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Metamath Proof Explorer

Theorem grpinvf

Description: The group inversion operation is a function on the base set. (Contributed by Mario Carneiro, 4-May-2015)

Ref Expression
Hypotheses grpinvcl.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
grpinvcl.n ⊢ 𝑁 = ( invg ‘ 𝐺 )
Assertion grpinvf ( 𝐺 ∈ Grp → 𝑁 : 𝐵 ⟶ 𝐵 )

Proof

Step Hyp Ref Expression
1 grpinvcl.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 grpinvcl.n ⊢ 𝑁 = ( invg ‘ 𝐺 )
3 eqid ⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
4 eqid ⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
5 1 3 4 grpinveu ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ) → ∃! 𝑦 ∈ 𝐵 ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) )
6 riotacl ⊢ ( ∃! 𝑦 ∈ 𝐵 ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) → ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) ) ∈ 𝐵 )
7 5 6 syl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ) → ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) ) ∈ 𝐵 )
8 1 3 4 2 grpinvfval ⊢ 𝑁 = ( 𝑥 ∈ 𝐵 ↦ ( ℩ 𝑦 ∈ 𝐵 ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) ) )
9 7 8 fmptd ⊢ ( 𝐺 ∈ Grp → 𝑁 : 𝐵 ⟶ 𝐵 )