Theorem signstf

Description: The zero skipping sign word is a 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	signstf	⊢ ( 𝐹 ∈ Word ℝ → ( 𝑇 ‘ 𝐹 ) ∈ Word ℝ )

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		neg1rr	⊢ - 1 ∈ ℝ
7		0re	⊢ 0 ∈ ℝ
8		1re	⊢ 1 ∈ ℝ
9		tpssi	⊢ ( ( - 1 ∈ ℝ ∧ 0 ∈ ℝ ∧ 1 ∈ ℝ ) → { - 1 , 0 , 1 } ⊆ ℝ )
10	6 7 8 9	mp3an	⊢ { - 1 , 0 , 1 } ⊆ ℝ
11	1 2	signswbase	⊢ { - 1 , 0 , 1 } = ( Base ‘ 𝑊 )
12	1 2	signswmnd	⊢ 𝑊 ∈ Mnd
13	12	a1i	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝑛 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → 𝑊 ∈ Mnd )
14		fzo0ssnn0	⊢ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⊆ ℕ0
15		nn0uz	⊢ ℕ0 = ( ℤ≥ ‘ 0 )
16	14 15	sseqtri	⊢ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⊆ ( ℤ≥ ‘ 0 )
17	16	a1i	⊢ ( 𝐹 ∈ Word ℝ → ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⊆ ( ℤ≥ ‘ 0 ) )
18	17	sselda	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝑛 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → 𝑛 ∈ ( ℤ≥ ‘ 0 ) )
19		wrdf	⊢ ( 𝐹 ∈ Word ℝ → 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ ℝ )
20	19	ad2antrr	⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝑛 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑛 ) ) → 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ ℝ )
21		fzssfzo	⊢ ( 𝑛 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 0 ... 𝑛 ) ⊆ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
22	21	adantl	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝑛 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 0 ... 𝑛 ) ⊆ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
23	22	sselda	⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝑛 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑛 ) ) → 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
24	20 23	ffvelrnd	⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝑛 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑛 ) ) → ( 𝐹 ‘ 𝑖 ) ∈ ℝ )
25	24	rexrd	⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝑛 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑛 ) ) → ( 𝐹 ‘ 𝑖 ) ∈ ℝ* )
26		sgncl	⊢ ( ( 𝐹 ‘ 𝑖 ) ∈ ℝ* → ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ∈ { - 1 , 0 , 1 } )
27	25 26	syl	⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝑛 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑛 ) ) → ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ∈ { - 1 , 0 , 1 } )
28	11 13 18 27	gsumncl	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝑛 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... 𝑛 ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ∈ { - 1 , 0 , 1 } )
29	10 28	sselid	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝑛 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... 𝑛 ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ∈ ℝ )
30	5 29	fmpt3d	⊢ ( 𝐹 ∈ Word ℝ → ( 𝑇 ‘ 𝐹 ) : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ ℝ )
31		iswrdi	⊢ ( ( 𝑇 ‘ 𝐹 ) : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ ℝ → ( 𝑇 ‘ 𝐹 ) ∈ Word ℝ )
32	30 31	syl	⊢ ( 𝐹 ∈ Word ℝ → ( 𝑇 ‘ 𝐹 ) ∈ Word ℝ )