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Metamath Proof Explorer

Theorem gaf

Description: The mapping of the group action operation. (Contributed by Jeff Hankins, 11-Aug-2009) (Revised by Mario Carneiro, 13-Jan-2015)

Ref Expression
Hypothesis gaf.1 ⊢ 𝑋 = ( Base ‘ 𝐺 )
Assertion gaf ( ⊕ ∈ ( 𝐺 GrpAct 𝑌 ) → ⊕ : ( 𝑋 × 𝑌 ) ⟶ 𝑌 )

Proof

Step Hyp Ref Expression
1 gaf.1 ⊢ 𝑋 = ( Base ‘ 𝐺 )
2 eqid ⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
3 eqid ⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
4 1 2 3 isga ⊢ ( ⊕ ∈ ( 𝐺 GrpAct 𝑌 ) ↔ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ V ) ∧ ( ⊕ : ( 𝑋 × 𝑌 ) ⟶ 𝑌 ∧ ∀ 𝑥 ∈ 𝑌 ( ( ( 0g ‘ 𝐺 ) ⊕ 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ⊕ 𝑥 ) = ( 𝑦 ⊕ ( 𝑧 ⊕ 𝑥 ) ) ) ) ) )
5 4 simprbi ⊢ ( ⊕ ∈ ( 𝐺 GrpAct 𝑌 ) → ( ⊕ : ( 𝑋 × 𝑌 ) ⟶ 𝑌 ∧ ∀ 𝑥 ∈ 𝑌 ( ( ( 0g ‘ 𝐺 ) ⊕ 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ⊕ 𝑥 ) = ( 𝑦 ⊕ ( 𝑧 ⊕ 𝑥 ) ) ) ) )
6 5 simpld ⊢ ( ⊕ ∈ ( 𝐺 GrpAct 𝑌 ) → ⊕ : ( 𝑋 × 𝑌 ) ⟶ 𝑌 )