signsvfn - Mathbox for Thierry Arnoux

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Theorem signsvfn 34688

Description: Number of changes in a word compared to a shorter word. (Contributed by Thierry Arnoux, 12-Oct-2018.)

**Hypotheses**
Ref	Expression
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 ↦ Σ𝑗 ∈ (1..^(♯‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0))

**Assertion**
Ref	Expression
signsvfn	⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (𝑉‘(𝐹 ++ ⟨“𝐾”⟩)) = ((𝑉‘𝐹) + if((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · 𝐾) < 0, 1, 0)))

Distinct variable groups: 𝑎,𝑏, ⨣ 𝑓,𝑖,𝑛,𝐹 𝑓,𝐾,𝑖,𝑛 𝑓,𝑊,𝑖,𝑛 𝑖,𝑎,𝑗,𝑛,𝐹,𝑏 𝐾,𝑎,𝑏,𝑗,𝑓 𝑇,𝑎 𝑓,𝑏,𝑇,𝑗,𝑛 Allowed substitution hints: ⨣ (𝑓,𝑖,𝑗,𝑛) 𝑇(𝑖) 𝑉(𝑓,𝑖,𝑗,𝑛,𝑎,𝑏) 𝑊(𝑗,𝑎,𝑏)

**Proof of Theorem signsvfn**
Step	Hyp	Ref	Expression
1		eldifi 4081	. . . . . 6 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → 𝐹 ∈ Word ℝ)
2		s1cl 14524	. . . . . 6 ⊢ (𝐾 ∈ ℝ → ⟨“𝐾”⟩ ∈ Word ℝ)
3		ccatcl 14495	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
4	1, 2, 3	syl2an 596	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
5		signsv.p	. . . . . 6 ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
6		signsv.w	. . . . . 6 ⊢ 𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+_g‘ndx), ⨣ ⟩}
7		signsv.t	. . . . . 6 ⊢ 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(♯‘𝑓)) ↦ (𝑊 Σ_g (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓‘𝑖))))))
8		signsv.v	. . . . . 6 ⊢ 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(♯‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0))
9	5, 6, 7, 8	signsvvfval 34684	. . . . 5 ⊢ ((𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ → (𝑉‘(𝐹 ++ ⟨“𝐾”⟩)) = Σ𝑗 ∈ (1..^(♯‘(𝐹 ++ ⟨“𝐾”⟩)))if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)), 1, 0))
10	4, 9	syl 17	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (𝑉‘(𝐹 ++ ⟨“𝐾”⟩)) = Σ𝑗 ∈ (1..^(♯‘(𝐹 ++ ⟨“𝐾”⟩)))if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)), 1, 0))
11		ccatlen 14496	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)))
12	1, 2, 11	syl2an 596	. . . . . . 7 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)))
13		s1len 14528	. . . . . . . 8 ⊢ (♯‘⟨“𝐾”⟩) = 1
14	13	oveq2i 7367	. . . . . . 7 ⊢ ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)) = ((♯‘𝐹) + 1)
15	12, 14	eqtrdi 2785	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + 1))
16	15	oveq2d 7372	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (1..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))) = (1..^((♯‘𝐹) + 1)))
17	16	sumeq1d 15621	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → Σ𝑗 ∈ (1..^(♯‘(𝐹 ++ ⟨“𝐾”⟩)))if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)), 1, 0) = Σ𝑗 ∈ (1..^((♯‘𝐹) + 1))if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)), 1, 0))
18		eldifsn 4740	. . . . . . . 8 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) ↔ (𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅))
19		lennncl 14455	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅) → (♯‘𝐹) ∈ ℕ)
20	18, 19	sylbi 217	. . . . . . 7 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (♯‘𝐹) ∈ ℕ)
21		nnuz 12788	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
22	20, 21	eleqtrdi 2844	. . . . . 6 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (♯‘𝐹) ∈ (ℤ_≥‘1))
23	22	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (♯‘𝐹) ∈ (ℤ_≥‘1))
24		1cnd 11125	. . . . . 6 ⊢ ((((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1...(♯‘𝐹))) ∧ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1))) → 1 ∈ ℂ)
25		0cnd 11123	. . . . . 6 ⊢ ((((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1...(♯‘𝐹))) ∧ ¬ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1))) → 0 ∈ ℂ)
