Metamath Proof Explorer

Theorem 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 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 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 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 signstfval ⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑁 ) = ( 𝑊 Σg ( 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 ffvelrnd ⊢ ( ( ( 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 Σg ( 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 } )