signsvfn - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > signsvfn		Structured version Visualization version GIF version

Theorem signsvfn 29985

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 472	. . . . . . 7 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 𝐹 ∈ (Word ℝ ∖ {∅}))
2	1	eldifad 3552	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 𝐹 ∈ Word ℝ)
3		simpr 476	. . . . . . 7 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 𝐾 ∈ ℝ)
4	3	s1cld 13236	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ⟨“𝐾”⟩ ∈ Word ℝ)
5		ccatcl 13212	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
6	2, 4, 5	syl2anc 691	. . . . 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 29981	. . . . 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 13213	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ) → (#‘(𝐹 ++ ⟨“𝐾”⟩)) = ((#‘𝐹) + (#‘⟨“𝐾”⟩)))
14	2, 4, 13	syl2anc 691	. . . . . . 7 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (#‘(𝐹 ++ ⟨“𝐾”⟩)) = ((#‘𝐹) + (#‘⟨“𝐾”⟩)))
15		s1len 13238	. . . . . . . 8 ⊢ (#‘⟨“𝐾”⟩) = 1
16	15	oveq2i 6560	. . . . . . 7 ⊢ ((#‘𝐹) + (#‘⟨“𝐾”⟩)) = ((#‘𝐹) + 1)
17	14, 16	syl6eq 2660	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (#‘(𝐹 ++ ⟨“𝐾”⟩)) = ((#‘𝐹) + 1))
18	17	oveq2d 6565	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (1..^(#‘(𝐹 ++ ⟨“𝐾”⟩))) = (1..^((#‘𝐹) + 1)))
19	18	sumeq1d 14279	. . . 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 4260	. . . . . . . 8 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) ↔ (𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅))
21		lennncl 13180	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅) → (#‘𝐹) ∈ ℕ)
22	20, 21	sylbi 206	. . . . . . 7 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (#‘𝐹) ∈ ℕ)
23		nnuz 11599	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
24	22, 23	syl6eleq 2698	. . . . . 6 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (#‘𝐹) ∈ (ℤ_≥‘1))
25	24	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (#‘𝐹) ∈ (ℤ_≥‘1))
26		1cnd 9935	. . . . . 6 ⊢ ((((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1...(#‘𝐹))) ∧ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1))) → 1 ∈ ℂ)
27		0cnd 9912	. . . . . 6 ⊢ ((((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1...(#‘𝐹))) ∧ ¬ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1))) → 0 ∈ ℂ)
28	26, 27	ifclda 4070	. . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1...(#‘𝐹))) → if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)), 1, 0) ∈ ℂ)
29		fveq2 6103	. . . . . . 7 ⊢ (𝑗 = (#‘𝐹) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) = ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(#‘𝐹)))
30		oveq1 6556	. . . . . . . 8 ⊢ (𝑗 = (#‘𝐹) → (𝑗 − 1) = ((#‘𝐹) − 1))
31	30	fveq2d 6107	. . . . . . 7 ⊢ (𝑗 = (#‘𝐹) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)) = ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((#‘𝐹) − 1)))
32	29, 31	neeq12d 2843	. . . . . 6 ⊢ (𝑗 = (#‘𝐹) → (((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)) ↔ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(#‘𝐹)) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((#‘𝐹) − 1))))
33	32	ifbid 4058	. . . . 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))
34	25, 28, 33	fzosump1 14325	. . . 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)))
35	12, 19, 34	3eqtrd 2648	. . 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)))
36	35	adantlr 747	. 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)))
37	2	adantr 480	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(#‘𝐹))) → 𝐹 ∈ Word ℝ)
38	3	adantr 480	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(#‘𝐹))) → 𝐾 ∈ ℝ)
39		fzo0ss1 12367	. . . . . . . . . . 11 ⊢ (1..^(#‘𝐹)) ⊆ (0..^(#‘𝐹))
40	39	a1i 11	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (1..^(#‘𝐹)) ⊆ (0..^(#‘𝐹)))
