signstfvn - Mathbox for Thierry Arnoux

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Theorem signstfvn 30646

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 30631	. . . 4 ⊢ {-1, 0, 1} = (Base‘𝑊)
4	1, 2	signswmnd 30634	. . . . 5 ⊢ 𝑊 ∈ Mnd
5	4	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 𝑊 ∈ Mnd)
6		eldifi 3732	. . . . . . . . 9 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → 𝐹 ∈ Word ℝ)
7		lencl 13324	. . . . . . . . 9 ⊢ (𝐹 ∈ Word ℝ → (#‘𝐹) ∈ ℕ₀)
8	6, 7	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (#‘𝐹) ∈ ℕ₀)
9		eldifsn 4317	. . . . . . . . 9 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) ↔ (𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅))
10		hasheq0 13154	. . . . . . . . . . 11 ⊢ (𝐹 ∈ Word ℝ → ((#‘𝐹) = 0 ↔ 𝐹 = ∅))
11	10	necon3bid 2838	. . . . . . . . . 10 ⊢ (𝐹 ∈ Word ℝ → ((#‘𝐹) ≠ 0 ↔ 𝐹 ≠ ∅))
12	11	biimpar 502	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅) → (#‘𝐹) ≠ 0)
13	9, 12	sylbi 207	. . . . . . . 8 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (#‘𝐹) ≠ 0)
14		elnnne0 11306	. . . . . . . 8 ⊢ ((#‘𝐹) ∈ ℕ ↔ ((#‘𝐹) ∈ ℕ₀ ∧ (#‘𝐹) ≠ 0))
15	8, 13, 14	sylanbrc 698	. . . . . . 7 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (#‘𝐹) ∈ ℕ)
16	15	adantr 481	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (#‘𝐹) ∈ ℕ)
17		nnm1nn0 11334	. . . . . 6 ⊢ ((#‘𝐹) ∈ ℕ → ((#‘𝐹) − 1) ∈ ℕ₀)
18	16, 17	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((#‘𝐹) − 1) ∈ ℕ₀)
19		nn0uz 11722	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
20	18, 19	syl6eleq 2711	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((#‘𝐹) − 1) ∈ (ℤ_≥‘0))
21		s1cl 13382	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℝ → ⟨“𝐾”⟩ ∈ Word ℝ)
22		ccatcl 13359	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
23	6, 21, 22	syl2an 494	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
24	23	adantr 481	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((#‘𝐹) − 1))) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
25		wrdf 13310	. . . . . . . 8 ⊢ ((𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ → (𝐹 ++ ⟨“𝐾”⟩):(0..^(#‘(𝐹 ++ ⟨“𝐾”⟩)))⟶ℝ)
26	24, 25	syl 17	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((#‘𝐹) − 1))) → (𝐹 ++ ⟨“𝐾”⟩):(0..^(#‘(𝐹 ++ ⟨“𝐾”⟩)))⟶ℝ)
27	8	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (#‘𝐹) ∈ ℕ₀)
28	27	nn0zd 11480	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (#‘𝐹) ∈ ℤ)
29		fzoval 12471	. . . . . . . . . . 11 ⊢ ((#‘𝐹) ∈ ℤ → (0..^(#‘𝐹)) = (0...((#‘𝐹) − 1)))
30	28, 29	syl 17	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (0..^(#‘𝐹)) = (0...((#‘𝐹) − 1)))
31		fzossfz 12488	. . . . . . . . . 10 ⊢ (0..^(#‘𝐹)) ⊆ (0...(#‘𝐹))
32	30, 31	syl6eqssr 3656	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (0...((#‘𝐹) − 1)) ⊆ (0...(#‘𝐹)))
33		ccatlen 13360	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ) → (#‘(𝐹 ++ ⟨“𝐾”⟩)) = ((#‘𝐹) + (#‘⟨“𝐾”⟩)))
34	6, 21, 33	syl2an 494	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (#‘(𝐹 ++ ⟨“𝐾”⟩)) = ((#‘𝐹) + (#‘⟨“𝐾”⟩)))
35		s1len 13385	. . . . . . . . . . . . 13 ⊢ (#‘⟨“𝐾”⟩) = 1
36	35	oveq2i 6661	. . . . . . . . . . . 12 ⊢ ((#‘𝐹) + (#‘⟨“𝐾”⟩)) = ((#‘𝐹) + 1)
37	34, 36	syl6eq 2672	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (#‘(𝐹 ++ ⟨“𝐾”⟩)) = ((#‘𝐹) + 1))
