signstfvn - Mathbox for Thierry Arnoux

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Theorem signstfvn 34567

Description: Zero-skipping sign in a word compared to a shorter word. (Contributed by Thierry Arnoux, 8-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
signstfvn	⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) = (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ⨣ (sgn‘𝐾)))

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

**Proof of Theorem signstfvn**
Step	Hyp	Ref	Expression
1		signsv.p	. . . . 5 ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
2		signsv.w	. . . . 5 ⊢ 𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+_g‘ndx), ⨣ ⟩}
3	1, 2	signswbase 34552	. . . 4 ⊢ {-1, 0, 1} = (Base‘𝑊)
4	1, 2	signswmnd 34555	. . . . 5 ⊢ 𝑊 ∈ Mnd
5	4	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 𝑊 ∈ Mnd)
6		eldifi 4097	. . . . . . . . 9 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → 𝐹 ∈ Word ℝ)
7		lencl 14505	. . . . . . . . 9 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℕ₀)
8	6, 7	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (♯‘𝐹) ∈ ℕ₀)
9		eldifsn 4753	. . . . . . . . 9 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) ↔ (𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅))
10		hasheq0 14335	. . . . . . . . . . 11 ⊢ (𝐹 ∈ Word ℝ → ((♯‘𝐹) = 0 ↔ 𝐹 = ∅))
11	10	necon3bid 2970	. . . . . . . . . 10 ⊢ (𝐹 ∈ Word ℝ → ((♯‘𝐹) ≠ 0 ↔ 𝐹 ≠ ∅))
12	11	biimpar 477	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅) → (♯‘𝐹) ≠ 0)
13	9, 12	sylbi 217	. . . . . . . 8 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (♯‘𝐹) ≠ 0)
14		elnnne0 12463	. . . . . . . 8 ⊢ ((♯‘𝐹) ∈ ℕ ↔ ((♯‘𝐹) ∈ ℕ₀ ∧ (♯‘𝐹) ≠ 0))
15	8, 13, 14	sylanbrc 583	. . . . . . 7 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (♯‘𝐹) ∈ ℕ)
16	15	adantr 480	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (♯‘𝐹) ∈ ℕ)
17		nnm1nn0 12490	. . . . . 6 ⊢ ((♯‘𝐹) ∈ ℕ → ((♯‘𝐹) − 1) ∈ ℕ₀)
18	16, 17	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((♯‘𝐹) − 1) ∈ ℕ₀)
19		nn0uz 12842	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
20	18, 19	eleqtrdi 2839	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((♯‘𝐹) − 1) ∈ (ℤ_≥‘0))
21		ccatws1cl 14588	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
22	21	adantr 480	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
23		wrdf 14490	. . . . . . . . 9 ⊢ ((𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ → (𝐹 ++ ⟨“𝐾”⟩):(0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩)))⟶ℝ)
24	22, 23	syl 17	. . . . . . . 8 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → (𝐹 ++ ⟨“𝐾”⟩):(0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩)))⟶ℝ)
25	7	nn0zd 12562	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℤ)
26		fzoval 13628	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) ∈ ℤ → (0..^(♯‘𝐹)) = (0...((♯‘𝐹) − 1)))
27	25, 26	syl 17	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ Word ℝ → (0..^(♯‘𝐹)) = (0...((♯‘𝐹) − 1)))
28	27	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0..^(♯‘𝐹)) = (0...((♯‘𝐹) − 1)))
29		fzossfz 13646	. . . . . . . . . . 11 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
30	28, 29	eqsstrrdi 3995	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0...((♯‘𝐹) − 1)) ⊆ (0...(♯‘𝐹)))
31		s1cl 14574	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℝ → ⟨“𝐾”⟩ ∈ Word ℝ)
32		ccatlen 14547	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)))
33	31, 32	sylan2 593	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)))
34		s1len 14578	. . . . . . . . . . . . . 14 ⊢ (♯‘⟨“𝐾”⟩) = 1
35	34	oveq2i 7401	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)) = ((♯‘𝐹) + 1)
36	33, 35	eqtrdi 2781	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + 1))
37	36	oveq2d 7406	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))) = (0..^((♯‘𝐹) + 1)))
38	25	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (♯‘𝐹) ∈ ℤ)
39	38	peano2zd 12648	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → ((♯‘𝐹) + 1) ∈ ℤ)
40		fzoval 13628	. . . . . . . . . . . 12 ⊢ (((♯‘𝐹) + 1) ∈ ℤ → (0..^((♯‘𝐹) + 1)) = (0...(((♯‘𝐹) + 1) − 1)))
41	39, 40	syl 17	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0..^((♯‘𝐹) + 1)) = (0...(((♯‘𝐹) + 1) − 1)))
42	7	nn0cnd 12512	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℂ)
43		1cnd 11176	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ Word ℝ → 1 ∈ ℂ)
44	42, 43	pncand 11541	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ Word ℝ → (((♯‘𝐹) + 1) − 1) = (♯‘𝐹))