41	40	sselda 3568	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(#‘𝐹))) → 𝑗 ∈ (0..^(#‘𝐹)))
42	7, 8, 9, 10	signstfvp 29974	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑗 ∈ (0..^(#‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) = ((𝑇‘𝐹)‘𝑗))
43	37, 38, 41, 42	syl3anc 1318	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(#‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) = ((𝑇‘𝐹)‘𝑗))
44		elfzoel2 12338	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1..^(#‘𝐹)) → (#‘𝐹) ∈ ℤ)
45	44	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(#‘𝐹))) → (#‘𝐹) ∈ ℤ)
46		1nn0 11185	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ₀
47		eluzmn 11570	. . . . . . . . . . . 12 ⊢ (((#‘𝐹) ∈ ℤ ∧ 1 ∈ ℕ₀) → (#‘𝐹) ∈ (ℤ_≥‘((#‘𝐹) − 1)))
48	45, 46, 47	sylancl 693	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(#‘𝐹))) → (#‘𝐹) ∈ (ℤ_≥‘((#‘𝐹) − 1)))
49		fzoss2 12365	. . . . . . . . . . 11 ⊢ ((#‘𝐹) ∈ (ℤ_≥‘((#‘𝐹) − 1)) → (0..^((#‘𝐹) − 1)) ⊆ (0..^(#‘𝐹)))
50	48, 49	syl 17	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(#‘𝐹))) → (0..^((#‘𝐹) − 1)) ⊆ (0..^(#‘𝐹)))
51		simpr 476	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(#‘𝐹))) → 𝑗 ∈ (1..^(#‘𝐹)))
52		elfzoelz 12339	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1..^(#‘𝐹)) → 𝑗 ∈ ℤ)
53	52	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(#‘𝐹))) → 𝑗 ∈ ℤ)
54		elfzom1b 12433	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ ℤ ∧ (#‘𝐹) ∈ ℤ) → (𝑗 ∈ (1..^(#‘𝐹)) ↔ (𝑗 − 1) ∈ (0..^((#‘𝐹) − 1))))
55	53, 45, 54	syl2anc 691	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(#‘𝐹))) → (𝑗 ∈ (1..^(#‘𝐹)) ↔ (𝑗 − 1) ∈ (0..^((#‘𝐹) − 1))))
56	51, 55	mpbid 221	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(#‘𝐹))) → (𝑗 − 1) ∈ (0..^((#‘𝐹) − 1)))
57	50, 56	sseldd 3569	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(#‘𝐹))) → (𝑗 − 1) ∈ (0..^(#‘𝐹)))
58	7, 8, 9, 10	signstfvp 29974	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ (𝑗 − 1) ∈ (0..^(#‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)) = ((𝑇‘𝐹)‘(𝑗 − 1)))
59	37, 38, 57, 58	syl3anc 1318	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑗 ∈ (1..^(#‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)) = ((𝑇‘𝐹)‘(𝑗 − 1)))
60	43, 59	neeq12d 2843	. . . . . . 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))))
61	60	ifbid 4058	. . . . . 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))
62	61	sumeq2dv 14281	. . . . 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))
63	7, 8, 9, 10	signsvvfval 29981	. . . . . 6 ⊢ (𝐹 ∈ Word ℝ → (𝑉‘𝐹) = Σ𝑗 ∈ (1..^(#‘𝐹))if(((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1)), 1, 0))
64	2, 63	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (𝑉‘𝐹) = Σ𝑗 ∈ (1..^(#‘𝐹))if(((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1)), 1, 0))
65	62, 64	eqtr4d 2647	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → Σ𝑗 ∈ (1..^(#‘𝐹))if(((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑗) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(𝑗 − 1)), 1, 0) = (𝑉‘𝐹))
66	65	adantlr 747	. . 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))
67	7, 8, 9, 10	signstfvn 29972	. . . . . . 7 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(#‘𝐹)) = (((𝑇‘𝐹)‘((#‘𝐹) − 1)) ⨣ (sgn‘𝐾)))
68	67	adantlr 747	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(#‘𝐹)) = (((𝑇‘𝐹)‘((#‘𝐹) − 1)) ⨣ (sgn‘𝐾)))
69	2	adantlr 747	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → 𝐹 ∈ Word ℝ)
70		simpr 476	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → 𝐾 ∈ ℝ)
71	22	ad2antrr 758	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (#‘𝐹) ∈ ℕ)
72		fzo0end 12426	. . . . . . . 8 ⊢ ((#‘𝐹) ∈ ℕ → ((#‘𝐹) − 1) ∈ (0..^(#‘𝐹)))
73	71, 72	syl 17	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((#‘𝐹) − 1) ∈ (0..^(#‘𝐹)))
