signstfvn - Mathbox for Thierry Arnoux

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Theorem signstfvn 34901

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 34886	. . . 4 ⊢ {-1, 0, 1} = (Base‘𝑊)
4	1, 2	signswmnd 34889	. . . . 5 ⊢ 𝑊 ∈ Mnd
5	4	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 𝑊 ∈ Mnd)
6		eldifi 4093	. . . . . . . . 9 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → 𝐹 ∈ Word ℝ)
7		lencl 14570	. . . . . . . . 9 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℕ₀)
8	6, 7	syl 18	. . . . . . . 8 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (♯‘𝐹) ∈ ℕ₀)
9		eldifsn 4758	. . . . . . . . 9 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) ↔ (𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅))
10		hasheq0 14399	. . . . . . . . . . 11 ⊢ (𝐹 ∈ Word ℝ → ((♯‘𝐹) = 0 ↔ 𝐹 = ∅))
11	10	necon3bid 3008	. . . . . . . . . 10 ⊢ (𝐹 ∈ Word ℝ → ((♯‘𝐹) ≠ 0 ↔ 𝐹 ≠ ∅))
12	11	biimpar 482	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅) → (♯‘𝐹) ≠ 0)
13	9, 12	sylbi 220	. . . . . . . 8 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (♯‘𝐹) ≠ 0)
14		elnnne0 12518	. . . . . . . 8 ⊢ ((♯‘𝐹) ∈ ℕ ↔ ((♯‘𝐹) ∈ ℕ₀ ∧ (♯‘𝐹) ≠ 0))
15	8, 13, 14	sylanbrc 594	. . . . . . 7 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (♯‘𝐹) ∈ ℕ)
16	15	adantr 485	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (♯‘𝐹) ∈ ℕ)
17		nnm1nn0 12545	. . . . . 6 ⊢ ((♯‘𝐹) ∈ ℕ → ((♯‘𝐹) − 1) ∈ ℕ₀)
18	16, 17	syl 18	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((♯‘𝐹) − 1) ∈ ℕ₀)
19		nn0uz 12900	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
20	18, 19	eleqtrdi 2879	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((♯‘𝐹) − 1) ∈ (ℤ_≥‘0))
21		ccatws1cl 14654	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
22	21	adantr 485	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
23		wrdf 14555	. . . . . . . . 9 ⊢ ((𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ → (𝐹 ++ ⟨“𝐾”⟩):(0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩)))⟶ℝ)
24	22, 23	syl 18	. . . . . . . 8 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → (𝐹 ++ ⟨“𝐾”⟩):(0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩)))⟶ℝ)
25	7	nn0zd 12616	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℤ)
26		fzoval 13688	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) ∈ ℤ → (0..^(♯‘𝐹)) = (0...((♯‘𝐹) − 1)))
27	25, 26	syl 18	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ Word ℝ → (0..^(♯‘𝐹)) = (0...((♯‘𝐹) − 1)))
28	27	adantr 485	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0..^(♯‘𝐹)) = (0...((♯‘𝐹) − 1)))
29		fzossfz 13707	. . . . . . . . . . 11 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
30	28, 29	eqsstrrdi 3990	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0...((♯‘𝐹) − 1)) ⊆ (0...(♯‘𝐹)))
31		s1cl 14640	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℝ → ⟨“𝐾”⟩ ∈ Word ℝ)
32		ccatlen 14612	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)))
33	31, 32	sylan2 604	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)))
34		s1len 14644	. . . . . . . . . . . . . 14 ⊢ (♯‘⟨“𝐾”⟩) = 1
35	34	oveq2i 7422	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)) = ((♯‘𝐹) + 1)
36	33, 35	eqtrdi 2820	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + 1))
37	36	oveq2d 7427	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))) = (0..^((♯‘𝐹) + 1)))
38	25	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (♯‘𝐹) ∈ ℤ)
39	38	peano2zd 12703	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → ((♯‘𝐹) + 1) ∈ ℤ)
40		fzoval 13688	. . . . . . . . . . . 12 ⊢ (((♯‘𝐹) + 1) ∈ ℤ → (0..^((♯‘𝐹) + 1)) = (0...(((♯‘𝐹) + 1) − 1)))