45	44	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (((♯‘𝐹) + 1) − 1) = (♯‘𝐹))
46	45	oveq2d 7406	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0...(((♯‘𝐹) + 1) − 1)) = (0...(♯‘𝐹)))
47	37, 41, 46	3eqtrd 2769	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))) = (0...(♯‘𝐹)))
48	30, 47	sseqtrrd 3987	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0...((♯‘𝐹) − 1)) ⊆ (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))))
49	48	sselda 3949	. . . . . . . 8 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → 𝑖 ∈ (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))))
50	24, 49	ffvelcdmd 7060	. . . . . . 7 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) ∈ ℝ)
51	6, 50	sylanl1 680	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) ∈ ℝ)
52	51	rexrd 11231	. . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) ∈ ℝ^*)
53		sgncl 32763	. . . . 5 ⊢ (((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) ∈ ℝ^* → (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)) ∈ {-1, 0, 1})
54	52, 53	syl 17	. . . 4 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)) ∈ {-1, 0, 1})
55	1, 2	signswplusg 34553	. . . 4 ⊢ ⨣ = (+_g‘𝑊)
56		rexr 11227	. . . . . 6 ⊢ (𝐾 ∈ ℝ → 𝐾 ∈ ℝ^*)
57		sgncl 32763	. . . . . 6 ⊢ (𝐾 ∈ ℝ^* → (sgn‘𝐾) ∈ {-1, 0, 1})
58	56, 57	syl 17	. . . . 5 ⊢ (𝐾 ∈ ℝ → (sgn‘𝐾) ∈ {-1, 0, 1})
59	58	adantl 481	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (sgn‘𝐾) ∈ {-1, 0, 1})
60		id 22	. . . . . . . . 9 ⊢ (𝑖 = (((♯‘𝐹) − 1) + 1) → 𝑖 = (((♯‘𝐹) − 1) + 1))
61	42, 43	npcand 11544	. . . . . . . . . 10 ⊢ (𝐹 ∈ Word ℝ → (((♯‘𝐹) − 1) + 1) = (♯‘𝐹))
62	61	adantr 480	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (((♯‘𝐹) − 1) + 1) = (♯‘𝐹))
63	60, 62	sylan9eqr 2787	. . . . . . . 8 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → 𝑖 = (♯‘𝐹))
64	63	fveq2d 6865	. . . . . . 7 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = ((𝐹 ++ ⟨“𝐾”⟩)‘(♯‘𝐹)))
65		ccatws1ls 14605	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → ((𝐹 ++ ⟨“𝐾”⟩)‘(♯‘𝐹)) = 𝐾)
66	65	adantr 480	. . . . . . 7 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘(♯‘𝐹)) = 𝐾)
67	64, 66	eqtrd 2765	. . . . . 6 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = 𝐾)
68	6, 67	sylanl1 680	. . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = 𝐾)
69	68	fveq2d 6865	. . . 4 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)) = (sgn‘𝐾))
70	3, 5, 20, 54, 55, 59, 69	gsumnunsn 34539	. . 3 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (𝑊 Σ_g (𝑖 ∈ (0...(((♯‘𝐹) − 1) + 1)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)))) = ((𝑊 Σ_g (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)))) ⨣ (sgn‘𝐾)))
71	6, 61	syl 17	. . . . . . 7 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (((♯‘𝐹) − 1) + 1) = (♯‘𝐹))
72	71	adantr 480	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (((♯‘𝐹) − 1) + 1) = (♯‘𝐹))
73	72	oveq2d 7406	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (0...(((♯‘𝐹) − 1) + 1)) = (0...(♯‘𝐹)))
74	73	mpteq1d 5200	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (𝑖 ∈ (0...(((♯‘𝐹) − 1) + 1)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖))) = (𝑖 ∈ (0...(♯‘𝐹)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖))))
75	74	oveq2d 7406	. . 3 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (𝑊 Σ_g (𝑖 ∈ (0...(((♯‘𝐹) − 1) + 1)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)))) = (𝑊 Σ_g (𝑖 ∈ (0...(♯‘𝐹)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)))))
76		simpll 766	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → 𝐹 ∈ Word ℝ)
77	31	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ⟨“𝐾”⟩ ∈ Word ℝ)
78	28	eleq2d 2815	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (𝑖 ∈ (0..^(♯‘𝐹)) ↔ 𝑖 ∈ (0...((♯‘𝐹) − 1))))
79	78	biimpar 477	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → 𝑖 ∈ (0..^(♯‘𝐹)))
80		ccatval1 14549	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = (𝐹‘𝑖))
81	76, 77, 79, 80	syl3anc 1373	. . . . . . . 8 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = (𝐹‘𝑖))
82	81	fveq2d 6865	. . . . . . 7 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)) = (sgn‘(𝐹‘𝑖)))
83	82	mpteq2dva 5203	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖))) = (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖))))
84	6, 83	sylan 580	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖))) = (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖))))
