signsvfn - Mathbox for Thierry Arnoux

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Theorem signsvfn 31269

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		simpl 476	. . . . . . 7 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 𝐹 ∈ (Word ℝ ∖ {∅}))
2	1	eldifad 3804	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 𝐹 ∈ Word ℝ)
3		simpr 479	. . . . . . 7 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 𝐾 ∈ ℝ)
4	3	s1cld 13699	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ⟨“𝐾”⟩ ∈ Word ℝ)
5		ccatcl 13670	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
6	2, 4, 5	syl2anc 579	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
7		signsv.p	. . . . . 6 ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
8		signsv.w	. . . . . 6 ⊢ 𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+_g‘ndx), ⨣ ⟩}
9		signsv.t	. . . . . 6 ⊢ 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(♯‘𝑓)) ↦ (𝑊 Σ_g (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓‘𝑖))))))
10		signsv.v	. . . . . 6 ⊢ 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(♯‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0))
11	7, 8, 9, 10	signsvvfval 31265	. . . . 5 ⊢ ((𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ → (𝑉‘(𝐹 ++ ⟨“𝐾”⟩)) = Σ𝑗 ∈ (1..^(♯‘(𝐹 ++ ⟨“𝐾”⟩)))if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)), 1, 0))
12	6, 11	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))
13		ccatlen 13671	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)))
14	2, 4, 13	syl2anc 579	. . . . . . 7 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)))
15		s1len 13702	. . . . . . . 8 ⊢ (♯‘⟨“𝐾”⟩) = 1
16	15	oveq2i 6935	. . . . . . 7 ⊢ ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)) = ((♯‘𝐹) + 1)
17	14, 16	syl6eq 2830	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + 1))
18	17	oveq2d 6940	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (1..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))) = (1..^((♯‘𝐹) + 1)))
19	18	sumeq1d 14848	. . . 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))
20		eldifsn 4550	. . . . . . . 8 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) ↔ (𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅))
21		lennncl 13628	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅) → (♯‘𝐹) ∈ ℕ)
22	20, 21	sylbi 209	. . . . . . 7 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (♯‘𝐹) ∈ ℕ)
23		nnuz 12034	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
24	22, 23	syl6eleq 2869	. . . . . 6 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (♯‘𝐹) ∈ (ℤ_≥‘1))
25	24	adantr 474	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (♯‘𝐹) ∈ (ℤ_≥‘1))
26		1cnd 10373	. . . . . 6 ⊢ ((((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1...(♯‘𝐹))) ∧ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1))) → 1 ∈ ℂ)
27		0cnd 10371	. . . . . 6 ⊢ ((((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1...(♯‘𝐹))) ∧ ¬ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1))) → 0 ∈ ℂ)
28	26, 27	ifclda 4341	. . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1...(♯‘𝐹))) → if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)), 1, 0) ∈ ℂ)
29		fveq2 6448	. . . . . . 7 ⊢ (𝑗 = (♯‘𝐹) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) = ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)))
30		fvoveq1 6947	. . . . . . 7 ⊢ (𝑗 = (♯‘𝐹) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)) = ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((♯‘𝐹) − 1)))
31	29, 30	neeq12d 3030	. . . . . 6 ⊢ (𝑗 = (♯‘𝐹) → (((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)) ↔ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((♯‘𝐹) − 1))))
32	31	ifbid 4329	. . . . 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))
33	25, 28, 32	fzosump1 14897	. . . 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)))
34	12, 19, 33	3eqtrd 2818	. . 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)))
35	34	adantlr 705	. 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)))
