signstfvn - Mathbox for Thierry Arnoux

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

Theorem signstfvn 34863

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 34848	. . . 4 ⊢ {-1, 0, 1} = (Base‘𝑊)
4	1, 2	signswmnd 34851	. . . . 5 ⊢ 𝑊 ∈ Mnd
5	4	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 𝑊 ∈ Mnd)
6		eldifi 4084	. . . . . . . . 9 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → 𝐹 ∈ Word ℝ)
7		lencl 14546	. . . . . . . . 9 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℕ₀)
8	6, 7	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (♯‘𝐹) ∈ ℕ₀)
9		eldifsn 4746	. . . . . . . . 9 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) ↔ (𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅))
10		hasheq0 14376	. . . . . . . . . . 11 ⊢ (𝐹 ∈ Word ℝ → ((♯‘𝐹) = 0 ↔ 𝐹 = ∅))
11	10	necon3bid 3001	. . . . . . . . . 10 ⊢ (𝐹 ∈ Word ℝ → ((♯‘𝐹) ≠ 0 ↔ 𝐹 ≠ ∅))
12	11	biimpar 481	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅) → (♯‘𝐹) ≠ 0)
13	9, 12	sylbi 219	. . . . . . . 8 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (♯‘𝐹) ≠ 0)
14		elnnne0 12495	. . . . . . . 8 ⊢ ((♯‘𝐹) ∈ ℕ ↔ ((♯‘𝐹) ∈ ℕ₀ ∧ (♯‘𝐹) ≠ 0))
15	8, 13, 14	sylanbrc 592	. . . . . . 7 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (♯‘𝐹) ∈ ℕ)
16	15	adantr 484	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (♯‘𝐹) ∈ ℕ)
17		nnm1nn0 12522	. . . . . 6 ⊢ ((♯‘𝐹) ∈ ℕ → ((♯‘𝐹) − 1) ∈ ℕ₀)
18	16, 17	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((♯‘𝐹) − 1) ∈ ℕ₀)
19		nn0uz 12877	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
20	18, 19	eleqtrdi 2872	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((♯‘𝐹) − 1) ∈ (ℤ_≥‘0))
21		ccatws1cl 14630	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
22	21	adantr 484	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
23		wrdf 14531	. . . . . . . . 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 12593	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℤ)
26		fzoval 13665	. . . . . . . . . . . . 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 484	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0..^(♯‘𝐹)) = (0...((♯‘𝐹) − 1)))
29		fzossfz 13684	. . . . . . . . . . 11 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
30	28, 29	eqsstrrdi 3981	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0...((♯‘𝐹) − 1)) ⊆ (0...(♯‘𝐹)))
31		s1cl 14616	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℝ → ⟨“𝐾”⟩ ∈ Word ℝ)
32		ccatlen 14588	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)))
33	31, 32	sylan2 602	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)))
34		s1len 14620	. . . . . . . . . . . . . 14 ⊢ (♯‘⟨“𝐾”⟩) = 1
35	34	oveq2i 7407	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)) = ((♯‘𝐹) + 1)
36	33, 35	eqtrdi 2813	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + 1))
37	36	oveq2d 7412	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))) = (0..^((♯‘𝐹) + 1)))
38	25	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (♯‘𝐹) ∈ ℤ)
39	38	peano2zd 12680	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → ((♯‘𝐹) + 1) ∈ ℤ)
40		fzoval 13665	. . . . . . . . . . . 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 12544	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℂ)
43		1cnd 11175	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ Word ℝ → 1 ∈ ℂ)
44	42, 43	pncand 11543	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ Word ℝ → (((♯‘𝐹) + 1) − 1) = (♯‘𝐹))
45	44	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (((♯‘𝐹) + 1) − 1) = (♯‘𝐹))
46	45	oveq2d 7412	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0...(((♯‘𝐹) + 1) − 1)) = (0...(♯‘𝐹)))
47	37, 41, 46	3eqtrd 2801	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))) = (0...(♯‘𝐹)))
48	30, 47	sseqtrrd 3973	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0...((♯‘𝐹) − 1)) ⊆ (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))))
49	48	sselda 3936	. . . . . . . 8 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → 𝑖 ∈ (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))))
50	24, 49	ffvelcdmd 7066	. . . . . . 7 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) ∈ ℝ)
51	6, 50	sylanl1 690	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) ∈ ℝ)
52	51	rexrd 11232	. . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) ∈ ℝ^*)
53		sgncl 15110	. . . . 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 34849	. . . 4 ⊢ ⨣ = (+_g‘𝑊)
56		rexr 11228	. . . . . 6 ⊢ (𝐾 ∈ ℝ → 𝐾 ∈ ℝ^*)
57		sgncl 15110	. . . . . 6 ⊢ (𝐾 ∈ ℝ^* → (sgn‘𝐾) ∈ {-1, 0, 1})
58	56, 57	syl 17	. . . . 5 ⊢ (𝐾 ∈ ℝ → (sgn‘𝐾) ∈ {-1, 0, 1})
59	58	adantl 485	. . . 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 11546	. . . . . . . . . 10 ⊢ (𝐹 ∈ Word ℝ → (((♯‘𝐹) − 1) + 1) = (♯‘𝐹))