74	7, 8, 9, 10	signstfvp 29974	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ ((#‘𝐹) − 1) ∈ (0..^(#‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((#‘𝐹) − 1)) = ((𝑇‘𝐹)‘((#‘𝐹) − 1)))
75	69, 70, 73, 74	syl3anc 1318	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((#‘𝐹) − 1)) = ((𝑇‘𝐹)‘((#‘𝐹) − 1)))
76	68, 75	neeq12d 2843	. . . . 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))))
77	7, 8, 9, 10	signstfvcl 29976	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ ((#‘𝐹) − 1) ∈ (0..^(#‘𝐹))) → ((𝑇‘𝐹)‘((#‘𝐹) − 1)) ∈ {-1, 1})
78	73, 77	syldan 486	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘𝐹)‘((#‘𝐹) − 1)) ∈ {-1, 1})
79	70	rexrd 9968	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → 𝐾 ∈ ℝ^*)
80		sgncl 29927	. . . . . . 7 ⊢ (𝐾 ∈ ℝ^* → (sgn‘𝐾) ∈ {-1, 0, 1})
81	79, 80	syl 17	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (sgn‘𝐾) ∈ {-1, 0, 1})
82	7, 8	signswch 29964	. . . . . 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))
83	78, 81, 82	syl2anc 691	. . . . 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))
84		sgnsgn 29937	. . . . . . . . 9 ⊢ (𝐾 ∈ ℝ^* → (sgn‘(sgn‘𝐾)) = (sgn‘𝐾))
85	79, 84	syl 17	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (sgn‘(sgn‘𝐾)) = (sgn‘𝐾))
86	85	oveq2d 6565	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((sgn‘((𝑇‘𝐹)‘((#‘𝐹) − 1))) · (sgn‘(sgn‘𝐾))) = ((sgn‘((𝑇‘𝐹)‘((#‘𝐹) − 1))) · (sgn‘𝐾)))
87	86	breq1d 4593	. . . . . 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))
88		neg1rr 11002	. . . . . . . . 9 ⊢ -1 ∈ ℝ
89		1re 9918	. . . . . . . . 9 ⊢ 1 ∈ ℝ
90		prssi 4293	. . . . . . . . 9 ⊢ ((-1 ∈ ℝ ∧ 1 ∈ ℝ) → {-1, 1} ⊆ ℝ)
91	88, 89, 90	mp2an 704	. . . . . . . 8 ⊢ {-1, 1} ⊆ ℝ
92	91, 78	sseldi 3566	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((𝑇‘𝐹)‘((#‘𝐹) − 1)) ∈ ℝ)
93		sgnclre 29928	. . . . . . . 8 ⊢ (𝐾 ∈ ℝ → (sgn‘𝐾) ∈ ℝ)
94	93	adantl 481	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (sgn‘𝐾) ∈ ℝ)
95		sgnmulsgn 29938	. . . . . . 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))
96	92, 94, 95	syl2anc 691	. . . . . 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))
97		sgnmulsgn 29938	. . . . . . 7 ⊢ ((((𝑇‘𝐹)‘((#‘𝐹) − 1)) ∈ ℝ ∧ 𝐾 ∈ ℝ) → ((((𝑇‘𝐹)‘((#‘𝐹) − 1)) · 𝐾) < 0 ↔ ((sgn‘((𝑇‘𝐹)‘((#‘𝐹) − 1))) · (sgn‘𝐾)) < 0))
98	92, 70, 97	syl2anc 691	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((((𝑇‘𝐹)‘((#‘𝐹) − 1)) · 𝐾) < 0 ↔ ((sgn‘((𝑇‘𝐹)‘((#‘𝐹) − 1))) · (sgn‘𝐾)) < 0))
99	87, 96, 98	3bitr4d 299	. . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → ((((𝑇‘𝐹)‘((#‘𝐹) − 1)) · (sgn‘𝐾)) < 0 ↔ (((𝑇‘𝐹)‘((#‘𝐹) − 1)) · 𝐾) < 0))
100	76, 83, 99	3bitrd 293	. . . 4 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(#‘𝐹)) ≠ ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘((#‘𝐹) − 1)) ↔ (((𝑇‘𝐹)‘((#‘𝐹) − 1)) · 𝐾) < 0))
101	100	ifbid 4058	. . 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))
102	66, 101	oveq12d 6567	. 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)))
103	36, 102	eqtrd 2644	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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 ⊆ wss 3540 ∅c0 3874 ifcif 4036 {csn 4125 {cpr 4127 {ctp 4129 ⟨cop 4131 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 ℝ^*cxr 9952 < clt 9953 − cmin 10145 -cneg 10146 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 ++ cconcat 13148 ⟨“cs1 13149 sgncsgn 13674 Σcsu 14264 ndxcnx 15692 Basecbs 15695 +_gcplusg 15768 Σ_g cgsu 15924

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-word 13154 df-lsw 13155 df-concat 13156 df-s1 13157 df-substr 13158 df-sgn 13675 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-0g 15925 df-gsum 15926 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mulg 17364 df-cntz 17573

This theorem is referenced by: signsvtp 29986 signsvtn 29987 signlem0 29990

W3C validator