38	37	oveq2d 6666	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (0..^(#‘(𝐹 ++ ⟨“𝐾”⟩))) = (0..^((#‘𝐹) + 1)))
39	28	peano2zd 11485	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((#‘𝐹) + 1) ∈ ℤ)
40		fzoval 12471	. . . . . . . . . . 11 ⊢ (((#‘𝐹) + 1) ∈ ℤ → (0..^((#‘𝐹) + 1)) = (0...(((#‘𝐹) + 1) − 1)))
41	39, 40	syl 17	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (0..^((#‘𝐹) + 1)) = (0...(((#‘𝐹) + 1) − 1)))
42	27	nn0cnd 11353	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (#‘𝐹) ∈ ℂ)
43		1cnd 10056	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 1 ∈ ℂ)
44	42, 43	pncand 10393	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (((#‘𝐹) + 1) − 1) = (#‘𝐹))
45	44	oveq2d 6666	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (0...(((#‘𝐹) + 1) − 1)) = (0...(#‘𝐹)))
46	38, 41, 45	3eqtrd 2660	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (0..^(#‘(𝐹 ++ ⟨“𝐾”⟩))) = (0...(#‘𝐹)))
47	32, 46	sseqtr4d 3642	. . . . . . . 8 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (0...((#‘𝐹) − 1)) ⊆ (0..^(#‘(𝐹 ++ ⟨“𝐾”⟩))))
48	47	sselda 3603	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((#‘𝐹) − 1))) → 𝑖 ∈ (0..^(#‘(𝐹 ++ ⟨“𝐾”⟩))))
49	26, 48	ffvelrnd 6360	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((#‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) ∈ ℝ)
50	49	rexrd 10089	. . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((#‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) ∈ ℝ^*)
51		sgncl 30600	. . . . 5 ⊢ (((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) ∈ ℝ^* → (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)) ∈ {-1, 0, 1})
52	50, 51	syl 17	. . . 4 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((#‘𝐹) − 1))) → (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)) ∈ {-1, 0, 1})
53	1, 2	signswplusg 30632	. . . 4 ⊢ ⨣ = (+_g‘𝑊)
54		simpr 477	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 𝐾 ∈ ℝ)
55	54	rexrd 10089	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 𝐾 ∈ ℝ^*)
56		sgncl 30600	. . . . 5 ⊢ (𝐾 ∈ ℝ^* → (sgn‘𝐾) ∈ {-1, 0, 1})
57	55, 56	syl 17	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (sgn‘𝐾) ∈ {-1, 0, 1})
58		simpr 477	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((#‘𝐹) − 1) + 1)) → 𝑖 = (((#‘𝐹) − 1) + 1))
59	42, 43	npcand 10396	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (((#‘𝐹) − 1) + 1) = (#‘𝐹))
60	59	adantr 481	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((#‘𝐹) − 1) + 1)) → (((#‘𝐹) − 1) + 1) = (#‘𝐹))
61	58, 60	eqtrd 2656	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((#‘𝐹) − 1) + 1)) → 𝑖 = (#‘𝐹))
62	61	fveq2d 6195	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((#‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = ((𝐹 ++ ⟨“𝐾”⟩)‘(#‘𝐹)))
63	6	adantr 481	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 𝐹 ∈ Word ℝ)
64	54, 21	syl 17	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ⟨“𝐾”⟩ ∈ Word ℝ)
65		c0ex 10034	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
66	65	snid 4208	. . . . . . . . . . . 12 ⊢ 0 ∈ {0}
67		fzo01 12550	. . . . . . . . . . . 12 ⊢ (0..^1) = {0}
68	66, 67	eleqtrri 2700	. . . . . . . . . . 11 ⊢ 0 ∈ (0..^1)
69	35	oveq2i 6661	. . . . . . . . . . 11 ⊢ (0..^(#‘⟨“𝐾”⟩)) = (0..^1)