36	2	adantr 474	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → 𝐹 ∈ Word ℝ)
37	3	adantr 474	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → 𝐾 ∈ ℝ)
38		fzo0ss1 12822	. . . . . . . . . . 11 ⊢ (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹))
39	38	a1i 11	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹)))
40	39	sselda 3821	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → 𝑗 ∈ (0..^(♯‘𝐹)))
41	7, 8, 9, 10	signstfvp 31257	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑗 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) = ((𝑇‘𝐹)‘𝑗))
42	36, 37, 40, 41	syl3anc 1439	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) = ((𝑇‘𝐹)‘𝑗))
43		elfzoel2 12793	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1..^(♯‘𝐹)) → (♯‘𝐹) ∈ ℤ)
44	43	adantl 475	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → (♯‘𝐹) ∈ ℤ)
45		1nn0 11665	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ₀
46		eluzmn 12004	. . . . . . . . . . . 12 ⊢ (((♯‘𝐹) ∈ ℤ ∧ 1 ∈ ℕ₀) → (♯‘𝐹) ∈ (ℤ_≥‘((♯‘𝐹) − 1)))
47	44, 45, 46	sylancl 580	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → (♯‘𝐹) ∈ (ℤ_≥‘((♯‘𝐹) − 1)))
48		fzoss2 12820	. . . . . . . . . . 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		simpr 479	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → 𝑗 ∈ (1..^(♯‘𝐹)))
51		elfzoelz 12794	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1..^(♯‘𝐹)) → 𝑗 ∈ ℤ)
52	51	adantl 475	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → 𝑗 ∈ ℤ)
53		elfzom1b 12891	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ ℤ ∧ (♯‘𝐹) ∈ ℤ) → (𝑗 ∈ (1..^(♯‘𝐹)) ↔ (𝑗 − 1) ∈ (0..^((♯‘𝐹) − 1))))
54	52, 44, 53	syl2anc 579	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → (𝑗 ∈ (1..^(♯‘𝐹)) ↔ (𝑗 − 1) ∈ (0..^((♯‘𝐹) − 1))))
55	50, 54	mpbid 224	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → (𝑗 − 1) ∈ (0..^((♯‘𝐹) − 1)))
56	49, 55	sseldd 3822	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → (𝑗 − 1) ∈ (0..^(♯‘𝐹)))
57	7, 8, 9, 10	signstfvp 31257	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ (𝑗 − 1) ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)) = ((𝑇‘𝐹)‘(𝑗 − 1)))
58	36, 37, 56, 57	syl3anc 1439	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)) = ((𝑇‘𝐹)‘(𝑗 − 1)))
59	42, 58	neeq12d 3030	. . . . . . 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))))
60	59	ifbid 4329	. . . . . 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))
61	60	sumeq2dv 14850	. . . . 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))
62	7, 8, 9, 10	signsvvfval 31265	. . . . . 6 ⊢ (𝐹 ∈ Word ℝ → (𝑉‘𝐹) = Σ𝑗 ∈ (1..^(♯‘𝐹))if(((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1)), 1, 0))
63	2, 62	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (𝑉‘𝐹) = Σ𝑗 ∈ (1..^(♯‘𝐹))if(((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1)), 1, 0))
64	61, 63	eqtr4d 2817	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → Σ𝑗 ∈ (1..^(♯‘𝐹))if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)), 1, 0) = (𝑉‘𝐹))
65	64	adantlr 705	. . 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))
66	7, 8, 9, 10	signstfvn 31254	. . . . . . 7 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) = (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ⨣ (sgn‘𝐾)))
67	66	adantlr 705	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) = (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ⨣ (sgn‘𝐾)))
68	2	adantlr 705	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → 𝐹 ∈ Word ℝ)
69		simpr 479	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → 𝐾 ∈ ℝ)
70	22	ad2antrr 716	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (♯‘𝐹) ∈ ℕ)
71		fzo0end 12884	. . . . . . . 8 ⊢ ((♯‘𝐹) ∈ ℕ → ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹)))
72	70, 71	syl 17	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹)))
73	7, 8, 9, 10	signstfvp 31257	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((♯‘𝐹) − 1)) = ((𝑇‘𝐹)‘((♯‘𝐹) − 1)))
74	68, 69, 72, 73	syl3anc 1439	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((♯‘𝐹) − 1)) = ((𝑇‘𝐹)‘((♯‘𝐹) − 1)))
