Metamath Proof Explorer

Theorem signstfvn

Description: Zero-skipping sign in a word compared to a shorter 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 signstfvn ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ 𝐾 ∈ ℝ ) → ( ( 𝑇 ‘ ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ) ‘ ( ♯ ‘ 𝐹 ) ) = ( ( ( 𝑇 ‘ 𝐹 ) ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ⨣ ( sgn ‘ 𝐾 ) ) )

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 signswbase ⊢ { - 1 , 0 , 1 } = ( Base ‘ 𝑊 )
6 1 2 signswmnd ⊢ 𝑊 ∈ Mnd
7 6 a1i ⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ 𝐾 ∈ ℝ ) → 𝑊 ∈ Mnd )
8 eldifi ⊢ ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) → 𝐹 ∈ Word ℝ )
9 lencl ⊢ ( 𝐹 ∈ Word ℝ → ( ♯ ‘ 𝐹 ) ∈ ℕ0 )
10 8 9 syl ⊢ ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) → ( ♯ ‘ 𝐹 ) ∈ ℕ0 )
11 eldifsn ⊢ ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ↔ ( 𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅ ) )
12 hasheq0 ⊢ ( 𝐹 ∈ Word ℝ → ( ( ♯ ‘ 𝐹 ) = 0 ↔ 𝐹 = ∅ ) )
13 12 necon3bid ⊢ ( 𝐹 ∈ Word ℝ → ( ( ♯ ‘ 𝐹 ) ≠ 0 ↔ 𝐹 ≠ ∅ ) )
14 13 biimpar ⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅ ) → ( ♯ ‘ 𝐹 ) ≠ 0 )
15 11 14 sylbi ⊢ ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) → ( ♯ ‘ 𝐹 ) ≠ 0 )
16 elnnne0 ⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ ↔ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( ♯ ‘ 𝐹 ) ≠ 0 ) )
17 10 15 16 sylanbrc ⊢ ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) → ( ♯ ‘ 𝐹 ) ∈ ℕ )
18 17 adantr ⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ 𝐾 ∈ ℝ ) → ( ♯ ‘ 𝐹 ) ∈ ℕ )
19 nnm1nn0 ⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ → ( ( ♯ ‘ 𝐹 ) − 1 ) ∈ ℕ0 )
20 18 19 syl ⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ 𝐾 ∈ ℝ ) → ( ( ♯ ‘ 𝐹 ) − 1 ) ∈ ℕ0 )
21 nn0uz ⊢ ℕ0 = ( ℤ≥ ‘ 0 )
22 20 21 eleqtrdi ⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ 𝐾 ∈ ℝ ) → ( ( ♯ ‘ 𝐹 ) − 1 ) ∈ ( ℤ≥ ‘ 0 ) )
23 ccatws1cl ⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) → ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ∈ Word ℝ )
24 23 adantr ⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) ∧ 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) → ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ∈ Word ℝ )
25 wrdf ⊢ ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ∈ Word ℝ → ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) : ( 0 ..^ ( ♯ ‘ ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ) ) ⟶ ℝ )
26 24 25 syl ⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) ∧ 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) → ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) : ( 0 ..^ ( ♯ ‘ ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ) ) ⟶ ℝ )
27 9 nn0zd ⊢ ( 𝐹 ∈ Word ℝ → ( ♯ ‘ 𝐹 ) ∈ ℤ )
28 fzoval ⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℤ → ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) )
29 27 28 syl ⊢ ( 𝐹 ∈ Word ℝ → ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) )
30 29 adantr ⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) → ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) )
31 fzossfz ⊢ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⊆ ( 0 ... ( ♯ ‘ 𝐹 ) )
32 30 31 eqsstrrdi ⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) → ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ⊆ ( 0 ... ( ♯ ‘ 𝐹 ) ) )
33 s1cl ⊢ ( 𝐾 ∈ ℝ → ⟨“ 𝐾 ”⟩ ∈ Word ℝ )
34 ccatlen ⊢ ( ( 𝐹 ∈ Word ℝ ∧ ⟨“ 𝐾 ”⟩ ∈ Word ℝ ) → ( ♯ ‘ ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ) = ( ( ♯ ‘ 𝐹 ) + ( ♯ ‘ ⟨“ 𝐾 ”⟩ ) ) )
35 33 34 sylan2 ⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) → ( ♯ ‘ ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ) = ( ( ♯ ‘ 𝐹 ) + ( ♯ ‘ ⟨“ 𝐾 ”⟩ ) ) )
36 s1len ⊢ ( ♯ ‘ ⟨“ 𝐾 ”⟩ ) = 1
37 36 oveq2i ⊢ ( ( ♯ ‘ 𝐹 ) + ( ♯ ‘ ⟨“ 𝐾 ”⟩ ) ) = ( ( ♯ ‘ 𝐹 ) + 1 )
38 35 37 eqtrdi ⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) → ( ♯ ‘ ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ) = ( ( ♯ ‘ 𝐹 ) + 1 ) )
