signstcl

Description: Closure of the zero skipping sign word. (Contributed by Thierry Arnoux, 9-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 Σ_g ( 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	signstcl	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑁 ) ∈ { - 1 , 0 , 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 Σ_g ( 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	signstfval	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑁 ) = ( 𝑊 Σ_g ( 𝑖 ∈ ( 0 ... 𝑁 ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) )
6	1 2	signswbase	⊢ { - 1 , 0 , 1 } = ( Base ‘ 𝑊 )
7	1 2	signswmnd	⊢ 𝑊 ∈ Mnd
8	7	a1i	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → 𝑊 ∈ Mnd )
9		fzo0ssnn0	⊢ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⊆ ℕ₀
10		nn0uz	⊢ ℕ₀ = ( ℤ_≥ ‘ 0 )
11	9 10	sseqtri	⊢ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⊆ ( ℤ_≥ ‘ 0 )
12	11	a1i	⊢ ( 𝐹 ∈ Word ℝ → ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⊆ ( ℤ_≥ ‘ 0 ) )
13	12	sselda	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → 𝑁 ∈ ( ℤ_≥ ‘ 0 ) )
14		wrdf	⊢ ( 𝐹 ∈ Word ℝ → 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ ℝ )
15	14	ad2antrr	⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑁 ) ) → 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ ℝ )
16		fzssfzo	⊢ ( 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 0 ... 𝑁 ) ⊆ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
17	16	adantl	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 0 ... 𝑁 ) ⊆ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
18	17	sselda	⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑁 ) ) → 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
19	15 18	ffvelcdmd	⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑖 ) ∈ ℝ )
20	19	rexrd	⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑖 ) ∈ ℝ^* )
21		sgncl	⊢ ( ( 𝐹 ‘ 𝑖 ) ∈ ℝ^* → ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ∈ { - 1 , 0 , 1 } )
22	20 21	syl	⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ 𝑖 ∈ ( 0 ... 𝑁 ) ) → ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ∈ { - 1 , 0 , 1 } )
23	6 8 13 22	gsumncl	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑊 Σ_g ( 𝑖 ∈ ( 0 ... 𝑁 ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ∈ { - 1 , 0 , 1 } )
24	5 23	eqeltrd	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑁 ) ∈ { - 1 , 0 , 1 } )

Metamath Proof Explorer

Theorem signstcl

Proof