signsvfn - Mathbox for Thierry Arnoux

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Theorem signsvfn 32461

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 4057	. . . . . 6 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → 𝐹 ∈ Word ℝ)
2		s1cl 14235	. . . . . 6 ⊢ (𝐾 ∈ ℝ → ⟨“𝐾”⟩ ∈ Word ℝ)
3		ccatcl 14205	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
4	1, 2, 3	syl2an 595	. . . . 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 32457	. . . . 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 14206	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)))
12	1, 2, 11	syl2an 595	. . . . . . 7 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)))
13		s1len 14239	. . . . . . . 8 ⊢ (♯‘⟨“𝐾”⟩) = 1
14	13	oveq2i 7266	. . . . . . 7 ⊢ ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)) = ((♯‘𝐹) + 1)
15	12, 14	eqtrdi 2795	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + 1))
16	15	oveq2d 7271	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (1..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))) = (1..^((♯‘𝐹) + 1)))
17	16	sumeq1d 15341	. . . 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 4717	. . . . . . . 8 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) ↔ (𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅))
19		lennncl 14165	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅) → (♯‘𝐹) ∈ ℕ)
20	18, 19	sylbi 216	. . . . . . 7 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (♯‘𝐹) ∈ ℕ)
21		nnuz 12550	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
22	20, 21	eleqtrdi 2849	. . . . . 6 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (♯‘𝐹) ∈ (ℤ_≥‘1))
23	22	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (♯‘𝐹) ∈ (ℤ_≥‘1))
24		1cnd 10901	. . . . . 6 ⊢ ((((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1...(♯‘𝐹))) ∧ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1))) → 1 ∈ ℂ)
25		0cnd 10899	. . . . . 6 ⊢ ((((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1...(♯‘𝐹))) ∧ ¬ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1))) → 0 ∈ ℂ)
26	24, 25	ifclda 4491	. . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1...(♯‘𝐹))) → if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)), 1, 0) ∈ ℂ)
27		fveq2 6756	. . . . . . 7 ⊢ (𝑗 = (♯‘𝐹) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) = ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)))
28		fvoveq1 7278	. . . . . . 7 ⊢ (𝑗 = (♯‘𝐹) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)) = ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((♯‘𝐹) − 1)))
29	27, 28	neeq12d 3004	. . . . . 6 ⊢ (𝑗 = (♯‘𝐹) → (((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)) ↔ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((♯‘𝐹) − 1))))
30	29	ifbid 4479	. . . . 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 15392	. . . 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 2782	. . 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 711	. 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 3895	. . . . . . . . . 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 765	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → 𝐾 ∈ ℝ)
38		fzo0ss1 13345	. . . . . . . . . . 11 ⊢ (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹))
39	38	a1i 11	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹)))
40	39	sselda 3917	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → 𝑗 ∈ (0..^(♯‘𝐹)))
41	5, 6, 7, 8	signstfvp 32450	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑗 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) = ((𝑇‘𝐹)‘𝑗))
42	36, 37, 40, 41	syl3anc 1369	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) = ((𝑇‘𝐹)‘𝑗))
43		elfzoel2 13315	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1..^(♯‘𝐹)) → (♯‘𝐹) ∈ ℤ)
44	43	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → (♯‘𝐹) ∈ ℤ)
45		1nn0 12179	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ₀
46		eluzmn 12518	. . . . . . . . . . . 12 ⊢ (((♯‘𝐹) ∈ ℤ ∧ 1 ∈ ℕ₀) → (♯‘𝐹) ∈ (ℤ_≥‘((♯‘𝐹) − 1)))
47	44, 45, 46	sylancl 585	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → (♯‘𝐹) ∈ (ℤ_≥‘((♯‘𝐹) − 1)))
48		fzoss2 13343	. . . . . . . . . . 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 13416	. . . . . . . . . . 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 3918	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → (𝑗 − 1) ∈ (0..^(♯‘𝐹)))
