signsvfn - Mathbox for Thierry Arnoux

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Theorem signsvfn 33283

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 4091	. . . . . 6 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → 𝐹 ∈ Word ℝ)
2		s1cl 14502	. . . . . 6 ⊢ (𝐾 ∈ ℝ → ⟨“𝐾”⟩ ∈ Word ℝ)
3		ccatcl 14474	. . . . . 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 33279	. . . . 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 14475	. . . . . . . 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 14506	. . . . . . . 8 ⊢ (♯‘⟨“𝐾”⟩) = 1
14	13	oveq2i 7373	. . . . . . 7 ⊢ ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)) = ((♯‘𝐹) + 1)
15	12, 14	eqtrdi 2787	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + 1))
16	15	oveq2d 7378	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (1..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))) = (1..^((♯‘𝐹) + 1)))
17	16	sumeq1d 15597	. . . 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 4752	. . . . . . . 8 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) ↔ (𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅))
19		lennncl 14434	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅) → (♯‘𝐹) ∈ ℕ)
20	18, 19	sylbi 216	. . . . . . 7 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (♯‘𝐹) ∈ ℕ)
21		nnuz 12815	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
22	20, 21	eleqtrdi 2842	. . . . . 6 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (♯‘𝐹) ∈ (ℤ_≥‘1))
23	22	adantr 481	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (♯‘𝐹) ∈ (ℤ_≥‘1))
24		1cnd 11159	. . . . . 6 ⊢ ((((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1...(♯‘𝐹))) ∧ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1))) → 1 ∈ ℂ)
25		0cnd 11157	. . . . . 6 ⊢ ((((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1...(♯‘𝐹))) ∧ ¬ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1))) → 0 ∈ ℂ)
26	24, 25	ifclda 4526	. . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1...(♯‘𝐹))) → if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)), 1, 0) ∈ ℂ)
27		fveq2 6847	. . . . . . 7 ⊢ (𝑗 = (♯‘𝐹) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) = ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)))
28		fvoveq1 7385	. . . . . . 7 ⊢ (𝑗 = (♯‘𝐹) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)) = ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((♯‘𝐹) − 1)))
29	27, 28	neeq12d 3001	. . . . . 6 ⊢ (𝑗 = (♯‘𝐹) → (((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)) ↔ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((♯‘𝐹) − 1))))
30	29	ifbid 4514	. . . . 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 15648	. . . 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 2775	. . 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 713	. 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 483	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 𝐹 ∈ (Word ℝ ∖ {∅}))
35	34	eldifad 3925	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 𝐹 ∈ Word ℝ)
36	35	adantr 481	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → 𝐹 ∈ Word ℝ)
37		simplr 767	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → 𝐾 ∈ ℝ)
38		fzo0ss1 13612	. . . . . . . . . . 11 ⊢ (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹))
39	38	a1i 11	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹)))
40	39	sselda 3947	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → 𝑗 ∈ (0..^(♯‘𝐹)))
41	5, 6, 7, 8	signstfvp 33272	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑗 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) = ((𝑇‘𝐹)‘𝑗))
42	36, 37, 40, 41	syl3anc 1371	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) = ((𝑇‘𝐹)‘𝑗))
43		elfzoel2 13581	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1..^(♯‘𝐹)) → (♯‘𝐹) ∈ ℤ)
44	43	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → (♯‘𝐹) ∈ ℤ)
45		1nn0 12438	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ₀
46		eluzmn 12779	. . . . . . . . . . . 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 13610	. . . . . . . . . . 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 13683	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (1..^(♯‘𝐹)) → (𝑗 − 1) ∈ (0..^((♯‘𝐹) − 1)))
51	50	adantl 482	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → (𝑗 − 1) ∈ (0..^((♯‘𝐹) − 1)))
52	49, 51	sseldd 3948	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → (𝑗 − 1) ∈ (0..^(♯‘𝐹)))
53	5, 6, 7, 8	signstfvp 33272	. . . . . . . . 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 1371	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)) = ((𝑇‘𝐹)‘(𝑗 − 1)))
55	42, 54	neeq12d 3001	. . . . . . 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 4514	. . . . . 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 15599	. . . . 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 33279	. . . . . 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 2774	. . . 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 713	. . 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 33270	. . . . . . 7 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) = (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ⨣ (sgn‘𝐾)))
63	62	adantlr 713	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) = (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ⨣ (sgn‘𝐾)))
64	35	adantlr 713	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → 𝐹 ∈ Word ℝ)
65		simpr 485	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → 𝐾 ∈ ℝ)
66		fzo0end 13674	. . . . . . . . 9 ⊢ ((♯‘𝐹) ∈ ℕ → ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹)))
67	20, 66	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹)))
68	67	ad2antrr 724	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹)))
69	5, 6, 7, 8	signstfvp 33272	. . . . . . 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 1371	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((♯‘𝐹) − 1)) = ((𝑇‘𝐹)‘((♯‘𝐹) − 1)))
71	63, 70	neeq12d 3001	. . . . 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 33274	. . . . . . 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 11210	. . . . . . . 8 ⊢ (𝐾 ∈ ℝ → 𝐾 ∈ ℝ^*)
75		sgncl 33227	. . . . . . . 8 ⊢ (𝐾 ∈ ℝ^* → (sgn‘𝐾) ∈ {-1, 0, 1})
76	74, 75	syl 17	. . . . . . 7 ⊢ (𝐾 ∈ ℝ → (sgn‘𝐾) ∈ {-1, 0, 1})
77	76	adantl 482	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (sgn‘𝐾) ∈ {-1, 0, 1})
78	5, 6	signswch 33262	. . . . . 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 11214	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → 𝐾 ∈ ℝ^*)
81		sgnsgn 33237	. . . . . . . . 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 7378	. . . . . . 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 5120	. . . . . 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 12277	. . . . . . . . 9 ⊢ -1 ∈ ℝ
86		1re 11164	. . . . . . . . 9 ⊢ 1 ∈ ℝ
87		prssi 4786	. . . . . . . . 9 ⊢ ((-1 ∈ ℝ ∧ 1 ∈ ℝ) → {-1, 1} ⊆ ℝ)
88	85, 86, 87	mp2an 690	. . . . . . . 8 ⊢ {-1, 1} ⊆ ℝ
89	88, 73	sselid 3945	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ∈ ℝ)
90		sgnclre 33228	. . . . . . . 8 ⊢ (𝐾 ∈ ℝ → (sgn‘𝐾) ∈ ℝ)
91	90	adantl 482	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (sgn‘𝐾) ∈ ℝ)
92		sgnmulsgn 33238	. . . . . . 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 33238	. . . . . . 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 310	. . . . 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 304	. . . 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 4514	. . 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 7380	. 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 2771	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 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ∖ cdif 3910 ⊆ wss 3913 ∅c0 4287 ifcif 4491 {csn 4591 {cpr 4593 {ctp 4595 ⟨cop 4597 class class class wbr 5110 ↦ cmpt 5193 ‘cfv 6501 (class class class)co 7362 ∈ cmpo 7364 ℂcc 11058 ℝcr 11059 0cc0 11060 1c1 11061 + caddc 11063 · cmul 11065 ℝ^*cxr 11197 < clt 11198 − cmin 11394 -cneg 11395 ℕcn 12162 ℕ₀cn0 12422 ℤcz 12508 ℤ_≥cuz 12772 ...cfz 13434 ..^cfzo 13577 ♯chash 14240 Word cword 14414 ++ cconcat 14470 ⟨“cs1 14495 sgncsgn 14983 Σcsu 15582 ndxcnx 17076 Basecbs 17094 +_gcplusg 17147 Σ_g cgsu 17336