62	61	adantr 484	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (((♯‘𝐹) − 1) + 1) = (♯‘𝐹))
63	60, 62	sylan9eqr 2819	. . . . . . . 8 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → 𝑖 = (♯‘𝐹))
64	63	fveq2d 6871	. . . . . . 7 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = ((𝐹 ++ ⟨“𝐾”⟩)‘(♯‘𝐹)))
65		ccatws1ls 14647	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → ((𝐹 ++ ⟨“𝐾”⟩)‘(♯‘𝐹)) = 𝐾)
66	65	adantr 484	. . . . . . 7 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘(♯‘𝐹)) = 𝐾)
67	64, 66	eqtrd 2797	. . . . . 6 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = 𝐾)
68	6, 67	sylanl1 690	. . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = 𝐾)
69	68	fveq2d 6871	. . . 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 34838	. . 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 484	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (((♯‘𝐹) − 1) + 1) = (♯‘𝐹))
73	72	oveq2d 7412	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (0...(((♯‘𝐹) − 1) + 1)) = (0...(♯‘𝐹)))
74	73	mpteq1d 5190	. . . 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 7412	. . 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 776	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → 𝐹 ∈ Word ℝ)
77	31	ad2antlr 737	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ⟨“𝐾”⟩ ∈ Word ℝ)
78	28	eleq2d 2848	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (𝑖 ∈ (0..^(♯‘𝐹)) ↔ 𝑖 ∈ (0...((♯‘𝐹) − 1))))
79	78	biimpar 481	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → 𝑖 ∈ (0..^(♯‘𝐹)))
80		ccatval1 14590	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = (𝐹‘𝑖))
81	76, 77, 79, 80	syl3anc 1390	. . . . . . . 8 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = (𝐹‘𝑖))
82	81	fveq2d 6871	. . . . . . 7 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)) = (sgn‘(𝐹‘𝑖)))
83	82	mpteq2dva 5193	. . . . . 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 589	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖))) = (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖))))
85	84	oveq2d 7412	. . . 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 7411	. . 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 2805	. 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 2762	. . . . . . . 8 ⊢ (♯‘𝐹) = (♯‘𝐹)
89	88	olci 877	. . . . . . 7 ⊢ ((♯‘𝐹) ∈ (0..^(♯‘𝐹)) ∨ (♯‘𝐹) = (♯‘𝐹))
90	7, 19	eleqtrdi 2872	. . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ (ℤ_≥‘0))
91		fzosplitsni 13785	. . . . . . . 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 260	. . . . . 6 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ (0..^((♯‘𝐹) + 1)))
94	93	adantr 484	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (♯‘𝐹) ∈ (0..^((♯‘𝐹) + 1)))
95	94, 37	eleqtrrd 2865	. . . 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 34858	. . . 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 593	. . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) = (𝑊 Σ_g (𝑖 ∈ (0...(♯‘𝐹)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)))))
100	6, 99	sylan 589	. 2 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) = (𝑊 Σ_g (𝑖 ∈ (0...(♯‘𝐹)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)))))
101		fzo0end 13764	. . . . . 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 34858	. . . . 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 593	. . . 4 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) = (𝑊 Σ_g (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖)))))
105	104	adantr 484	. . 3 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) = (𝑊 Σ_g (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖)))))
106	105	oveq1d 7411	. 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 2807	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 208 ∧ wa 399 ∨ wo 858 = wceq 1560 ∈ wcel 2142 ≠ wne 2957 ∖ cdif 3901 ∅c0 4285 ifcif 4480 {csn 4582 {cpr 4584 {ctp 4586 ⟨cop 4588 ↦ cmpt 5181 ⟶wf 6517 ‘cfv 6521 (class class class)co 7396 ∈ cmpo 7398 ℝcr 11072 0cc0 11073 1c1 11074 + caddc 11076 ℝ^*cxr 11215 − cmin 11414 -cneg 11415 ℕcn 12210 ℕ₀cn0 12481 ℤcz 12568 ℤ_≥cuz 12839 ...cfz 13512 ..^cfzo 13659 ♯chash 14343 Word cword 14526 ++ cconcat 14583 ⟨“cs1 14609 sgncsgn 15099 Σcsu 15713 ndxcnx 17229 Basecbs 17245 +_gcplusg 17286 Σ_g cgsu 17469 Mndcmnd 18768

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-er 8678 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-nn 12211 df-2 12280 df-n0 12482 df-z 12569 df-uz 12840 df-fz 13513 df-fzo 13660 df-seq 14015 df-hash 14344 df-word 14527 df-concat 14584 df-s1 14610 df-sgn 15100 df-struct 17183 df-slot 17218 df-ndx 17230 df-base 17246 df-plusg 17299 df-0g 17470 df-gsum 17471 df-mgm 18674 df-sgrp 18753 df-mnd 18769

This theorem is referenced by: signsvtn0 34864 signstfvneq0 34866 signstfveq0 34871 signsvfn 34876

W3C validator