39 38 oveq2d ⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) → ( 0 ..^ ( ♯ ‘ ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ) ) = ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) )
40 27 adantr ⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) → ( ♯ ‘ 𝐹 ) ∈ ℤ )
41 40 peano2zd ⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) → ( ( ♯ ‘ 𝐹 ) + 1 ) ∈ ℤ )
42 fzoval ⊢ ( ( ( ♯ ‘ 𝐹 ) + 1 ) ∈ ℤ → ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) = ( 0 ... ( ( ( ♯ ‘ 𝐹 ) + 1 ) − 1 ) ) )
43 41 42 syl ⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) → ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) = ( 0 ... ( ( ( ♯ ‘ 𝐹 ) + 1 ) − 1 ) ) )
44 9 nn0cnd ⊢ ( 𝐹 ∈ Word ℝ → ( ♯ ‘ 𝐹 ) ∈ ℂ )
45 1cnd ⊢ ( 𝐹 ∈ Word ℝ → 1 ∈ ℂ )
46 44 45 pncand ⊢ ( 𝐹 ∈ Word ℝ → ( ( ( ♯ ‘ 𝐹 ) + 1 ) − 1 ) = ( ♯ ‘ 𝐹 ) )
47 46 adantr ⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) → ( ( ( ♯ ‘ 𝐹 ) + 1 ) − 1 ) = ( ♯ ‘ 𝐹 ) )
48 47 oveq2d ⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) → ( 0 ... ( ( ( ♯ ‘ 𝐹 ) + 1 ) − 1 ) ) = ( 0 ... ( ♯ ‘ 𝐹 ) ) )
49 39 43 48 3eqtrd ⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) → ( 0 ..^ ( ♯ ‘ ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ) ) = ( 0 ... ( ♯ ‘ 𝐹 ) ) )
50 32 49 sseqtrrd ⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) → ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ⊆ ( 0 ..^ ( ♯ ‘ ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ) ) )
51 50 sselda ⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) ∧ 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) → 𝑖 ∈ ( 0 ..^ ( ♯ ‘ ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ) ) )
52 26 51 ffvelrnd ⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) ∧ 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) → ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) ∈ ℝ )
53 8 52 sylanl1 ⊢ ( ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ 𝐾 ∈ ℝ ) ∧ 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) → ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) ∈ ℝ )
54 53 rexrd ⊢ ( ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ 𝐾 ∈ ℝ ) ∧ 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) → ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) ∈ ℝ* )
55 sgncl ⊢ ( ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) ∈ ℝ* → ( sgn ‘ ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) ) ∈ { - 1 , 0 , 1 } )
56 54 55 syl ⊢ ( ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ 𝐾 ∈ ℝ ) ∧ 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) → ( sgn ‘ ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) ) ∈ { - 1 , 0 , 1 } )
57 1 2 signswplusg ⊢ ⨣ = ( +g ‘ 𝑊 )
58 rexr ⊢ ( 𝐾 ∈ ℝ → 𝐾 ∈ ℝ* )
59 sgncl ⊢ ( 𝐾 ∈ ℝ* → ( sgn ‘ 𝐾 ) ∈ { - 1 , 0 , 1 } )
60 58 59 syl ⊢ ( 𝐾 ∈ ℝ → ( sgn ‘ 𝐾 ) ∈ { - 1 , 0 , 1 } )
61 60 adantl ⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ 𝐾 ∈ ℝ ) → ( sgn ‘ 𝐾 ) ∈ { - 1 , 0 , 1 } )
62 id ⊢ ( 𝑖 = ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) → 𝑖 = ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) )
63 44 45 npcand ⊢ ( 𝐹 ∈ Word ℝ → ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) = ( ♯ ‘ 𝐹 ) )
64 63 adantr ⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) → ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) = ( ♯ ‘ 𝐹 ) )
65 62 64 sylan9eqr ⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) ∧ 𝑖 = ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) ) → 𝑖 = ( ♯ ‘ 𝐹 ) )
66 65 fveq2d ⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) ∧ 𝑖 = ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) ) → ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) = ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ ( ♯ ‘ 𝐹 ) ) )
67 ccatws1ls ⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) → ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ ( ♯ ‘ 𝐹 ) ) = 𝐾 )
68 67 adantr ⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) ∧ 𝑖 = ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) ) → ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ ( ♯ ‘ 𝐹 ) ) = 𝐾 )
