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Metamath Proof Explorer

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 )