85	84	oveq2d 7406	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (𝑊 Σ_g (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)))) = (𝑊 Σ_g (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖)))))
86	85	oveq1d 7405	. . 3 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝑊 Σ_g (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)))) ⨣ (sgn‘𝐾)) = ((𝑊 Σ_g (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖)))) ⨣ (sgn‘𝐾)))
87	70, 75, 86	3eqtr3d 2773	. 2 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (𝑊 Σ_g (𝑖 ∈ (0...(♯‘𝐹)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)))) = ((𝑊 Σ_g (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖)))) ⨣ (sgn‘𝐾)))
88		eqid 2730	. . . . . . . 8 ⊢ (♯‘𝐹) = (♯‘𝐹)
89	88	olci 866	. . . . . . 7 ⊢ ((♯‘𝐹) ∈ (0..^(♯‘𝐹)) ∨ (♯‘𝐹) = (♯‘𝐹))
90	7, 19	eleqtrdi 2839	. . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ (ℤ_≥‘0))
91		fzosplitsni 13746	. . . . . . . 8 ⊢ ((♯‘𝐹) ∈ (ℤ_≥‘0) → ((♯‘𝐹) ∈ (0..^((♯‘𝐹) + 1)) ↔ ((♯‘𝐹) ∈ (0..^(♯‘𝐹)) ∨ (♯‘𝐹) = (♯‘𝐹))))
92	90, 91	syl 17	. . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → ((♯‘𝐹) ∈ (0..^((♯‘𝐹) + 1)) ↔ ((♯‘𝐹) ∈ (0..^(♯‘𝐹)) ∨ (♯‘𝐹) = (♯‘𝐹))))
93	89, 92	mpbiri 258	. . . . . 6 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ (0..^((♯‘𝐹) + 1)))
94	93	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (♯‘𝐹) ∈ (0..^((♯‘𝐹) + 1)))
95	94, 37	eleqtrrd 2832	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (♯‘𝐹) ∈ (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))))
96		signsv.t	. . . . 5 ⊢ 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(♯‘𝑓)) ↦ (𝑊 Σ_g (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓‘𝑖))))))
97		signsv.v	. . . . 5 ⊢ 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(♯‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0))
98	1, 2, 96, 97	signstfval 34562	. . . 4 ⊢ (((𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ ∧ (♯‘𝐹) ∈ (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩)))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) = (𝑊 Σ_g (𝑖 ∈ (0...(♯‘𝐹)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)))))
99	21, 95, 98	syl2anc 584	. . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) = (𝑊 Σ_g (𝑖 ∈ (0...(♯‘𝐹)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)))))
100	6, 99	sylan 580	. 2 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) = (𝑊 Σ_g (𝑖 ∈ (0...(♯‘𝐹)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)))))
101		fzo0end 13726	. . . . . 6 ⊢ ((♯‘𝐹) ∈ ℕ → ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹)))
102	15, 101	syl 17	. . . . 5 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹)))
103	1, 2, 96, 97	signstfval 34562	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹))) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) = (𝑊 Σ_g (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖)))))
104	6, 102, 103	syl2anc 584	. . . 4 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) = (𝑊 Σ_g (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖)))))
105	104	adantr 480	. . 3 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) = (𝑊 Σ_g (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖)))))
106	105	oveq1d 7405	. 2 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ⨣ (sgn‘𝐾)) = ((𝑊 Σ_g (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖)))) ⨣ (sgn‘𝐾)))
107	87, 100, 106	3eqtr4d 2775	1 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) = (((𝑇‘𝐹)‘((♯‘𝐹) − 1)) ⨣ (sgn‘𝐾)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∖ cdif 3914 ∅c0 4299 ifcif 4491 {csn 4592 {cpr 4594 {ctp 4596 ⟨cop 4598 ↦ cmpt 5191 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ∈ cmpo 7392 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 ℝ^*cxr 11214 − cmin 11412 -cneg 11413 ℕcn 12193 ℕ₀cn0 12449 ℤcz 12536 ℤ_≥cuz 12800 ...cfz 13475 ..^cfzo 13622 ♯chash 14302 Word cword 14485 ++ cconcat 14542 ⟨“cs1 14567 sgncsgn 15059 Σcsu 15659 ndxcnx 17170 Basecbs 17186 +_gcplusg 17227 Σ_g cgsu 17410 Mndcmnd 18668

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476 df-fzo 13623 df-seq 13974 df-hash 14303 df-word 14486 df-concat 14543 df-s1 14568 df-sgn 15060 df-struct 17124 df-slot 17159 df-ndx 17171 df-base 17187 df-plusg 17240 df-0g 17411 df-gsum 17412 df-mgm 18574 df-sgrp 18653 df-mnd 18669

This theorem is referenced by: signsvtn0 34568 signstfvneq0 34570 signstfveq0 34575 signsvfn 34580

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