signshf

Description: H , corresponding to the word F multiplied by ( x - C ) , as a function. (Contributed by Thierry Arnoux, 29-Sep-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 ) )
	signs.h	⊢ 𝐻 = ( ( ⟨“ 0 ”⟩ ++ 𝐹 ) ∘_f − ( ( 𝐹 ++ ⟨“ 0 ”⟩ ) ∘_f/c · 𝐶 ) )
Assertion	signshf	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺ ) → 𝐻 : ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ⟶ 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 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		signs.h	⊢ 𝐻 = ( ( ⟨“ 0 ”⟩ ++ 𝐹 ) ∘_f − ( ( 𝐹 ++ ⟨“ 0 ”⟩ ) ∘_f/c · 𝐶 ) )
6		resubcl	⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 − 𝑦 ) ∈ ℝ )
7	6	adantl	⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺ ) ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) → ( 𝑥 − 𝑦 ) ∈ ℝ )
8		0re	⊢ 0 ∈ ℝ
9		s1cl	⊢ ( 0 ∈ ℝ → ⟨“ 0 ”⟩ ∈ Word ℝ )
10	8 9	ax-mp	⊢ ⟨“ 0 ”⟩ ∈ Word ℝ
11		ccatcl	⊢ ( ( ⟨“ 0 ”⟩ ∈ Word ℝ ∧ 𝐹 ∈ Word ℝ ) → ( ⟨“ 0 ”⟩ ++ 𝐹 ) ∈ Word ℝ )
12	10 11	mpan	⊢ ( 𝐹 ∈ Word ℝ → ( ⟨“ 0 ”⟩ ++ 𝐹 ) ∈ Word ℝ )
13		wrdf	⊢ ( ( ⟨“ 0 ”⟩ ++ 𝐹 ) ∈ Word ℝ → ( ⟨“ 0 ”⟩ ++ 𝐹 ) : ( 0 ..^ ( ♯ ‘ ( ⟨“ 0 ”⟩ ++ 𝐹 ) ) ) ⟶ ℝ )
14	12 13	syl	⊢ ( 𝐹 ∈ Word ℝ → ( ⟨“ 0 ”⟩ ++ 𝐹 ) : ( 0 ..^ ( ♯ ‘ ( ⟨“ 0 ”⟩ ++ 𝐹 ) ) ) ⟶ ℝ )
15		1cnd	⊢ ( 𝐹 ∈ Word ℝ → 1 ∈ ℂ )
16		lencl	⊢ ( 𝐹 ∈ Word ℝ → ( ♯ ‘ 𝐹 ) ∈ ℕ₀ )
17	16	nn0cnd	⊢ ( 𝐹 ∈ Word ℝ → ( ♯ ‘ 𝐹 ) ∈ ℂ )
18		ccatlen	⊢ ( ( ⟨“ 0 ”⟩ ∈ Word ℝ ∧ 𝐹 ∈ Word ℝ ) → ( ♯ ‘ ( ⟨“ 0 ”⟩ ++ 𝐹 ) ) = ( ( ♯ ‘ ⟨“ 0 ”⟩ ) + ( ♯ ‘ 𝐹 ) ) )
19	10 18	mpan	⊢ ( 𝐹 ∈ Word ℝ → ( ♯ ‘ ( ⟨“ 0 ”⟩ ++ 𝐹 ) ) = ( ( ♯ ‘ ⟨“ 0 ”⟩ ) + ( ♯ ‘ 𝐹 ) ) )
20		s1len	⊢ ( ♯ ‘ ⟨“ 0 ”⟩ ) = 1
21	20	oveq1i	⊢ ( ( ♯ ‘ ⟨“ 0 ”⟩ ) + ( ♯ ‘ 𝐹 ) ) = ( 1 + ( ♯ ‘ 𝐹 ) )
22	19 21	eqtrdi	⊢ ( 𝐹 ∈ Word ℝ → ( ♯ ‘ ( ⟨“ 0 ”⟩ ++ 𝐹 ) ) = ( 1 + ( ♯ ‘ 𝐹 ) ) )
23	15 17 22	comraddd	⊢ ( 𝐹 ∈ Word ℝ → ( ♯ ‘ ( ⟨“ 0 ”⟩ ++ 𝐹 ) ) = ( ( ♯ ‘ 𝐹 ) + 1 ) )
24	23	oveq2d	⊢ ( 𝐹 ∈ Word ℝ → ( 0 ..^ ( ♯ ‘ ( ⟨“ 0 ”⟩ ++ 𝐹 ) ) ) = ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) )
25	24	feq2d	⊢ ( 𝐹 ∈ Word ℝ → ( ( ⟨“ 0 ”⟩ ++ 𝐹 ) : ( 0 ..^ ( ♯ ‘ ( ⟨“ 0 ”⟩ ++ 𝐹 ) ) ) ⟶ ℝ ↔ ( ⟨“ 0 ”⟩ ++ 𝐹 ) : ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ⟶ ℝ ) )
26	14 25	mpbid	⊢ ( 𝐹 ∈ Word ℝ → ( ⟨“ 0 ”⟩ ++ 𝐹 ) : ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ⟶ ℝ )
27	26	adantr	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺ ) → ( ⟨“ 0 ”⟩ ++ 𝐹 ) : ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ⟶ ℝ )