41	39, 40	syl 18	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0..^((♯‘𝐹) + 1)) = (0...(((♯‘𝐹) + 1) − 1)))
42	7	nn0cnd 12567	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℂ)
43		1cnd 11202	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ Word ℝ → 1 ∈ ℂ)
44	42, 43	pncand 11570	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ Word ℝ → (((♯‘𝐹) + 1) − 1) = (♯‘𝐹))
45	44	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (((♯‘𝐹) + 1) − 1) = (♯‘𝐹))
46	45	oveq2d 7427	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0...(((♯‘𝐹) + 1) − 1)) = (0...(♯‘𝐹)))
47	37, 41, 46	3eqtrd 2808	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))) = (0...(♯‘𝐹)))
48	30, 47	sseqtrrd 3982	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0...((♯‘𝐹) − 1)) ⊆ (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))))
49	48	sselda 3945	. . . . . . . 8 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → 𝑖 ∈ (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))))
50	24, 49	ffvelcdmd 7081	. . . . . . 7 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) ∈ ℝ)
51	6, 50	sylanl1 692	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) ∈ ℝ)
52	51	rexrd 11259	. . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) ∈ ℝ^*)
53		sgncl 15134	. . . . 5 ⊢ (((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) ∈ ℝ^* → (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)) ∈ {-1, 0, 1})
54	52, 53	syl 18	. . . 4 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)) ∈ {-1, 0, 1})
55	1, 2	signswplusg 34887	. . . 4 ⊢ ⨣ = (+_g‘𝑊)
56		rexr 11255	. . . . . 6 ⊢ (𝐾 ∈ ℝ → 𝐾 ∈ ℝ^*)
57		sgncl 15134	. . . . . 6 ⊢ (𝐾 ∈ ℝ^* → (sgn‘𝐾) ∈ {-1, 0, 1})
58	56, 57	syl 18	. . . . 5 ⊢ (𝐾 ∈ ℝ → (sgn‘𝐾) ∈ {-1, 0, 1})
59	58	adantl 486	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (sgn‘𝐾) ∈ {-1, 0, 1})
60		id 23	. . . . . . . . 9 ⊢ (𝑖 = (((♯‘𝐹) − 1) + 1) → 𝑖 = (((♯‘𝐹) − 1) + 1))
61	42, 43	npcand 11573	. . . . . . . . . 10 ⊢ (𝐹 ∈ Word ℝ → (((♯‘𝐹) − 1) + 1) = (♯‘𝐹))
62	61	adantr 485	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (((♯‘𝐹) − 1) + 1) = (♯‘𝐹))
63	60, 62	sylan9eqr 2826	. . . . . . . 8 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → 𝑖 = (♯‘𝐹))
64	63	fveq2d 6886	. . . . . . 7 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = ((𝐹 ++ ⟨“𝐾”⟩)‘(♯‘𝐹)))
65		ccatws1ls 14671	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → ((𝐹 ++ ⟨“𝐾”⟩)‘(♯‘𝐹)) = 𝐾)
66	65	adantr 485	. . . . . . 7 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘(♯‘𝐹)) = 𝐾)
67	64, 66	eqtrd 2804	. . . . . 6 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = 𝐾)
68	6, 67	sylanl1 692	. . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = 𝐾)
69	68	fveq2d 6886	. . . 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 34876	. . 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 18	. . . . . . 7 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (((♯‘𝐹) − 1) + 1) = (♯‘𝐹))
72	71	adantr 485	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (((♯‘𝐹) − 1) + 1) = (♯‘𝐹))
73	72	oveq2d 7427	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (0...(((♯‘𝐹) − 1) + 1)) = (0...(♯‘𝐹)))
74	73	mpteq1d 5205	. . . 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 7427	. . 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 778	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → 𝐹 ∈ Word ℝ)
77	31	ad2antlr 739	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ⟨“𝐾”⟩ ∈ Word ℝ)
78	28	eleq2d 2855	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (𝑖 ∈ (0..^(♯‘𝐹)) ↔ 𝑖 ∈ (0...((♯‘𝐹) − 1))))
79	78	biimpar 482	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → 𝑖 ∈ (0..^(♯‘𝐹)))
