Theorem signstfval

Description: Value of the zero-skipping sign word. (Contributed by Thierry Arnoux, 8-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	signstfval	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑁 ) = ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... 𝑁 ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) )

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	1 2 3 4	signstfv	⊢ ( 𝐹 ∈ Word ℝ → ( 𝑇 ‘ 𝐹 ) = ( 𝑛 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↦ ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... 𝑛 ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) )
6	5	adantr	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑇 ‘ 𝐹 ) = ( 𝑛 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↦ ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... 𝑛 ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) )
7		simpr	⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ 𝑛 = 𝑁 ) → 𝑛 = 𝑁 )
8	7	oveq2d	⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ 𝑛 = 𝑁 ) → ( 0 ... 𝑛 ) = ( 0 ... 𝑁 ) )
9	8	mpteq1d	⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ 𝑛 = 𝑁 ) → ( 𝑖 ∈ ( 0 ... 𝑛 ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) = ( 𝑖 ∈ ( 0 ... 𝑁 ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) )
10	9	oveq2d	⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ 𝑛 = 𝑁 ) → ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... 𝑛 ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) = ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... 𝑁 ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) )
11		simpr	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
12		ovexd	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... 𝑁 ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ∈ V )
13	6 10 11 12	fvmptd	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑁 ) = ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... 𝑁 ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) )