signstfvn - Mathbox for Thierry Arnoux

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Theorem signstfvn 29972

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 29957	. . . 4 ⊢ {-1, 0, 1} = (Base‘𝑊)
4	1, 2	signswmnd 29960	. . . . 5 ⊢ 𝑊 ∈ Mnd
5	4	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 𝑊 ∈ Mnd)
6		eldifi 3694	. . . . . . . . 9 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → 𝐹 ∈ Word ℝ)
7		lencl 13179	. . . . . . . . 9 ⊢ (𝐹 ∈ Word ℝ → (#‘𝐹) ∈ ℕ₀)
8	6, 7	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (#‘𝐹) ∈ ℕ₀)
9		eldifsn 4260	. . . . . . . . 9 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) ↔ (𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅))
10		hasheq0 13015	. . . . . . . . . . 11 ⊢ (𝐹 ∈ Word ℝ → ((#‘𝐹) = 0 ↔ 𝐹 = ∅))
11	10	necon3bid 2826	. . . . . . . . . 10 ⊢ (𝐹 ∈ Word ℝ → ((#‘𝐹) ≠ 0 ↔ 𝐹 ≠ ∅))
12	11	biimpar 501	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅) → (#‘𝐹) ≠ 0)
13	9, 12	sylbi 206	. . . . . . . 8 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (#‘𝐹) ≠ 0)
14		elnnne0 11183	. . . . . . . 8 ⊢ ((#‘𝐹) ∈ ℕ ↔ ((#‘𝐹) ∈ ℕ₀ ∧ (#‘𝐹) ≠ 0))
15	8, 13, 14	sylanbrc 695	. . . . . . 7 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (#‘𝐹) ∈ ℕ)
16	15	adantr 480	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (#‘𝐹) ∈ ℕ)
17		nnm1nn0 11211	. . . . . 6 ⊢ ((#‘𝐹) ∈ ℕ → ((#‘𝐹) − 1) ∈ ℕ₀)
18	16, 17	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((#‘𝐹) − 1) ∈ ℕ₀)
19		nn0uz 11598	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
20	18, 19	syl6eleq 2698	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((#‘𝐹) − 1) ∈ (ℤ_≥‘0))
21		s1cl 13235	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℝ → ⟨“𝐾”⟩ ∈ Word ℝ)
22		ccatcl 13212	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
23	6, 21, 22	syl2an 493	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
24	23	adantr 480	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((#‘𝐹) − 1))) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
25		wrdf 13165	. . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (#‘𝐹) ∈ ℕ₀)
28	27	nn0zd 11356	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (#‘𝐹) ∈ ℤ)
29		fzoval 12340	. . . . . . . . . . 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 12357	. . . . . . . . . 10 ⊢ (0..^(#‘𝐹)) ⊆ (0...(#‘𝐹))
32	30, 31	syl6eqssr 3619	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (0...((#‘𝐹) − 1)) ⊆ (0...(#‘𝐹)))
33		ccatlen 13213	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ) → (#‘(𝐹 ++ ⟨“𝐾”⟩)) = ((#‘𝐹) + (#‘⟨“𝐾”⟩)))
34	6, 21, 33	syl2an 493	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (#‘(𝐹 ++ ⟨“𝐾”⟩)) = ((#‘𝐹) + (#‘⟨“𝐾”⟩)))
35		s1len 13238	. . . . . . . . . . . . 13 ⊢ (#‘⟨“𝐾”⟩) = 1
36	35	oveq2i 6560	. . . . . . . . . . . 12 ⊢ ((#‘𝐹) + (#‘⟨“𝐾”⟩)) = ((#‘𝐹) + 1)
37	34, 36	syl6eq 2660	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (#‘(𝐹 ++ ⟨“𝐾”⟩)) = ((#‘𝐹) + 1))
38	37	oveq2d 6565	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (0..^(#‘(𝐹 ++ ⟨“𝐾”⟩))) = (0..^((#‘𝐹) + 1)))
39	28	peano2zd 11361	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((#‘𝐹) + 1) ∈ ℤ)
40		fzoval 12340	. . . . . . . . . . 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 11230	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (#‘𝐹) ∈ ℂ)
43		1cnd 9935	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 1 ∈ ℂ)
44	42, 43	pncand 10272	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (((#‘𝐹) + 1) − 1) = (#‘𝐹))
45	44	oveq2d 6565	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (0...(((#‘𝐹) + 1) − 1)) = (0...(#‘𝐹)))
46	38, 41, 45	3eqtrd 2648	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (0..^(#‘(𝐹 ++ ⟨“𝐾”⟩))) = (0...(#‘𝐹)))
47	32, 46	sseqtr4d 3605	. . . . . . . 8 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (0...((#‘𝐹) − 1)) ⊆ (0..^(#‘(𝐹 ++ ⟨“𝐾”⟩))))
48	47	sselda 3568	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((#‘𝐹) − 1))) → 𝑖 ∈ (0..^(#‘(𝐹 ++ ⟨“𝐾”⟩))))
49	26, 48	ffvelrnd 6268	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((#‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) ∈ ℝ)
50	49	rexrd 9968	. . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((#‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) ∈ ℝ^*)
51		sgncl 29927	. . . . 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 29958	. . . 4 ⊢ ⨣ = (+_g‘𝑊)
54		simpr 476	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 𝐾 ∈ ℝ)
55	54	rexrd 9968	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 𝐾 ∈ ℝ^*)
56		sgncl 29927	. . . . 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 476	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((#‘𝐹) − 1) + 1)) → 𝑖 = (((#‘𝐹) − 1) + 1))
59	42, 43	npcand 10275	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (((#‘𝐹) − 1) + 1) = (#‘𝐹))
60	59	adantr 480	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((#‘𝐹) − 1) + 1)) → (((#‘𝐹) − 1) + 1) = (#‘𝐹))
61	58, 60	eqtrd 2644	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((#‘𝐹) − 1) + 1)) → 𝑖 = (#‘𝐹))
62	61	fveq2d 6107	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((#‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = ((𝐹 ++ ⟨“𝐾”⟩)‘(#‘𝐹)))
63	6	adantr 480	. . . . . . . . 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 9913	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
66	65	snid 4155	. . . . . . . . . . . 12 ⊢ 0 ∈ {0}