80		ccatval1 14614	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = (𝐹‘𝑖))
81	76, 77, 79, 80	syl3anc 1396	. . . . . . . 8 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = (𝐹‘𝑖))
82	81	fveq2d 6886	. . . . . . 7 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)) = (sgn‘(𝐹‘𝑖)))
83	82	mpteq2dva 5208	. . . . . 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 591	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖))) = (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖))))
85	84	oveq2d 7427	. . . 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 7426	. . 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 2812	. 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 2769	. . . . . . . 8 ⊢ (♯‘𝐹) = (♯‘𝐹)
89	88	olci 879	. . . . . . 7 ⊢ ((♯‘𝐹) ∈ (0..^(♯‘𝐹)) ∨ (♯‘𝐹) = (♯‘𝐹))
90	7, 19	eleqtrdi 2879	. . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ (ℤ_≥‘0))
91		fzosplitsni 13808	. . . . . . . 8 ⊢ ((♯‘𝐹) ∈ (ℤ_≥‘0) → ((♯‘𝐹) ∈ (0..^((♯‘𝐹) + 1)) ↔ ((♯‘𝐹) ∈ (0..^(♯‘𝐹)) ∨ (♯‘𝐹) = (♯‘𝐹))))
92	90, 91	syl 18	. . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → ((♯‘𝐹) ∈ (0..^((♯‘𝐹) + 1)) ↔ ((♯‘𝐹) ∈ (0..^(♯‘𝐹)) ∨ (♯‘𝐹) = (♯‘𝐹))))
93	89, 92	mpbiri 261	. . . . . 6 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ (0..^((♯‘𝐹) + 1)))
94	93	adantr 485	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (♯‘𝐹) ∈ (0..^((♯‘𝐹) + 1)))
95	94, 37	eleqtrrd 2872	. . . 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 34896	. . . 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 595	. . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) = (𝑊 Σ_g (𝑖 ∈ (0...(♯‘𝐹)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)))))
100	6, 99	sylan 591	. 2 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) = (𝑊 Σ_g (𝑖 ∈ (0...(♯‘𝐹)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)))))
101		fzo0end 13787	. . . . . 6 ⊢ ((♯‘𝐹) ∈ ℕ → ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹)))
102	15, 101	syl 18	. . . . 5 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → ((♯‘𝐹) − 1) ∈ (0..^(♯‘𝐹)))
103	1, 2, 96, 97	signstfval 34896	. . . . 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 595	. . . 4 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) = (𝑊 Σ_g (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖)))))
105	104	adantr 485	. . 3 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) = (𝑊 Σ_g (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖)))))
106	105	oveq1d 7426	. 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 2814	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 209 ∧ wa 400 ∨ wo 860 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∖ cdif 3910 ∅c0 4294 ifcif 4492 {csn 4594 {cpr 4596 {ctp 4598 ⟨cop 4600 ↦ cmpt 5196 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 ∈ cmpo 7413 ℝcr 11099 0cc0 11100 1c1 11101 + caddc 11103 ℝ^*cxr 11242 − cmin 11441 -cneg 11442 ℕcn 12233 ℕ₀cn0 12504 ℤcz 12591 ℤ_≥cuz 12862 ...cfz 13535 ..^cfzo 13682 ♯chash 14366 Word cword 14550 ++ cconcat 14607 ⟨“cs1 14633 sgncsgn 15123 Σcsu 15737 ndxcnx 17253 Basecbs 17269 +_gcplusg 17310 Σ_g cgsu 17493 Mndcmnd 18792

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8694 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-nn 12234 df-2 12303 df-n0 12505 df-z 12592 df-uz 12863 df-fz 13536 df-fzo 13683 df-seq 14038 df-hash 14367 df-word 14551 df-concat 14608 df-s1 14634 df-sgn 15124 df-struct 17207 df-slot 17242 df-ndx 17254 df-base 17270 df-plusg 17323 df-0g 17494 df-gsum 17495 df-mgm 18698 df-sgrp 18777 df-mnd 18793

This theorem is referenced by: signsvtn0 34902 signstfvneq0 34904 signstfveq0 34909 signsvfn 34914

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