# Metamath Proof Explorer

## Theorem ablodiv32

Description: Swap the second and third terms in a double division. (Contributed by NM, 29-Feb-2008) (New usage is discouraged.)

Ref Expression
Hypotheses abldiv.1 𝑋 = ran 𝐺
abldiv.3 𝐷 = ( /𝑔𝐺 )
Assertion ablodiv32 ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴𝑋𝐵𝑋𝐶𝑋 ) ) → ( ( 𝐴 𝐷 𝐵 ) 𝐷 𝐶 ) = ( ( 𝐴 𝐷 𝐶 ) 𝐷 𝐵 ) )

### Proof

Step Hyp Ref Expression
1 abldiv.1 𝑋 = ran 𝐺
2 abldiv.3 𝐷 = ( /𝑔𝐺 )
3 1 ablocom ( ( 𝐺 ∈ AbelOp ∧ 𝐵𝑋𝐶𝑋 ) → ( 𝐵 𝐺 𝐶 ) = ( 𝐶 𝐺 𝐵 ) )
4 3 3adant3r1 ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴𝑋𝐵𝑋𝐶𝑋 ) ) → ( 𝐵 𝐺 𝐶 ) = ( 𝐶 𝐺 𝐵 ) )
5 4 oveq2d ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴𝑋𝐵𝑋𝐶𝑋 ) ) → ( 𝐴 𝐷 ( 𝐵 𝐺 𝐶 ) ) = ( 𝐴 𝐷 ( 𝐶 𝐺 𝐵 ) ) )
6 1 2 ablodivdiv4 ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴𝑋𝐵𝑋𝐶𝑋 ) ) → ( ( 𝐴 𝐷 𝐵 ) 𝐷 𝐶 ) = ( 𝐴 𝐷 ( 𝐵 𝐺 𝐶 ) ) )
7 3ancomb ( ( 𝐴𝑋𝐵𝑋𝐶𝑋 ) ↔ ( 𝐴𝑋𝐶𝑋𝐵𝑋 ) )
8 1 2 ablodivdiv4 ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴𝑋𝐶𝑋𝐵𝑋 ) ) → ( ( 𝐴 𝐷 𝐶 ) 𝐷 𝐵 ) = ( 𝐴 𝐷 ( 𝐶 𝐺 𝐵 ) ) )
9 7 8 sylan2b ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴𝑋𝐵𝑋𝐶𝑋 ) ) → ( ( 𝐴 𝐷 𝐶 ) 𝐷 𝐵 ) = ( 𝐴 𝐷 ( 𝐶 𝐺 𝐵 ) ) )
10 5 6 9 3eqtr4d ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴𝑋𝐵𝑋𝐶𝑋 ) ) → ( ( 𝐴 𝐷 𝐵 ) 𝐷 𝐶 ) = ( ( 𝐴 𝐷 𝐶 ) 𝐷 𝐵 ) )