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9586 ax-cnex 11116 ax-resscn 11117 ax-1cn 11118 ax-icn 11119 ax-addcl 11120 ax-addrcl 11121 ax-mulcl 11122 ax-mulrcl 11123 ax-mulcom 11124 ax-addass 11125 ax-mulass 11126 ax-distr 11127 ax-i2m1 11128 ax-1ne0 11129 ax-1rid 11130 ax-rnegex 11131 ax-rrecex 11132 ax-cnre 11133 ax-pre-lttri 11134 ax-pre-lttrn 11135 ax-pre-ltadd 11136 ax-pre-mulgt0 11137 ax-pre-sup 11138

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3448 df-sbc 3743 df-csb 3859 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9387 df-oi 9455 df-card 9884 df-pnf 11200 df-mnf 11201 df-xr 11202 df-ltxr 11203 df-le 11204 df-sub 11396 df-neg 11397 df-div 11822 df-nn 12163 df-2 12225 df-3 12226 df-n0 12423 df-xnn0 12495 df-z 12509 df-uz 12773 df-rp 12925 df-fz 13435 df-fzo 13578 df-seq 13917 df-exp 13978 df-hash 14241 df-word 14415 df-lsw 14463 df-concat 14471 df-s1 14496 df-substr 14541 df-pfx 14571 df-sgn 14984 df-cj 14996 df-re 14997 df-im 14998 df-sqrt 15132 df-abs 15133 df-clim 15382 df-sum 15583 df-struct 17030 df-slot 17065 df-ndx 17077 df-base 17095 df-plusg 17160 df-0g 17337 df-gsum 17338 df-mgm 18511 df-sgrp 18560 df-mnd 18571 df-mulg 18887 df-cntz 19111

This theorem is referenced by: signsvtp 33284 signsvtn 33285 signlem0 33288

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