69 66 68 eqtrd ⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) ∧ 𝑖 = ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) ) → ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) = 𝐾 )
70 8 69 sylanl1 ⊢ ( ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ 𝐾 ∈ ℝ ) ∧ 𝑖 = ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) ) → ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) = 𝐾 )
71 70 fveq2d ⊢ ( ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ 𝐾 ∈ ℝ ) ∧ 𝑖 = ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) ) → ( sgn ‘ ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) ) = ( sgn ‘ 𝐾 ) )
72 5 7 22 56 57 61 71 gsumnunsn ⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ 𝐾 ∈ ℝ ) → ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) ) ↦ ( sgn ‘ ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) ) ) ) = ( ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ↦ ( sgn ‘ ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) ) ) ) ⨣ ( sgn ‘ 𝐾 ) ) )
73 8 63 syl ⊢ ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) → ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) = ( ♯ ‘ 𝐹 ) )
74 73 adantr ⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ 𝐾 ∈ ℝ ) → ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) = ( ♯ ‘ 𝐹 ) )
75 74 oveq2d ⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ 𝐾 ∈ ℝ ) → ( 0 ... ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) ) = ( 0 ... ( ♯ ‘ 𝐹 ) ) )
76 75 mpteq1d ⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ 𝐾 ∈ ℝ ) → ( 𝑖 ∈ ( 0 ... ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) ) ↦ ( sgn ‘ ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) ) ) = ( 𝑖 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ↦ ( sgn ‘ ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) ) ) )
77 76 oveq2d ⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ 𝐾 ∈ ℝ ) → ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... ( ( ( ♯ ‘ 𝐹 ) − 1 ) + 1 ) ) ↦ ( sgn ‘ ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) ) ) ) = ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ↦ ( sgn ‘ ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) ) ) ) )
78 simpll ⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) ∧ 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) → 𝐹 ∈ Word ℝ )
79 33 ad2antlr ⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) ∧ 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) → ⟨“ 𝐾 ”⟩ ∈ Word ℝ )
80 30 eleq2d ⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) → ( 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↔ 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) )
81 80 biimpar ⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) ∧ 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) → 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
82 ccatval1 ⊢ ( ( 𝐹 ∈ Word ℝ ∧ ⟨“ 𝐾 ”⟩ ∈ Word ℝ ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) = ( 𝐹 ‘ 𝑖 ) )
83 78 79 81 82 syl3anc ⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) ∧ 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) → ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) = ( 𝐹 ‘ 𝑖 ) )
84 83 fveq2d ⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) ∧ 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ) → ( sgn ‘ ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) ) = ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) )
85 84 mpteq2dva ⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) → ( 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ↦ ( sgn ‘ ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) ) ) = ( 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) )
86 8 85 sylan ⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ 𝐾 ∈ ℝ ) → ( 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ↦ ( sgn ‘ ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) ) ) = ( 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) )
87 86 oveq2d ⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ 𝐾 ∈ ℝ ) → ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ↦ ( sgn ‘ ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) ) ) ) = ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) )