53	5, 6, 7, 8	signstfvp 32450	. . . . . . . . 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 1369	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)) = ((𝑇‘𝐹)‘(𝑗 − 1)))
55	42, 54	neeq12d 3004	. . . . . . 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 4479	. . . . . 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 15343	. . . . 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 32457	. . . . . 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 2781	. . . 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 711	. . 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 32448	. . . . . . 7 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) = (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ⨣ (sgn‘𝐾)))
63	62	adantlr 711	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) = (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ⨣ (sgn‘𝐾)))
64	35	adantlr 711	. . . . . . 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 13407	. . . . . . . . 9 ⊢ ((♯‘𝐹) ∈ ℕ → ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹)))
67	20, 66	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹)))
68	67	ad2antrr 722	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹)))
69	5, 6, 7, 8	signstfvp 32450	. . . . . . 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 1369	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((♯‘𝐹) − 1)) = ((𝑇‘𝐹)‘((♯‘𝐹) − 1)))
71	63, 70	neeq12d 3004	. . . . 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 32452	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹))) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ∈ {-1, 1})
73	68, 72	syldan 590	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ∈ {-1, 1})
74		rexr 10952	. . . . . . . 8 ⊢ (𝐾 ∈ ℝ → 𝐾 ∈ ℝ^*)
75		sgncl 32405	. . . . . . . 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 32440	. . . . . 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 583	. . . . 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 10956	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → 𝐾 ∈ ℝ^*)
81		sgnsgn 32415	. . . . . . . . 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 7271	. . . . . . 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 5080	. . . . . 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 12018	. . . . . . . . 9 ⊢ -1 ∈ ℝ
86		1re 10906	. . . . . . . . 9 ⊢ 1 ∈ ℝ
87		prssi 4751	. . . . . . . . 9 ⊢ ((-1 ∈ ℝ ∧ 1 ∈ ℝ) → {-1, 1} ⊆ ℝ)
88	85, 86, 87	mp2an 688	. . . . . . . 8 ⊢ {-1, 1} ⊆ ℝ
89	88, 73	sselid 3915	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ∈ ℝ)
90		sgnclre 32406	. . . . . . . 8 ⊢ (𝐾 ∈ ℝ → (sgn‘𝐾) ∈ ℝ)
91	90	adantl 481	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (sgn‘𝐾) ∈ ℝ)
92		sgnmulsgn 32416	. . . . . . 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 583	. . . . . 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 32416	. . . . . . 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 587	. . . . . 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 4479	. . 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 7273	. 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 2778	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 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∖ cdif 3880 ⊆ wss 3883 ∅c0 4253 ifcif 4456 {csn 4558 {cpr 4560 {ctp 4562 ⟨cop 4564 class class class wbr 5070 ↦ cmpt 5153 ‘cfv 6418 (class class class)co 7255 ∈ cmpo 7257 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 ℝ^*cxr 10939 < clt 10940 − cmin 11135 -cneg 11136 ℕcn 11903 ℕ₀cn0 12163 ℤcz 12249 ℤ_≥cuz 12511 ...cfz 13168 ..^cfzo 13311 ♯chash 13972 Word cword 14145 ++ cconcat 14201 ⟨“cs1 14228 sgncsgn 14725 Σcsu 15325 ndxcnx 16822 Basecbs 16840 +_gcplusg 16888 Σ_g cgsu 17068

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-xnn0 12236 df-z 12250 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-hash 13973 df-word 14146 df-lsw 14194 df-concat 14202 df-s1 14229 df-substr 14282 df-pfx 14312 df-sgn 14726 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-sum 15326 df-struct 16776 df-slot 16811 df-ndx 16823 df-base 16841 df-plusg 16901 df-0g 17069 df-gsum 17070 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mulg 18616 df-cntz 18838

This theorem is referenced by: signsvtp 32462 signsvtn 32463 signlem0 32466

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