26	24, 25	ifclda 4513	. . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1...(♯‘𝐹))) → if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)), 1, 0) ∈ ℂ)
27		fveq2 6832	. . . . . . 7 ⊢ (𝑗 = (♯‘𝐹) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) = ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)))
28		fvoveq1 7379	. . . . . . 7 ⊢ (𝑗 = (♯‘𝐹) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)) = ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((♯‘𝐹) − 1)))
29	27, 28	neeq12d 2991	. . . . . 6 ⊢ (𝑗 = (♯‘𝐹) → (((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)) ↔ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((♯‘𝐹) − 1))))
30	29	ifbid 4501	. . . . 5 ⊢ (𝑗 = (♯‘𝐹) → if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)), 1, 0) = if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((♯‘𝐹) − 1)), 1, 0))
31	23, 26, 30	fzosump1 15673	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → Σ𝑗 ∈ (1..^((♯‘𝐹) + 1))if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)), 1, 0) = (Σ𝑗 ∈ (1..^(♯‘𝐹))if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)), 1, 0) + if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((♯‘𝐹) − 1)), 1, 0)))
32	10, 17, 31	3eqtrd 2773	. . 3 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (𝑉‘(𝐹 ++ ⟨“𝐾”⟩)) = (Σ𝑗 ∈ (1..^(♯‘𝐹))if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)), 1, 0) + if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((♯‘𝐹) − 1)), 1, 0)))
33	32	adantlr 715	. 2 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (𝑉‘(𝐹 ++ ⟨“𝐾”⟩)) = (Σ𝑗 ∈ (1..^(♯‘𝐹))if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)), 1, 0) + if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((♯‘𝐹) − 1)), 1, 0)))
34		simpl 482	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 𝐹 ∈ (Word ℝ ∖ {∅}))
35	34	eldifad 3911	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 𝐹 ∈ Word ℝ)
36	35	adantr 480	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → 𝐹 ∈ Word ℝ)
37		simplr 768	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → 𝐾 ∈ ℝ)
38		fzo0ss1 13603	. . . . . . . . . . 11 ⊢ (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹))
39	38	a1i 11	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹)))
40	39	sselda 3931	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → 𝑗 ∈ (0..^(♯‘𝐹)))
41	5, 6, 7, 8	signstfvp 34677	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑗 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) = ((𝑇‘𝐹)‘𝑗))
42	36, 37, 40, 41	syl3anc 1373	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) = ((𝑇‘𝐹)‘𝑗))
43		elfzoel2 13572	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1..^(♯‘𝐹)) → (♯‘𝐹) ∈ ℤ)
44	43	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → (♯‘𝐹) ∈ ℤ)
45		1nn0 12415	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ₀
46		eluzmn 12756	. . . . . . . . . . . 12 ⊢ (((♯‘𝐹) ∈ ℤ ∧ 1 ∈ ℕ₀) → (♯‘𝐹) ∈ (ℤ_≥‘((♯‘𝐹) − 1)))
47	44, 45, 46	sylancl 586	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → (♯‘𝐹) ∈ (ℤ_≥‘((♯‘𝐹) − 1)))
48		fzoss2 13601	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) ∈ (ℤ_≥‘((♯‘𝐹) − 1)) → (0..^((♯‘𝐹) − 1)) ⊆ (0..^(♯‘𝐹)))
49	47, 48	syl 17	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → (0..^((♯‘𝐹) − 1)) ⊆ (0..^(♯‘𝐹)))
50		elfzo1elm1fzo0 13682	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (1..^(♯‘𝐹)) → (𝑗 − 1) ∈ (0..^((♯‘𝐹) − 1)))
51	50	adantl 481	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → (𝑗 − 1) ∈ (0..^((♯‘𝐹) − 1)))
52	49, 51	sseldd 3932	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → (𝑗 − 1) ∈ (0..^(♯‘𝐹)))
53	5, 6, 7, 8	signstfvp 34677	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ (𝑗 − 1) ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)) = ((𝑇‘𝐹)‘(𝑗 − 1)))
54	36, 37, 52, 53	syl3anc 1373	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)) = ((𝑇‘𝐹)‘(𝑗 − 1)))