70	68, 69	eleqtrri 2700	. . . . . . . . . 10 ⊢ 0 ∈ (0..^(#‘⟨“𝐾”⟩))
71	70	a1i 11	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 0 ∈ (0..^(#‘⟨“𝐾”⟩)))
72		ccatval3 13363	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ ∧ 0 ∈ (0..^(#‘⟨“𝐾”⟩))) → ((𝐹 ++ ⟨“𝐾”⟩)‘(0 + (#‘𝐹))) = (⟨“𝐾”⟩‘0))
73	63, 64, 71, 72	syl3anc 1326	. . . . . . . 8 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝐹 ++ ⟨“𝐾”⟩)‘(0 + (#‘𝐹))) = (⟨“𝐾”⟩‘0))
74	42	addid2d 10237	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (0 + (#‘𝐹)) = (#‘𝐹))
75	74	fveq2d 6195	. . . . . . . 8 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝐹 ++ ⟨“𝐾”⟩)‘(0 + (#‘𝐹))) = ((𝐹 ++ ⟨“𝐾”⟩)‘(#‘𝐹)))
76		s1fv 13390	. . . . . . . . 9 ⊢ (𝐾 ∈ ℝ → (⟨“𝐾”⟩‘0) = 𝐾)
77	54, 76	syl 17	. . . . . . . 8 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (⟨“𝐾”⟩‘0) = 𝐾)
78	73, 75, 77	3eqtr3d 2664	. . . . . . 7 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝐹 ++ ⟨“𝐾”⟩)‘(#‘𝐹)) = 𝐾)
79	78	adantr 481	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((#‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘(#‘𝐹)) = 𝐾)
80	62, 79	eqtrd 2656	. . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((#‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = 𝐾)
81	80	fveq2d 6195	. . . 4 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((#‘𝐹) − 1) + 1)) → (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)) = (sgn‘𝐾))
82	3, 5, 20, 52, 53, 57, 81	gsumnunsn 30615	. . 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)))
83	59	oveq2d 6666	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (0...(((#‘𝐹) − 1) + 1)) = (0...(#‘𝐹)))
84	83	mpteq1d 4738	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (𝑖 ∈ (0...(((#‘𝐹) − 1) + 1)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖))) = (𝑖 ∈ (0...(#‘𝐹)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖))))
85	84	oveq2d 6666	. . 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)))))
86	63	adantr 481	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((#‘𝐹) − 1))) → 𝐹 ∈ Word ℝ)
87	64	adantr 481	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((#‘𝐹) − 1))) → ⟨“𝐾”⟩ ∈ Word ℝ)
88	30	eleq2d 2687	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (𝑖 ∈ (0..^(#‘𝐹)) ↔ 𝑖 ∈ (0...((#‘𝐹) − 1))))
89	88	biimpar 502	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((#‘𝐹) − 1))) → 𝑖 ∈ (0..^(#‘𝐹)))
90		ccatval1 13361	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ ∧ 𝑖 ∈ (0..^(#‘𝐹))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = (𝐹‘𝑖))
91	86, 87, 89, 90	syl3anc 1326	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((#‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = (𝐹‘𝑖))
92	91	fveq2d 6195	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((#‘𝐹) − 1))) → (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)) = (sgn‘(𝐹‘𝑖)))
93	92	mpteq2dva 4744	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (𝑖 ∈ (0...((#‘𝐹) − 1)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖))) = (𝑖 ∈ (0...((#‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖))))
94	93	oveq2d 6666	. . . 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)))))
95	94	oveq1d 6665	. . 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)))
96	82, 85, 95	3eqtr3d 2664	. 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)))
