Theorem signswbase

Description: The base of W is the unordered triple reprensenting the possible signs. (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	signswbase	⊢ { - 1 , 0 , 1 } = ( Base ‘ 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	2	grpbase	⊢ ( { - 1 , 0 , 1 } ∈ V → { - 1 , 0 , 1 } = ( Base ‘ 𝑊 ) )
5	3 4	ax-mp	⊢ { - 1 , 0 , 1 } = ( Base ‘ 𝑊 )