55	42, 54	neeq12d 2991	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → (((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)) ↔ ((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1))))
56	55	ifbid 4501	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)), 1, 0) = if(((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1)), 1, 0))
57	56	sumeq2dv 15623	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → Σ𝑗 ∈ (1..^(♯‘𝐹))if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)), 1, 0) = Σ𝑗 ∈ (1..^(♯‘𝐹))if(((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1)), 1, 0))
58	5, 6, 7, 8	signsvvfval 34684	. . . . . 6 ⊢ (𝐹 ∈ Word ℝ → (𝑉‘𝐹) = Σ𝑗 ∈ (1..^(♯‘𝐹))if(((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1)), 1, 0))
59	35, 58	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (𝑉‘𝐹) = Σ𝑗 ∈ (1..^(♯‘𝐹))if(((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1)), 1, 0))
60	57, 59	eqtr4d 2772	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → Σ𝑗 ∈ (1..^(♯‘𝐹))if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)), 1, 0) = (𝑉‘𝐹))
61	60	adantlr 715	. . 3 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → Σ𝑗 ∈ (1..^(♯‘𝐹))if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)), 1, 0) = (𝑉‘𝐹))
62	5, 6, 7, 8	signstfvn 34675	. . . . . . 7 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) = (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ⨣ (sgn‘𝐾)))
63	62	adantlr 715	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) = (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ⨣ (sgn‘𝐾)))
64	35	adantlr 715	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → 𝐹 ∈ Word ℝ)
65		simpr 484	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → 𝐾 ∈ ℝ)
66		fzo0end 13672	. . . . . . . . 9 ⊢ ((♯‘𝐹) ∈ ℕ → ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹)))
67	20, 66	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹)))
68	67	ad2antrr 726	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹)))
69	5, 6, 7, 8	signstfvp 34677	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((♯‘𝐹) − 1)) = ((𝑇‘𝐹)‘((♯‘𝐹) − 1)))
70	64, 65, 68, 69	syl3anc 1373	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((♯‘𝐹) − 1)) = ((𝑇‘𝐹)‘((♯‘𝐹) − 1)))
71	63, 70	neeq12d 2991	. . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((♯‘𝐹) − 1)) ↔ (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ⨣ (sgn‘𝐾)) ≠ ((𝑇‘𝐹)‘((♯‘𝐹) − 1))))
72	5, 6, 7, 8	signstfvcl 34679	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹))) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ∈ {-1, 1})
73	68, 72	syldan 591	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ∈ {-1, 1})
74		rexr 11176	. . . . . . . 8 ⊢ (𝐾 ∈ ℝ → 𝐾 ∈ ℝ^*)
75		sgncl 32861	. . . . . . . 8 ⊢ (𝐾 ∈ ℝ^* → (sgn‘𝐾) ∈ {-1, 0, 1})
76	74, 75	syl 17	. . . . . . 7 ⊢ (𝐾 ∈ ℝ → (sgn‘𝐾) ∈ {-1, 0, 1})
77	76	adantl 481	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (sgn‘𝐾) ∈ {-1, 0, 1})
78	5, 6	signswch 34667	. . . . . 6 ⊢ ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ∈ {-1, 1} ∧ (sgn‘𝐾) ∈ {-1, 0, 1}) → ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ⨣ (sgn‘𝐾)) ≠ ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ↔ (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · (sgn‘𝐾)) < 0))
79	73, 77, 78	syl2anc 584	. . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ⨣ (sgn‘𝐾)) ≠ ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ↔ (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · (sgn‘𝐾)) < 0))
80	65	rexrd 11180	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → 𝐾 ∈ ℝ^*)
81		sgnsgn 32871	. . . . . . . . 9 ⊢ (𝐾 ∈ ℝ^* → (sgn‘(sgn‘𝐾)) = (sgn‘𝐾))
82	80, 81	syl 17	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (sgn‘(sgn‘𝐾)) = (sgn‘𝐾))
