Theorem signstlen

Description: Length 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	signstlen	⊢ ( 𝐹 ∈ Word ℝ → ( ♯ ‘ ( 𝑇 ‘ 𝐹 ) ) = ( ♯ ‘ 𝐹 ) )

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		ovex	⊢ ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... 𝑛 ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ∈ V
6		eqid	⊢ ( 𝑛 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↦ ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... 𝑛 ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) = ( 𝑛 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↦ ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... 𝑛 ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) )
7	5 6	fnmpti	⊢ ( 𝑛 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↦ ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... 𝑛 ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) Fn ( 0 ..^ ( ♯ ‘ 𝐹 ) )
8	1 2 3 4	signstfv	⊢ ( 𝐹 ∈ Word ℝ → ( 𝑇 ‘ 𝐹 ) = ( 𝑛 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↦ ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... 𝑛 ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) )
9	8	fneq1d	⊢ ( 𝐹 ∈ Word ℝ → ( ( 𝑇 ‘ 𝐹 ) Fn ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↔ ( 𝑛 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↦ ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... 𝑛 ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) Fn ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) )
10	7 9	mpbiri	⊢ ( 𝐹 ∈ Word ℝ → ( 𝑇 ‘ 𝐹 ) Fn ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
11		hashfn	⊢ ( ( 𝑇 ‘ 𝐹 ) Fn ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( ♯ ‘ ( 𝑇 ‘ 𝐹 ) ) = ( ♯ ‘ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) )
12	10 11	syl	⊢ ( 𝐹 ∈ Word ℝ → ( ♯ ‘ ( 𝑇 ‘ 𝐹 ) ) = ( ♯ ‘ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) )
13		lencl	⊢ ( 𝐹 ∈ Word ℝ → ( ♯ ‘ 𝐹 ) ∈ ℕ0 )
14		hashfzo0	⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 → ( ♯ ‘ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) = ( ♯ ‘ 𝐹 ) )
15	13 14	syl	⊢ ( 𝐹 ∈ Word ℝ → ( ♯ ‘ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) = ( ♯ ‘ 𝐹 ) )
16	12 15	eqtrd	⊢ ( 𝐹 ∈ Word ℝ → ( ♯ ‘ ( 𝑇 ‘ 𝐹 ) ) = ( ♯ ‘ 𝐹 ) )