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Metamath Proof Explorer

Theorem grplactf1o

Description: The left group action of element A of group G maps the underlying set X of G one-to-one onto itself. (Contributed by Paul Chapman, 18-Mar-2008) (Proof shortened by Mario Carneiro, 14-Aug-2015)

Ref Expression
Hypotheses grplact.1 ⊢ 𝐹 = ( 𝑔 ∈ 𝑋 ↦ ( 𝑎 ∈ 𝑋 ↦ ( 𝑔 + 𝑎 ) ) )
grplact.2 ⊢ 𝑋 = ( Base ‘ 𝐺 )
grplact.3 ⊢ + = ( +g ‘ 𝐺 )
Assertion grplactf1o ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( 𝐹 ‘ 𝐴 ) : 𝑋 –1-1-onto→ 𝑋 )

Proof

Step Hyp Ref Expression
1 grplact.1 ⊢ 𝐹 = ( 𝑔 ∈ 𝑋 ↦ ( 𝑎 ∈ 𝑋 ↦ ( 𝑔 + 𝑎 ) ) )
2 grplact.2 ⊢ 𝑋 = ( Base ‘ 𝐺 )
3 grplact.3 ⊢ + = ( +g ‘ 𝐺 )
4 eqid ⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 )
5 1 2 3 4 grplactcnv ⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝐹 ‘ 𝐴 ) : 𝑋 –1-1-onto→ 𝑋 ∧ ◡ ( 𝐹 ‘ 𝐴 ) = ( 𝐹 ‘ ( ( invg ‘ 𝐺 ) ‘ 𝐴 ) ) ) )
6 5 simpld ⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( 𝐹 ‘ 𝐴 ) : 𝑋 –1-1-onto→ 𝑋 )