Theorem signstfvcl

Description: Closure of the zero skipping sign in case the first letter is not zero. (Contributed by Thierry Arnoux, 10-Oct-2018)

	Ref	Expression
Hypotheses	signsv.p	⊢ ⨣ = ( 𝑎 ∈ { - 1 , 0 , 1 } , 𝑏 ∈ { - 1 , 0 , 1 } ↦ if ( 𝑏 = 0 , 𝑎 , 𝑏 ) )
	signsv.w	⊢ 𝑊 = { ⟨ ( Base ‘ ndx ) , { - 1 , 0 , 1 } ⟩ , ⟨ ( +g ‘ ndx ) , ⨣ ⟩ }
	signsv.t	⊢ 𝑇 = ( 𝑓 ∈ Word ℝ ↦ ( 𝑛 ∈ ( 0 ..^ ( ♯ ‘ 𝑓 ) ) ↦ ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... 𝑛 ) ↦ ( sgn ‘ ( 𝑓 ‘ 𝑖 ) ) ) ) ) )
	signsv.v	⊢ 𝑉 = ( 𝑓 ∈ Word ℝ ↦ Σ 𝑗 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) if ( ( ( 𝑇 ‘ 𝑓 ) ‘ 𝑗 ) ≠ ( ( 𝑇 ‘ 𝑓 ) ‘ ( 𝑗 − 1 ) ) , 1 , 0 ) )
Assertion	signstfvcl	⊢ ( ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑁 ) ∈ { - 1 , 1 } )

Proof

Step	Hyp	Ref	Expression
1		signsv.p	⊢ ⨣ = ( 𝑎 ∈ { - 1 , 0 , 1 } , 𝑏 ∈ { - 1 , 0 , 1 } ↦ if ( 𝑏 = 0 , 𝑎 , 𝑏 ) )
2		signsv.w	⊢ 𝑊 = { ⟨ ( Base ‘ ndx ) , { - 1 , 0 , 1 } ⟩ , ⟨ ( +g ‘ ndx ) , ⨣ ⟩ }
3		signsv.t	⊢ 𝑇 = ( 𝑓 ∈ Word ℝ ↦ ( 𝑛 ∈ ( 0 ..^ ( ♯ ‘ 𝑓 ) ) ↦ ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... 𝑛 ) ↦ ( sgn ‘ ( 𝑓 ‘ 𝑖 ) ) ) ) ) )
4		signsv.v	⊢ 𝑉 = ( 𝑓 ∈ Word ℝ ↦ Σ 𝑗 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) if ( ( ( 𝑇 ‘ 𝑓 ) ‘ 𝑗 ) ≠ ( ( 𝑇 ‘ 𝑓 ) ‘ ( 𝑗 − 1 ) ) , 1 , 0 ) )
5		simpll	⊢ ( ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) )
6	5	eldifad	⊢ ( ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → 𝐹 ∈ Word ℝ )
7	1 2 3 4	signstcl	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑁 ) ∈ { - 1 , 0 , 1 } )
8	6 7	sylancom	⊢ ( ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑁 ) ∈ { - 1 , 0 , 1 } )
9	1 2 3 4	signstfvneq0	⊢ ( ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑁 ) ≠ 0 )
10		eldifsn	⊢ ( ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑁 ) ∈ ( { - 1 , 0 , 1 } ∖ { 0 } ) ↔ ( ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑁 ) ∈ { - 1 , 0 , 1 } ∧ ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑁 ) ≠ 0 ) )
11	8 9 10	sylanbrc	⊢ ( ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑁 ) ∈ ( { - 1 , 0 , 1 } ∖ { 0 } ) )
12		tpcomb	⊢ { - 1 , 0 , 1 } = { - 1 , 1 , 0 }
13	12	difeq1i	⊢ ( { - 1 , 0 , 1 } ∖ { 0 } ) = ( { - 1 , 1 , 0 } ∖ { 0 } )
14		neg1ne0	⊢ - 1 ≠ 0
15		ax-1ne0	⊢ 1 ≠ 0
16		diftpsn3	⊢ ( ( - 1 ≠ 0 ∧ 1 ≠ 0 ) → ( { - 1 , 1 , 0 } ∖ { 0 } ) = { - 1 , 1 } )
17	14 15 16	mp2an	⊢ ( { - 1 , 1 , 0 } ∖ { 0 } ) = { - 1 , 1 }
18	13 17	eqtri	⊢ ( { - 1 , 0 , 1 } ∖ { 0 } ) = { - 1 , 1 }
19	11 18	eleqtrdi	⊢ ( ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑁 ) ∈ { - 1 , 1 } )