Theorem signswn0

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	signswn0	⊢ ( ( ( 𝑋 ∈ { - 1 , 0 , 1 } ∧ 𝑌 ∈ { - 1 , 0 , 1 } ) ∧ 𝑋 ≠ 0 ) → ( 𝑋 ⨣ 𝑌 ) ≠ 0 )

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	1	signspval	⊢ ( ( 𝑋 ∈ { - 1 , 0 , 1 } ∧ 𝑌 ∈ { - 1 , 0 , 1 } ) → ( 𝑋 ⨣ 𝑌 ) = if ( 𝑌 = 0 , 𝑋 , 𝑌 ) )
4	3	adantr	⊢ ( ( ( 𝑋 ∈ { - 1 , 0 , 1 } ∧ 𝑌 ∈ { - 1 , 0 , 1 } ) ∧ 𝑋 ≠ 0 ) → ( 𝑋 ⨣ 𝑌 ) = if ( 𝑌 = 0 , 𝑋 , 𝑌 ) )
5		neeq1	⊢ ( 𝑋 = if ( 𝑌 = 0 , 𝑋 , 𝑌 ) → ( 𝑋 ≠ 0 ↔ if ( 𝑌 = 0 , 𝑋 , 𝑌 ) ≠ 0 ) )
6		neeq1	⊢ ( 𝑌 = if ( 𝑌 = 0 , 𝑋 , 𝑌 ) → ( 𝑌 ≠ 0 ↔ if ( 𝑌 = 0 , 𝑋 , 𝑌 ) ≠ 0 ) )
7		simplr	⊢ ( ( ( ( 𝑋 ∈ { - 1 , 0 , 1 } ∧ 𝑌 ∈ { - 1 , 0 , 1 } ) ∧ 𝑋 ≠ 0 ) ∧ 𝑌 = 0 ) → 𝑋 ≠ 0 )
8		simpr	⊢ ( ( ( ( 𝑋 ∈ { - 1 , 0 , 1 } ∧ 𝑌 ∈ { - 1 , 0 , 1 } ) ∧ 𝑋 ≠ 0 ) ∧ ¬ 𝑌 = 0 ) → ¬ 𝑌 = 0 )
9	8	neqned	⊢ ( ( ( ( 𝑋 ∈ { - 1 , 0 , 1 } ∧ 𝑌 ∈ { - 1 , 0 , 1 } ) ∧ 𝑋 ≠ 0 ) ∧ ¬ 𝑌 = 0 ) → 𝑌 ≠ 0 )
10	5 6 7 9	ifbothda	⊢ ( ( ( 𝑋 ∈ { - 1 , 0 , 1 } ∧ 𝑌 ∈ { - 1 , 0 , 1 } ) ∧ 𝑋 ≠ 0 ) → if ( 𝑌 = 0 , 𝑋 , 𝑌 ) ≠ 0 )
11	4 10	eqnetrd	⊢ ( ( ( 𝑋 ∈ { - 1 , 0 , 1 } ∧ 𝑌 ∈ { - 1 , 0 , 1 } ) ∧ 𝑋 ≠ 0 ) → ( 𝑋 ⨣ 𝑌 ) ≠ 0 )