67		fzo01 12417	. . . . . . . . . . . 12 ⊢ (0..^1) = {0}
68	66, 67	eleqtrri 2687	. . . . . . . . . . 11 ⊢ 0 ∈ (0..^1)
69	35	oveq2i 6560	. . . . . . . . . . 11 ⊢ (0..^(#‘⟨“𝐾”⟩)) = (0..^1)
70	68, 69	eleqtrri 2687	. . . . . . . . . 10 ⊢ 0 ∈ (0..^(#‘⟨“𝐾”⟩))
71	70	a1i 11	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 0 ∈ (0..^(#‘⟨“𝐾”⟩)))
72		ccatval3 13216	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ ∧ 0 ∈ (0..^(#‘⟨“𝐾”⟩))) → ((𝐹 ++ ⟨“𝐾”⟩)‘(0 + (#‘𝐹))) = (⟨“𝐾”⟩‘0))
73	63, 64, 71, 72	syl3anc 1318	. . . . . . . 8 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝐹 ++ ⟨“𝐾”⟩)‘(0 + (#‘𝐹))) = (⟨“𝐾”⟩‘0))
74	42	addid2d 10116	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (0 + (#‘𝐹)) = (#‘𝐹))
75	74	fveq2d 6107	. . . . . . . 8 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝐹 ++ ⟨“𝐾”⟩)‘(0 + (#‘𝐹))) = ((𝐹 ++ ⟨“𝐾”⟩)‘(#‘𝐹)))
76		s1fv 13243	. . . . . . . . 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 2652	. . . . . . 7 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝐹 ++ ⟨“𝐾”⟩)‘(#‘𝐹)) = 𝐾)
79	78	adantr 480	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((#‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘(#‘𝐹)) = 𝐾)
80	62, 79	eqtrd 2644	. . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((#‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = 𝐾)
81	80	fveq2d 6107	. . . 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 29942	. . 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 6565	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (0...(((#‘𝐹) − 1) + 1)) = (0...(#‘𝐹)))
84	83	mpteq1d 4666	. . . 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 6565	. . 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 480	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((#‘𝐹) − 1))) → 𝐹 ∈ Word ℝ)
87	64	adantr 480	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((#‘𝐹) − 1))) → ⟨“𝐾”⟩ ∈ Word ℝ)
88	30	eleq2d 2673	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (𝑖 ∈ (0..^(#‘𝐹)) ↔ 𝑖 ∈ (0...((#‘𝐹) − 1))))
89	88	biimpar 501	. . . . . . . 8 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((#‘𝐹) − 1))) → 𝑖 ∈ (0..^(#‘𝐹)))
90		ccatval1 13214	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ ∧ 𝑖 ∈ (0..^(#‘𝐹))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = (𝐹‘𝑖))
91	86, 87, 89, 90	syl3anc 1318	. . . . . . 7 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((#‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = (𝐹‘𝑖))
92	91	fveq2d 6107	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((#‘𝐹) − 1))) → (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)) = (sgn‘(𝐹‘𝑖)))
93	92	mpteq2dva 4672	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (𝑖 ∈ (0...((#‘𝐹) − 1)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖))) = (𝑖 ∈ (0...((#‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖))))
94	93	oveq2d 6565	. . . 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 6564	. . 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 2652	. 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 2611	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (#‘𝐹) = (#‘𝐹))
98	97	olcd 407	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((#‘𝐹) ∈ (0..^(#‘𝐹)) ∨ (#‘𝐹) = (#‘𝐹)))
99	27, 19	syl6eleq 2698	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (#‘𝐹) ∈ (ℤ_≥‘0))
100		fzosplitsni 12444	. . . . . 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 246	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (#‘𝐹) ∈ (0..^((#‘𝐹) + 1)))
103	102, 38	eleqtrrd 2691	. . 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 29967	. . 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 691	. 2 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(#‘𝐹)) = (𝑊 Σ_g (𝑖 ∈ (0...(#‘𝐹)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)))))
108		fzo0end 12426	. . . . . 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 29967	. . . . 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 691	. . . 4 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → ((𝑇‘𝐹)‘((#‘𝐹) − 1)) = (𝑊 Σ_g (𝑖 ∈ (0...((#‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖)))))
112	111	adantr 480	. . 3 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝑇‘𝐹)‘((#‘𝐹) − 1)) = (𝑊 Σ_g (𝑖 ∈ (0...((#‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖)))))
113	112	oveq1d 6564	. 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 2654	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 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 ∅c0 3874 ifcif 4036 {csn 4125 {cpr 4127 {ctp 4129 ⟨cop 4131 ↦ cmpt 4643 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 ℝ^*cxr 9952 − 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 Mndcmnd 17117

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-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

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 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-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-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 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-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-sgn 13675 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

This theorem is referenced by: signsvtn0 29973 signstfvneq0 29975 signstfveq0 29980 signsvfn 29985

W3C validator