Theorem signswplusg

Description: The operation of W . (Contributed by Thierry Arnoux, 9-Sep-2018)

	Ref	Expression
Hypotheses	signsw.p	⊢ ⨣ = ( 𝑎 ∈ { - 1 , 0 , 1 } , 𝑏 ∈ { - 1 , 0 , 1 } ↦ if ( 𝑏 = 0 , 𝑎 , 𝑏 ) )
	signsw.w	⊢ 𝑊 = { ⟨ ( Base ‘ ndx ) , { - 1 , 0 , 1 } ⟩ , ⟨ ( +g ‘ ndx ) , ⨣ ⟩ }
Assertion	signswplusg	⊢ ⨣ = ( +g ‘ 𝑊 )

Proof

Step	Hyp	Ref	Expression
1		signsw.p	⊢ ⨣ = ( 𝑎 ∈ { - 1 , 0 , 1 } , 𝑏 ∈ { - 1 , 0 , 1 } ↦ if ( 𝑏 = 0 , 𝑎 , 𝑏 ) )
2		signsw.w	⊢ 𝑊 = { ⟨ ( Base ‘ ndx ) , { - 1 , 0 , 1 } ⟩ , ⟨ ( +g ‘ ndx ) , ⨣ ⟩ }
3		tpex	⊢ { - 1 , 0 , 1 } ∈ V
4	3 3	mpoex	⊢ ( 𝑎 ∈ { - 1 , 0 , 1 } , 𝑏 ∈ { - 1 , 0 , 1 } ↦ if ( 𝑏 = 0 , 𝑎 , 𝑏 ) ) ∈ V
5	1 4	eqeltri	⊢ ⨣ ∈ V
6	2	grpplusg	⊢ ( ⨣ ∈ V → ⨣ = ( +g ‘ 𝑊 ) )
7	5 6	ax-mp	⊢ ⨣ = ( +g ‘ 𝑊 )