83	82	oveq2d 7372	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((sgn‘((𝑇‘𝐹)‘((♯‘𝐹) − 1))) · (sgn‘(sgn‘𝐾))) = ((sgn‘((𝑇‘𝐹)‘((♯‘𝐹) − 1))) · (sgn‘𝐾)))
84	83	breq1d 5106	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (((sgn‘((𝑇‘𝐹)‘((♯‘𝐹) − 1))) · (sgn‘(sgn‘𝐾))) < 0 ↔ ((sgn‘((𝑇‘𝐹)‘((♯‘𝐹) − 1))) · (sgn‘𝐾)) < 0))
85		neg1rr 12129	. . . . . . . . 9 ⊢ -1 ∈ ℝ
86		1re 11130	. . . . . . . . 9 ⊢ 1 ∈ ℝ
87		prssi 4775	. . . . . . . . 9 ⊢ ((-1 ∈ ℝ ∧ 1 ∈ ℝ) → {-1, 1} ⊆ ℝ)
88	85, 86, 87	mp2an 692	. . . . . . . 8 ⊢ {-1, 1} ⊆ ℝ
89	88, 73	sselid 3929	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ∈ ℝ)
90		sgnclre 32862	. . . . . . . 8 ⊢ (𝐾 ∈ ℝ → (sgn‘𝐾) ∈ ℝ)
91	90	adantl 481	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (sgn‘𝐾) ∈ ℝ)
92		sgnmulsgn 32872	. . . . . . 7 ⊢ ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ∈ ℝ ∧ (sgn‘𝐾) ∈ ℝ) → ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · (sgn‘𝐾)) < 0 ↔ ((sgn‘((𝑇‘𝐹)‘((♯‘𝐹) − 1))) · (sgn‘(sgn‘𝐾))) < 0))
93	89, 91, 92	syl2anc 584	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · (sgn‘𝐾)) < 0 ↔ ((sgn‘((𝑇‘𝐹)‘((♯‘𝐹) − 1))) · (sgn‘(sgn‘𝐾))) < 0))
94		sgnmulsgn 32872	. . . . . . 7 ⊢ ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ∈ ℝ ∧ 𝐾 ∈ ℝ) → ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · 𝐾) < 0 ↔ ((sgn‘((𝑇‘𝐹)‘((♯‘𝐹) − 1))) · (sgn‘𝐾)) < 0))
95	89, 94	sylancom 588	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · 𝐾) < 0 ↔ ((sgn‘((𝑇‘𝐹)‘((♯‘𝐹) − 1))) · (sgn‘𝐾)) < 0))
96	84, 93, 95	3bitr4d 311	. . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · (sgn‘𝐾)) < 0 ↔ (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · 𝐾) < 0))
97	71, 79, 96	3bitrd 305	. . . 4 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((♯‘𝐹) − 1)) ↔ (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · 𝐾) < 0))
98	97	ifbid 4501	. . 3 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((♯‘𝐹) − 1)), 1, 0) = if((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · 𝐾) < 0, 1, 0))
99	61, 98	oveq12d 7374	. 2 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (Σ𝑗 ∈ (1..^(♯‘𝐹))if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)), 1, 0) + if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((♯‘𝐹) − 1)), 1, 0)) = ((𝑉‘𝐹) + if((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · 𝐾) < 0, 1, 0)))
100	33, 99	eqtrd 2769	1 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (𝑉‘(𝐹 ++ ⟨“𝐾”⟩)) = ((𝑉‘𝐹) + if((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · 𝐾) < 0, 1, 0)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2930 ∖ cdif 3896 ⊆ wss 3899 ∅c0 4283 ifcif 4477 {csn 4578 {cpr 4580 {ctp 4582 ⟨cop 4584 class class class wbr 5096 ↦ cmpt 5177 ‘cfv 6490 (class class class)co 7356 ∈ cmpo 7358 ℂcc 11022 ℝcr 11023 0cc0 11024 1c1 11025 + caddc 11027 · cmul 11029 ℝ^*cxr 11163 < clt 11164 − cmin 11362 -cneg 11363 ℕcn 12143 ℕ₀cn0 12399 ℤcz 12486 ℤ_≥cuz 12749 ...cfz 13421 ..^cfzo 13568 ♯chash 14251 Word cword 14434 ++ cconcat 14491 ⟨“cs1 14517 sgncsgn 15007 Σcsu 15607 ndxcnx 17118 Basecbs 17134 +_gcplusg 17175 Σ_g cgsu 17358

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-sup 9343 df-oi 9413 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-n0 12400 df-xnn0 12473 df-z 12487 df-uz 12750 df-rp 12904 df-fz 13422 df-fzo 13569 df-seq 13923 df-exp 13983 df-hash 14252 df-word 14435 df-lsw 14484 df-concat 14492 df-s1 14518 df-substr 14563 df-pfx 14593 df-sgn 15008 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-clim 15409 df-sum 15608 df-struct 17072 df-slot 17107 df-ndx 17119 df-base 17135 df-plusg 17188 df-0g 17359 df-gsum 17360 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mulg 18996 df-cntz 19244

This theorem is referenced by: signsvtp 34689 signsvtn 34690 signlem0 34693

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