Theorem signswrid

Description: The zero-skipping operation propagages nonzeros. (Contributed by Thierry Arnoux, 11-Oct-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	signswrid	⊢ ( 𝑋 ∈ { - 1 , 0 , 1 } → ( 𝑋 ⨣ 0 ) = 𝑋 )

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		c0ex	⊢ 0 ∈ V
4	3	tpid2	⊢ 0 ∈ { - 1 , 0 , 1 }
5	1	signspval	⊢ ( ( 𝑋 ∈ { - 1 , 0 , 1 } ∧ 0 ∈ { - 1 , 0 , 1 } ) → ( 𝑋 ⨣ 0 ) = if ( 0 = 0 , 𝑋 , 0 ) )
6	4 5	mpan2	⊢ ( 𝑋 ∈ { - 1 , 0 , 1 } → ( 𝑋 ⨣ 0 ) = if ( 0 = 0 , 𝑋 , 0 ) )
7		eqid	⊢ 0 = 0
8		iftrue	⊢ ( 0 = 0 → if ( 0 = 0 , 𝑋 , 0 ) = 𝑋 )
9	7 8	mp1i	⊢ ( 𝑋 ∈ { - 1 , 0 , 1 } → if ( 0 = 0 , 𝑋 , 0 ) = 𝑋 )
10	6 9	eqtrd	⊢ ( 𝑋 ∈ { - 1 , 0 , 1 } → ( 𝑋 ⨣ 0 ) = 𝑋 )