Theorem gagrpid

Description: The identity of the group does not alter the base set. (Contributed by Jeff Hankins, 11-Aug-2009) (Revised by Mario Carneiro, 13-Jan-2015)

	Ref	Expression
Hypothesis	gagrpid.1	⊢ 0 = ( 0g ‘ 𝐺 )
Assertion	gagrpid	⊢ ( ( ⊕ ∈ ( 𝐺 GrpAct 𝑌 ) ∧ 𝐴 ∈ 𝑌 ) → ( 0 ⊕ 𝐴 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		gagrpid.1	⊢ 0 = ( 0g ‘ 𝐺 )
2		eqid	⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
3		eqid	⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
4	2 3 1	isga	⊢ ( ⊕ ∈ ( 𝐺 GrpAct 𝑌 ) ↔ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ V ) ∧ ( ⊕ : ( ( Base ‘ 𝐺 ) × 𝑌 ) ⟶ 𝑌 ∧ ∀ 𝑥 ∈ 𝑌 ( ( 0 ⊕ 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐺 ) ∀ 𝑧 ∈ ( Base ‘ 𝐺 ) ( ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ⊕ 𝑥 ) = ( 𝑦 ⊕ ( 𝑧 ⊕ 𝑥 ) ) ) ) ) )
5	4	simprbi	⊢ ( ⊕ ∈ ( 𝐺 GrpAct 𝑌 ) → ( ⊕ : ( ( Base ‘ 𝐺 ) × 𝑌 ) ⟶ 𝑌 ∧ ∀ 𝑥 ∈ 𝑌 ( ( 0 ⊕ 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐺 ) ∀ 𝑧 ∈ ( Base ‘ 𝐺 ) ( ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ⊕ 𝑥 ) = ( 𝑦 ⊕ ( 𝑧 ⊕ 𝑥 ) ) ) ) )
6		simpl	⊢ ( ( ( 0 ⊕ 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐺 ) ∀ 𝑧 ∈ ( Base ‘ 𝐺 ) ( ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ⊕ 𝑥 ) = ( 𝑦 ⊕ ( 𝑧 ⊕ 𝑥 ) ) ) → ( 0 ⊕ 𝑥 ) = 𝑥 )
7	6	ralimi	⊢ ( ∀ 𝑥 ∈ 𝑌 ( ( 0 ⊕ 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐺 ) ∀ 𝑧 ∈ ( Base ‘ 𝐺 ) ( ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ⊕ 𝑥 ) = ( 𝑦 ⊕ ( 𝑧 ⊕ 𝑥 ) ) ) → ∀ 𝑥 ∈ 𝑌 ( 0 ⊕ 𝑥 ) = 𝑥 )
8	5 7	simpl2im	⊢ ( ⊕ ∈ ( 𝐺 GrpAct 𝑌 ) → ∀ 𝑥 ∈ 𝑌 ( 0 ⊕ 𝑥 ) = 𝑥 )
9		oveq2	⊢ ( 𝑥 = 𝐴 → ( 0 ⊕ 𝑥 ) = ( 0 ⊕ 𝐴 ) )
10		id	⊢ ( 𝑥 = 𝐴 → 𝑥 = 𝐴 )
11	9 10	eqeq12d	⊢ ( 𝑥 = 𝐴 → ( ( 0 ⊕ 𝑥 ) = 𝑥 ↔ ( 0 ⊕ 𝐴 ) = 𝐴 ) )
12	11	rspccva	⊢ ( ( ∀ 𝑥 ∈ 𝑌 ( 0 ⊕ 𝑥 ) = 𝑥 ∧ 𝐴 ∈ 𝑌 ) → ( 0 ⊕ 𝐴 ) = 𝐴 )
13	8 12	sylan	⊢ ( ( ⊕ ∈ ( 𝐺 GrpAct 𝑌 ) ∧ 𝐴 ∈ 𝑌 ) → ( 0 ⊕ 𝐴 ) = 𝐴 )