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Metamath Proof Explorer

Theorem grpodivf

Description: Mapping for group division. (Contributed by NM, 10-Apr-2008) (Revised by Mario Carneiro, 15-Dec-2013) (New usage is discouraged.)

Ref Expression
Hypotheses grpdivf.1 ⊢ 𝑋 = ran 𝐺
grpdivf.3 ⊢ 𝐷 = ( /𝑔 ‘ 𝐺 )
Assertion grpodivf ( 𝐺 ∈ GrpOp → 𝐷 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 )

Proof

Step Hyp Ref Expression
1 grpdivf.1 ⊢ 𝑋 = ran 𝐺
2 grpdivf.3 ⊢ 𝐷 = ( /𝑔 ‘ 𝐺 )
3 eqid ⊢ ( inv ‘ 𝐺 ) = ( inv ‘ 𝐺 )
4 1 3 grpoinvcl ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝑦 ∈ 𝑋 ) → ( ( inv ‘ 𝐺 ) ‘ 𝑦 ) ∈ 𝑋 )
5 4 3adant2 ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) → ( ( inv ‘ 𝐺 ) ‘ 𝑦 ) ∈ 𝑋 )
6 1 grpocl ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋 ∧ ( ( inv ‘ 𝐺 ) ‘ 𝑦 ) ∈ 𝑋 ) → ( 𝑥 𝐺 ( ( inv ‘ 𝐺 ) ‘ 𝑦 ) ) ∈ 𝑋 )
7 5 6 syld3an3 ⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) → ( 𝑥 𝐺 ( ( inv ‘ 𝐺 ) ‘ 𝑦 ) ) ∈ 𝑋 )
8 7 3expib ⊢ ( 𝐺 ∈ GrpOp → ( ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) → ( 𝑥 𝐺 ( ( inv ‘ 𝐺 ) ‘ 𝑦 ) ) ∈ 𝑋 ) )
9 8 ralrimivv ⊢ ( 𝐺 ∈ GrpOp → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝐺 ( ( inv ‘ 𝐺 ) ‘ 𝑦 ) ) ∈ 𝑋 )
10 eqid ⊢ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( 𝑥 𝐺 ( ( inv ‘ 𝐺 ) ‘ 𝑦 ) ) ) = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( 𝑥 𝐺 ( ( inv ‘ 𝐺 ) ‘ 𝑦 ) ) )
11 10 fmpo ⊢ ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝐺 ( ( inv ‘ 𝐺 ) ‘ 𝑦 ) ) ∈ 𝑋 ↔ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( 𝑥 𝐺 ( ( inv ‘ 𝐺 ) ‘ 𝑦 ) ) ) : ( 𝑋 × 𝑋 ) ⟶ 𝑋 )
12 9 11 sylib ⊢ ( 𝐺 ∈ GrpOp → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( 𝑥 𝐺 ( ( inv ‘ 𝐺 ) ‘ 𝑦 ) ) ) : ( 𝑋 × 𝑋 ) ⟶ 𝑋 )
13 1 3 2 grpodivfval ⊢ ( 𝐺 ∈ GrpOp → 𝐷 = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( 𝑥 𝐺 ( ( inv ‘ 𝐺 ) ‘ 𝑦 ) ) ) )
14 13 feq1d ⊢ ( 𝐺 ∈ GrpOp → ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ↔ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( 𝑥 𝐺 ( ( inv ‘ 𝐺 ) ‘ 𝑦 ) ) ) : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ) )
15 12 14 mpbird ⊢ ( 𝐺 ∈ GrpOp → 𝐷 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 )