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Metamath Proof Explorer

Theorem ablodivdiv

Description: Law for double group division. (Contributed by NM, 29-Feb-2008) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 abldiv.1 ⊢ 𝑋 = ran 𝐺
2 abldiv.3 ⊢ 𝐷 = ( /𝑔 ‘ 𝐺 )
3 ablogrpo ⊢ ( 𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp )
4 1 2 grpodivdiv ⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 𝐷 ( 𝐵 𝐷 𝐶 ) ) = ( 𝐴 𝐺 ( 𝐶 𝐷 𝐵 ) ) )
5 3 4 sylan ⊢ ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 𝐷 ( 𝐵 𝐷 𝐶 ) ) = ( 𝐴 𝐺 ( 𝐶 𝐷 𝐵 ) ) )
6 3ancomb ⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ↔ ( 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) )
7 1 2 grpomuldivass ⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐶 ) 𝐷 𝐵 ) = ( 𝐴 𝐺 ( 𝐶 𝐷 𝐵 ) ) )
8 3 7 sylan ⊢ ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐶 ) 𝐷 𝐵 ) = ( 𝐴 𝐺 ( 𝐶 𝐷 𝐵 ) ) )
9 1 2 ablomuldiv ⊢ ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐶 ) 𝐷 𝐵 ) = ( ( 𝐴 𝐷 𝐵 ) 𝐺 𝐶 ) )
10 8 9 eqtr3d ⊢ ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐴 𝐺 ( 𝐶 𝐷 𝐵 ) ) = ( ( 𝐴 𝐷 𝐵 ) 𝐺 𝐶 ) )
11 6 10 sylan2b ⊢ ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 𝐺 ( 𝐶 𝐷 𝐵 ) ) = ( ( 𝐴 𝐷 𝐵 ) 𝐺 𝐶 ) )
12 5 11 eqtrd ⊢ ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 𝐷 ( 𝐵 𝐷 𝐶 ) ) = ( ( 𝐴 𝐷 𝐵 ) 𝐺 𝐶 ) )