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Metamath Proof Explorer

Theorem ablomuldiv

Description: Law for group multiplication and division. (Contributed by NM, 15-Feb-2008) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 abldiv.1 ⊢ 𝑋 = ran 𝐺
2 abldiv.3 ⊢ 𝐷 = ( /𝑔 ‘ 𝐺 )
3 1 ablocom ⊢ ( ( 𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐺 𝐵 ) = ( 𝐵 𝐺 𝐴 ) )
4 3 3adant3r3 ⊢ ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 𝐺 𝐵 ) = ( 𝐵 𝐺 𝐴 ) )
5 4 oveq1d ⊢ ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝐷 𝐶 ) = ( ( 𝐵 𝐺 𝐴 ) 𝐷 𝐶 ) )
6 3ancoma ⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ↔ ( 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) )
7 ablogrpo ⊢ ( 𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp )
8 1 2 grpomuldivass ⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐵 𝐺 𝐴 ) 𝐷 𝐶 ) = ( 𝐵 𝐺 ( 𝐴 𝐷 𝐶 ) ) )
9 7 8 sylan ⊢ ( ( 𝐺 ∈ AbelOp ∧ ( 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐵 𝐺 𝐴 ) 𝐷 𝐶 ) = ( 𝐵 𝐺 ( 𝐴 𝐷 𝐶 ) ) )
10 6 9 sylan2b ⊢ ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐵 𝐺 𝐴 ) 𝐷 𝐶 ) = ( 𝐵 𝐺 ( 𝐴 𝐷 𝐶 ) ) )
11 simpr2 ⊢ ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → 𝐵 ∈ 𝑋 )
12 1 2 grpodivcl ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐶 ) ∈ 𝑋 )
13 7 12 syl3an1 ⊢ ( ( 𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐶 ) ∈ 𝑋 )
14 13 3adant3r2 ⊢ ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 𝐷 𝐶 ) ∈ 𝑋 )
15 11 14 jca ⊢ ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐵 ∈ 𝑋 ∧ ( 𝐴 𝐷 𝐶 ) ∈ 𝑋 ) )
16 1 ablocom ⊢ ( ( 𝐺 ∈ AbelOp ∧ 𝐵 ∈ 𝑋 ∧ ( 𝐴 𝐷 𝐶 ) ∈ 𝑋 ) → ( 𝐵 𝐺 ( 𝐴 𝐷 𝐶 ) ) = ( ( 𝐴 𝐷 𝐶 ) 𝐺 𝐵 ) )
17 16 3expb ⊢ ( ( 𝐺 ∈ AbelOp ∧ ( 𝐵 ∈ 𝑋 ∧ ( 𝐴 𝐷 𝐶 ) ∈ 𝑋 ) ) → ( 𝐵 𝐺 ( 𝐴 𝐷 𝐶 ) ) = ( ( 𝐴 𝐷 𝐶 ) 𝐺 𝐵 ) )
18 15 17 syldan ⊢ ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐵 𝐺 ( 𝐴 𝐷 𝐶 ) ) = ( ( 𝐴 𝐷 𝐶 ) 𝐺 𝐵 ) )
19 5 10 18 3eqtrd ⊢ ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝐷 𝐶 ) = ( ( 𝐴 𝐷 𝐶 ) 𝐺 𝐵 ) )