75	67, 74	neeq12d 3030	. . . . 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))))
76	7, 8, 9, 10	signstfvcl 31259	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹))) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ∈ {-1, 1})
77	72, 76	syldan 585	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ∈ {-1, 1})
78	69	rexrd 10428	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → 𝐾 ∈ ℝ^*)
79		sgncl 31207	. . . . . . 7 ⊢ (𝐾 ∈ ℝ^* → (sgn‘𝐾) ∈ {-1, 0, 1})
80	78, 79	syl 17	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (sgn‘𝐾) ∈ {-1, 0, 1})
81	7, 8	signswch 31246	. . . . . 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))
82	77, 80, 81	syl2anc 579	. . . . 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))
83		sgnsgn 31217	. . . . . . . . 9 ⊢ (𝐾 ∈ ℝ^* → (sgn‘(sgn‘𝐾)) = (sgn‘𝐾))
84	78, 83	syl 17	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (sgn‘(sgn‘𝐾)) = (sgn‘𝐾))
85	84	oveq2d 6940	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((sgn‘((𝑇‘𝐹)‘((♯‘𝐹) − 1))) · (sgn‘(sgn‘𝐾))) = ((sgn‘((𝑇‘𝐹)‘((♯‘𝐹) − 1))) · (sgn‘𝐾)))
86	85	breq1d 4898	. . . . . 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))
87		neg1rr 11502	. . . . . . . . 9 ⊢ -1 ∈ ℝ
88		1re 10378	. . . . . . . . 9 ⊢ 1 ∈ ℝ
89		prssi 4585	. . . . . . . . 9 ⊢ ((-1 ∈ ℝ ∧ 1 ∈ ℝ) → {-1, 1} ⊆ ℝ)
90	87, 88, 89	mp2an 682	. . . . . . . 8 ⊢ {-1, 1} ⊆ ℝ
91	90, 77	sseldi 3819	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ∈ ℝ)
92		sgnclre 31208	. . . . . . . 8 ⊢ (𝐾 ∈ ℝ → (sgn‘𝐾) ∈ ℝ)
93	92	adantl 475	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (sgn‘𝐾) ∈ ℝ)
94		sgnmulsgn 31218	. . . . . . 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))
95	91, 93, 94	syl2anc 579	. . . . . 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))
96		sgnmulsgn 31218	. . . . . . 7 ⊢ ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ∈ ℝ ∧ 𝐾 ∈ ℝ) → ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · 𝐾) < 0 ↔ ((sgn‘((𝑇‘𝐹)‘((♯‘𝐹) − 1))) · (sgn‘𝐾)) < 0))
97	91, 69, 96	syl2anc 579	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · 𝐾) < 0 ↔ ((sgn‘((𝑇‘𝐹)‘((♯‘𝐹) − 1))) · (sgn‘𝐾)) < 0))
98	86, 95, 97	3bitr4d 303	. . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · (sgn‘𝐾)) < 0 ↔ (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · 𝐾) < 0))
99	75, 82, 98	3bitrd 297	. . . 4 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((♯‘𝐹) − 1)) ↔ (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) · 𝐾) < 0))
100	99	ifbid 4329	. . 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))
101	65, 100	oveq12d 6942	. 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)))
102	35, 101	eqtrd 2814	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 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∖ cdif 3789 ⊆ wss 3792 ∅c0 4141 ifcif 4307 {csn 4398 {cpr 4400 {ctp 4402 ⟨cop 4404 class class class wbr 4888 ↦ cmpt 4967 ‘cfv 6137 (class class class)co 6924 ↦ cmpt2 6926 ℂcc 10272 ℝcr 10273 0cc0 10274 1c1 10275 + caddc 10277 · cmul 10279 ℝ^*cxr 10412 < clt 10413 − cmin 10608 -cneg 10609 ℕcn 11379 ℕ₀cn0 11647 ℤcz 11733 ℤ_≥cuz 11997 ...cfz 12648 ..^cfzo 12789 ♯chash 13441 Word cword 13605 ++ cconcat 13666 ⟨“cs1 13691 sgncsgn 14239 Σcsu 14833 ndxcnx 16263 Basecbs 16266 +_gcplusg 16349 Σ_g cgsu 16498

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-sup 8638 df-oi 8706 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-n0 11648 df-xnn0 11720 df-z 11734 df-uz 11998 df-rp 12143 df-fz 12649 df-fzo 12790 df-seq 13125 df-exp 13184 df-hash 13442 df-word 13606 df-lsw 13659 df-concat 13667 df-s1 13692 df-substr 13737 df-pfx 13786 df-sgn 14240 df-cj 14252 df-re 14253 df-im 14254 df-sqrt 14388 df-abs 14389 df-clim 14636 df-sum 14834 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-plusg 16362 df-0g 16499 df-gsum 16500 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-mulg 17939 df-cntz 18144

This theorem is referenced by: signsvtp 31270 signsvtn 31271 signlem0 31274

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