28		remulcl	⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 · 𝑦 ) ∈ ℝ )
29	28	adantl	⊢ ( ( ( 𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺ ) ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) → ( 𝑥 · 𝑦 ) ∈ ℝ )
30		ccatcl	⊢ ( ( 𝐹 ∈ Word ℝ ∧ ⟨“ 0 ”⟩ ∈ Word ℝ ) → ( 𝐹 ++ ⟨“ 0 ”⟩ ) ∈ Word ℝ )
31	10 30	mpan2	⊢ ( 𝐹 ∈ Word ℝ → ( 𝐹 ++ ⟨“ 0 ”⟩ ) ∈ Word ℝ )
32		wrdf	⊢ ( ( 𝐹 ++ ⟨“ 0 ”⟩ ) ∈ Word ℝ → ( 𝐹 ++ ⟨“ 0 ”⟩ ) : ( 0 ..^ ( ♯ ‘ ( 𝐹 ++ ⟨“ 0 ”⟩ ) ) ) ⟶ ℝ )
33	31 32	syl	⊢ ( 𝐹 ∈ Word ℝ → ( 𝐹 ++ ⟨“ 0 ”⟩ ) : ( 0 ..^ ( ♯ ‘ ( 𝐹 ++ ⟨“ 0 ”⟩ ) ) ) ⟶ ℝ )
34		ccatws1len	⊢ ( 𝐹 ∈ Word ℝ → ( ♯ ‘ ( 𝐹 ++ ⟨“ 0 ”⟩ ) ) = ( ( ♯ ‘ 𝐹 ) + 1 ) )
35	34	oveq2d	⊢ ( 𝐹 ∈ Word ℝ → ( 0 ..^ ( ♯ ‘ ( 𝐹 ++ ⟨“ 0 ”⟩ ) ) ) = ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) )
36	35	feq2d	⊢ ( 𝐹 ∈ Word ℝ → ( ( 𝐹 ++ ⟨“ 0 ”⟩ ) : ( 0 ..^ ( ♯ ‘ ( 𝐹 ++ ⟨“ 0 ”⟩ ) ) ) ⟶ ℝ ↔ ( 𝐹 ++ ⟨“ 0 ”⟩ ) : ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ⟶ ℝ ) )
37	33 36	mpbid	⊢ ( 𝐹 ∈ Word ℝ → ( 𝐹 ++ ⟨“ 0 ”⟩ ) : ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ⟶ ℝ )
38	37	adantr	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺ ) → ( 𝐹 ++ ⟨“ 0 ”⟩ ) : ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ⟶ ℝ )
39		ovexd	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺ ) → ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ∈ V )
40		rpre	⊢ ( 𝐶 ∈ ℝ⁺ → 𝐶 ∈ ℝ )
41	40	adantl	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺ ) → 𝐶 ∈ ℝ )
42	29 38 39 41	ofcf	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺ ) → ( ( 𝐹 ++ ⟨“ 0 ”⟩ ) ∘_f/c · 𝐶 ) : ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ⟶ ℝ )
43		inidm	⊢ ( ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ∩ ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ) = ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) )
44	7 27 42 39 39 43	off	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺ ) → ( ( ⟨“ 0 ”⟩ ++ 𝐹 ) ∘_f − ( ( 𝐹 ++ ⟨“ 0 ”⟩ ) ∘_f/c · 𝐶 ) ) : ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ⟶ ℝ )
45	5	feq1i	⊢ ( 𝐻 : ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ⟶ ℝ ↔ ( ( ⟨“ 0 ”⟩ ++ 𝐹 ) ∘_f − ( ( 𝐹 ++ ⟨“ 0 ”⟩ ) ∘_f/c · 𝐶 ) ) : ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ⟶ ℝ )
46	44 45	sylibr	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺ ) → 𝐻 : ( 0 ..^ ( ( ♯ ‘ 𝐹 ) + 1 ) ) ⟶ ℝ )

Metamath Proof Explorer

Theorem signshf

Proof