Theorem signsvvfval

Description: The value of V , which represents the number of times the sign changes in a word. (Contributed by Thierry Arnoux, 7-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	signsvvfval	⊢ ( 𝐹 ∈ Word ℝ → ( 𝑉 ‘ 𝐹 ) = Σ 𝑗 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) if ( ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑗 ) ≠ ( ( 𝑇 ‘ 𝐹 ) ‘ ( 𝑗 − 1 ) ) , 1 , 0 ) )

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		fveq2	⊢ ( 𝑓 = 𝐹 → ( ♯ ‘ 𝑓 ) = ( ♯ ‘ 𝐹 ) )
6	5	oveq2d	⊢ ( 𝑓 = 𝐹 → ( 1 ..^ ( ♯ ‘ 𝑓 ) ) = ( 1 ..^ ( ♯ ‘ 𝐹 ) ) )
7		fveq2	⊢ ( 𝑓 = 𝐹 → ( 𝑇 ‘ 𝑓 ) = ( 𝑇 ‘ 𝐹 ) )
8	7	fveq1d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑇 ‘ 𝑓 ) ‘ 𝑗 ) = ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑗 ) )
9	7	fveq1d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑇 ‘ 𝑓 ) ‘ ( 𝑗 − 1 ) ) = ( ( 𝑇 ‘ 𝐹 ) ‘ ( 𝑗 − 1 ) ) )
10	8 9	neeq12d	⊢ ( 𝑓 = 𝐹 → ( ( ( 𝑇 ‘ 𝑓 ) ‘ 𝑗 ) ≠ ( ( 𝑇 ‘ 𝑓 ) ‘ ( 𝑗 − 1 ) ) ↔ ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑗 ) ≠ ( ( 𝑇 ‘ 𝐹 ) ‘ ( 𝑗 − 1 ) ) ) )
11	10	ifbid	⊢ ( 𝑓 = 𝐹 → if ( ( ( 𝑇 ‘ 𝑓 ) ‘ 𝑗 ) ≠ ( ( 𝑇 ‘ 𝑓 ) ‘ ( 𝑗 − 1 ) ) , 1 , 0 ) = if ( ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑗 ) ≠ ( ( 𝑇 ‘ 𝐹 ) ‘ ( 𝑗 − 1 ) ) , 1 , 0 ) )
12	11	adantr	⊢ ( ( 𝑓 = 𝐹 ∧ 𝑗 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) → if ( ( ( 𝑇 ‘ 𝑓 ) ‘ 𝑗 ) ≠ ( ( 𝑇 ‘ 𝑓 ) ‘ ( 𝑗 − 1 ) ) , 1 , 0 ) = if ( ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑗 ) ≠ ( ( 𝑇 ‘ 𝐹 ) ‘ ( 𝑗 − 1 ) ) , 1 , 0 ) )
13	6 12	sumeq12dv	⊢ ( 𝑓 = 𝐹 → Σ 𝑗 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) if ( ( ( 𝑇 ‘ 𝑓 ) ‘ 𝑗 ) ≠ ( ( 𝑇 ‘ 𝑓 ) ‘ ( 𝑗 − 1 ) ) , 1 , 0 ) = Σ 𝑗 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) if ( ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑗 ) ≠ ( ( 𝑇 ‘ 𝐹 ) ‘ ( 𝑗 − 1 ) ) , 1 , 0 ) )
14		sumex	⊢ Σ 𝑗 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) if ( ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑗 ) ≠ ( ( 𝑇 ‘ 𝐹 ) ‘ ( 𝑗 − 1 ) ) , 1 , 0 ) ∈ V
15	13 4 14	fvmpt	⊢ ( 𝐹 ∈ Word ℝ → ( 𝑉 ‘ 𝐹 ) = Σ 𝑗 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) if ( ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑗 ) ≠ ( ( 𝑇 ‘ 𝐹 ) ‘ ( 𝑗 − 1 ) ) , 1 , 0 ) )