97		eqidd 2623	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (#‘𝐹) = (#‘𝐹))
98	97	olcd 408	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((#‘𝐹) ∈ (0..^(#‘𝐹)) ∨ (#‘𝐹) = (#‘𝐹)))
99	27, 19	syl6eleq 2711	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (#‘𝐹) ∈ (ℤ_≥‘0))
100		fzosplitsni 12579	. . . . . 6 ⊢ ((#‘𝐹) ∈ (ℤ_≥‘0) → ((#‘𝐹) ∈ (0..^((#‘𝐹) + 1)) ↔ ((#‘𝐹) ∈ (0..^(#‘𝐹)) ∨ (#‘𝐹) = (#‘𝐹))))
101	99, 100	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((#‘𝐹) ∈ (0..^((#‘𝐹) + 1)) ↔ ((#‘𝐹) ∈ (0..^(#‘𝐹)) ∨ (#‘𝐹) = (#‘𝐹))))
102	98, 101	mpbird 247	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (#‘𝐹) ∈ (0..^((#‘𝐹) + 1)))
103	102, 38	eleqtrrd 2704	. . 3 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (#‘𝐹) ∈ (0..^(#‘(𝐹 ++ ⟨“𝐾”⟩))))
104		signsv.t	. . . 4 ⊢ 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σ_g (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓‘𝑖))))))
105		signsv.v	. . . 4 ⊢ 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0))
106	1, 2, 104, 105	signstfval 30641	. . 3 ⊢ (((𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ ∧ (#‘𝐹) ∈ (0..^(#‘(𝐹 ++ ⟨“𝐾”⟩)))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(#‘𝐹)) = (𝑊 Σ_g (𝑖 ∈ (0...(#‘𝐹)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)))))
107	23, 103, 106	syl2anc 693	. 2 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(#‘𝐹)) = (𝑊 Σ_g (𝑖 ∈ (0...(#‘𝐹)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)))))
108		fzo0end 12560	. . . . . 6 ⊢ ((#‘𝐹) ∈ ℕ → ((#‘𝐹) − 1) ∈ (0..^(#‘𝐹)))
109	15, 108	syl 17	. . . . 5 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → ((#‘𝐹) − 1) ∈ (0..^(#‘𝐹)))
110	1, 2, 104, 105	signstfval 30641	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ ((#‘𝐹) − 1) ∈ (0..^(#‘𝐹))) → ((𝑇‘𝐹)‘((#‘𝐹) − 1)) = (𝑊 Σ_g (𝑖 ∈ (0...((#‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖)))))
111	6, 109, 110	syl2anc 693	. . . 4 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → ((𝑇‘𝐹)‘((#‘𝐹) − 1)) = (𝑊 Σ_g (𝑖 ∈ (0...((#‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖)))))
112	111	adantr 481	. . 3 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝑇‘𝐹)‘((#‘𝐹) − 1)) = (𝑊 Σ_g (𝑖 ∈ (0...((#‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖)))))
113	112	oveq1d 6665	. 2 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (((𝑇‘𝐹)‘((#‘𝐹) − 1)) ⨣ (sgn‘𝐾)) = ((𝑊 Σ_g (𝑖 ∈ (0...((#‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖)))) ⨣ (sgn‘𝐾)))
114	96, 107, 113	3eqtr4d 2666	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 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 ∅c0 3915 ifcif 4086 {csn 4177 {cpr 4179 {ctp 4181 ⟨cop 4183 ↦ cmpt 4729 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 ℝ^*cxr 10073 − cmin 10266 -cneg 10267 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 ++ cconcat 13293 ⟨“cs1 13294 sgncsgn 13826 Σcsu 14416 ndxcnx 15854 Basecbs 15857 +_gcplusg 15941 Σ_g cgsu 16101 Mndcmnd 17294

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-sgn 13827 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-0g 16102 df-gsum 16103 df-mgm 17242 df-sgrp 17284 df-mnd 17295

This theorem is referenced by: signsvtn0 30647 signstfvneq0 30649 signstfveq0 30654 signsvfn 30659

W3C validator