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 )