88 87 oveq1d ⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ 𝐾 ∈ ℝ ) → ( ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ↦ ( sgn ‘ ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) ) ) ) ⨣ ( sgn ‘ 𝐾 ) ) = ( ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ⨣ ( sgn ‘ 𝐾 ) ) )
89 72 77 88 3eqtr3d ⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ 𝐾 ∈ ℝ ) → ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ↦ ( sgn ‘ ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) ) ) ) = ( ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ⨣ ( sgn ‘ 𝐾 ) ) )
90 eqid ⊢ ( ♯ ‘ 𝐹 ) = ( ♯ ‘ 𝐹 )
91 90 olci ⊢ ( ( ♯ ‘ 𝐹 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∨ ( ♯ ‘ 𝐹 ) = ( ♯ ‘ 𝐹 ) )
92 9 21 eleqtrdi ⊢ ( 𝐹 ∈ Word ℝ → ( ♯ ‘ 𝐹 ) ∈ ( ℤ≥ ‘ 0 ) )
93 fzosplitsni ⊢ ( ( ♯ ‘ 𝐹 ) ∈ ( ℤ≥ ‘ 0 ) → ( ( ♯ ‘ 𝐹 ) ∈ ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ↔ ( ( ♯ ‘ 𝐹 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∨ ( ♯ ‘ 𝐹 ) = ( ♯ ‘ 𝐹 ) ) ) )
94 92 93 syl ⊢ ( 𝐹 ∈ Word ℝ → ( ( ♯ ‘ 𝐹 ) ∈ ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ↔ ( ( ♯ ‘ 𝐹 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∨ ( ♯ ‘ 𝐹 ) = ( ♯ ‘ 𝐹 ) ) ) )
95 91 94 mpbiri ⊢ ( 𝐹 ∈ Word ℝ → ( ♯ ‘ 𝐹 ) ∈ ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) )
96 95 adantr ⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) → ( ♯ ‘ 𝐹 ) ∈ ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) )
97 96 39 eleqtrrd ⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) → ( ♯ ‘ 𝐹 ) ∈ ( 0 ..^ ( ♯ ‘ ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ) ) )
98 1 2 3 4 signstfval ⊢ ( ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ∈ Word ℝ ∧ ( ♯ ‘ 𝐹 ) ∈ ( 0 ..^ ( ♯ ‘ ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ) ) ) → ( ( 𝑇 ‘ ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ) ‘ ( ♯ ‘ 𝐹 ) ) = ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ↦ ( sgn ‘ ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) ) ) ) )
99 23 97 98 syl2anc ⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ) → ( ( 𝑇 ‘ ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ) ‘ ( ♯ ‘ 𝐹 ) ) = ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ↦ ( sgn ‘ ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) ) ) ) )
100 8 99 sylan ⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ 𝐾 ∈ ℝ ) → ( ( 𝑇 ‘ ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ) ‘ ( ♯ ‘ 𝐹 ) ) = ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ↦ ( sgn ‘ ( ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ‘ 𝑖 ) ) ) ) )
101 fzo0end ⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ → ( ( ♯ ‘ 𝐹 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
102 17 101 syl ⊢ ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) → ( ( ♯ ‘ 𝐹 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
103 1 2 3 4 signstfval ⊢ ( ( 𝐹 ∈ Word ℝ ∧ ( ( ♯ ‘ 𝐹 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑇 ‘ 𝐹 ) ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) = ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) )
104 8 102 103 syl2anc ⊢ ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) → ( ( 𝑇 ‘ 𝐹 ) ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) = ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) )
105 104 adantr ⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ 𝐾 ∈ ℝ ) → ( ( 𝑇 ‘ 𝐹 ) ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) = ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) )
106 105 oveq1d ⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ 𝐾 ∈ ℝ ) → ( ( ( 𝑇 ‘ 𝐹 ) ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ⨣ ( sgn ‘ 𝐾 ) ) = ( ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... ( ( ♯ ‘ 𝐹 ) − 1 ) ) ↦ ( sgn ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ⨣ ( sgn ‘ 𝐾 ) ) )
107 89 100 106 3eqtr4d ⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ 𝐾 ∈ ℝ ) → ( ( 𝑇 ‘ ( 𝐹 ++ ⟨“ 𝐾 ”⟩ ) ) ‘ ( ♯ ‘ 𝐹 ) ) = ( ( ( 𝑇 ‘ 𝐹 ) ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ⨣ ( sgn ‘ 𝐾 ) ) )