ 𝐵 ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) ↔ ( ( 𝐴 𝐺 𝐵 ) 𝐺 ( ( 𝑁 ‘ 𝐵 ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) ) = ( GId ‘ 𝐺 ) ) )
30 3 28 11 29 syl3anc ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝑁 ‘ ( 𝐴 𝐺 𝐵 ) ) = ( ( 𝑁 ‘ 𝐵 ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) ↔ ( ( 𝐴 𝐺 𝐵 ) 𝐺 ( ( 𝑁 ‘ 𝐵 ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) ) = ( GId ‘ 𝐺 ) ) )
31 27 30 mpbird ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝑁 ‘ ( 𝐴 𝐺 𝐵 ) ) = ( ( 𝑁 ‘ 𝐵 ) 𝐺 ( 𝑁 ‘ 𝐴 ) ) )