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Metamath Proof Explorer

Theorem ablonncan

Description: Cancellation law for group division. ( nncan analog.) (Contributed by NM, 7-Mar-2008) (New usage is discouraged.)

Ref Expression
Hypotheses abldiv.1 ⊢ 𝑋 = ran 𝐺
abldiv.3 ⊢ 𝐷 = ( /𝑔 ‘ 𝐺 )
Assertion ablonncan ( ( 𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 ( 𝐴 𝐷 𝐵 ) ) = 𝐵 )

Proof

Step Hyp Ref Expression
1 abldiv.1 ⊢ 𝑋 = ran 𝐺
2 abldiv.3 ⊢ 𝐷 = ( /𝑔 ‘ 𝐺 )
3 id ⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) )
4 3 3anidm12 ⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) )
5 1 2 ablodivdiv ⊢ ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐴 𝐷 ( 𝐴 𝐷 𝐵 ) ) = ( ( 𝐴 𝐷 𝐴 ) 𝐺 𝐵 ) )
6 4 5 sylan2 ⊢ ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐴 𝐷 ( 𝐴 𝐷 𝐵 ) ) = ( ( 𝐴 𝐷 𝐴 ) 𝐺 𝐵 ) )
7 6 3impb ⊢ ( ( 𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 ( 𝐴 𝐷 𝐵 ) ) = ( ( 𝐴 𝐷 𝐴 ) 𝐺 𝐵 ) )
8 ablogrpo ⊢ ( 𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp )
9 eqid ⊢ ( GId ‘ 𝐺 ) = ( GId ‘ 𝐺 )
10 1 2 9 grpodivid ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐴 ) = ( GId ‘ 𝐺 ) )
11 8 10 sylan ⊢ ( ( 𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐴 ) = ( GId ‘ 𝐺 ) )
12 11 3adant3 ⊢ ( ( 𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐴 ) = ( GId ‘ 𝐺 ) )
13 12 oveq1d ⊢ ( ( 𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 𝐷 𝐴 ) 𝐺 𝐵 ) = ( ( GId ‘ 𝐺 ) 𝐺 𝐵 ) )
14 1 9 grpolid ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ) → ( ( GId ‘ 𝐺 ) 𝐺 𝐵 ) = 𝐵 )
15 8 14 sylan ⊢ ( ( 𝐺 ∈ AbelOp ∧ 𝐵 ∈ 𝑋 ) → ( ( GId ‘ 𝐺 ) 𝐺 𝐵 ) = 𝐵 )
16 15 3adant2 ⊢ ( ( 𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( GId ‘ 𝐺 ) 𝐺 𝐵 ) = 𝐵 )
17 7 13 16 3eqtrd ⊢ ( ( 𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 ( 𝐴 𝐷 𝐵 ) ) = 𝐵 )