signstfvn - Mathbox for Thierry Arnoux

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Theorem signstfvn 33878

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 33863	. . . 4 ⊢ {-1, 0, 1} = (Base‘𝑊)
4	1, 2	signswmnd 33866	. . . . 5 ⊢ 𝑊 ∈ Mnd
5	4	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 𝑊 ∈ Mnd)
6		eldifi 4125	. . . . . . . . 9 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → 𝐹 ∈ Word ℝ)
7		lencl 14487	. . . . . . . . 9 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℕ₀)
8	6, 7	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (♯‘𝐹) ∈ ℕ₀)
9		eldifsn 4789	. . . . . . . . 9 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) ↔ (𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅))
10		hasheq0 14327	. . . . . . . . . . 11 ⊢ (𝐹 ∈ Word ℝ → ((♯‘𝐹) = 0 ↔ 𝐹 = ∅))
11	10	necon3bid 2983	. . . . . . . . . 10 ⊢ (𝐹 ∈ Word ℝ → ((♯‘𝐹) ≠ 0 ↔ 𝐹 ≠ ∅))
12	11	biimpar 476	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅) → (♯‘𝐹) ≠ 0)
13	9, 12	sylbi 216	. . . . . . . 8 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (♯‘𝐹) ≠ 0)
14		elnnne0 12490	. . . . . . . 8 ⊢ ((♯‘𝐹) ∈ ℕ ↔ ((♯‘𝐹) ∈ ℕ₀ ∧ (♯‘𝐹) ≠ 0))
15	8, 13, 14	sylanbrc 581	. . . . . . 7 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (♯‘𝐹) ∈ ℕ)
16	15	adantr 479	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (♯‘𝐹) ∈ ℕ)
17		nnm1nn0 12517	. . . . . 6 ⊢ ((♯‘𝐹) ∈ ℕ → ((♯‘𝐹) − 1) ∈ ℕ₀)
18	16, 17	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((♯‘𝐹) − 1) ∈ ℕ₀)
19		nn0uz 12868	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
20	18, 19	eleqtrdi 2841	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((♯‘𝐹) − 1) ∈ (ℤ_≥‘0))
21		ccatws1cl 14570	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
22	21	adantr 479	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
23		wrdf 14473	. . . . . . . . 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 12588	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℤ)
26		fzoval 13637	. . . . . . . . . . . . 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 479	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0..^(♯‘𝐹)) = (0...((♯‘𝐹) − 1)))
29		fzossfz 13655	. . . . . . . . . . 11 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
30	28, 29	eqsstrrdi 4036	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0...((♯‘𝐹) − 1)) ⊆ (0...(♯‘𝐹)))
31		s1cl 14556	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℝ → ⟨“𝐾”⟩ ∈ Word ℝ)
32		ccatlen 14529	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)))
33	31, 32	sylan2 591	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)))
34		s1len 14560	. . . . . . . . . . . . . 14 ⊢ (♯‘⟨“𝐾”⟩) = 1
35	34	oveq2i 7422	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)) = ((♯‘𝐹) + 1)
36	33, 35	eqtrdi 2786	. . . . . . . . . . . 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 479	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (♯‘𝐹) ∈ ℤ)
39	38	peano2zd 12673	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → ((♯‘𝐹) + 1) ∈ ℤ)
40		fzoval 13637	. . . . . . . . . . . 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 12538	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℂ)
43		1cnd 11213	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ Word ℝ → 1 ∈ ℂ)
44	42, 43	pncand 11576	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ Word ℝ → (((♯‘𝐹) + 1) − 1) = (♯‘𝐹))
45	44	adantr 479	. . . . . . . . . . . 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 2774	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))) = (0...(♯‘𝐹)))
48	30, 47	sseqtrrd 4022	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0...((♯‘𝐹) − 1)) ⊆ (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))))
49	48	sselda 3981	. . . . . . . 8 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → 𝑖 ∈ (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))))
50	24, 49	ffvelcdmd 7086	. . . . . . 7 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) ∈ ℝ)
51	6, 50	sylanl1 676	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) ∈ ℝ)
52	51	rexrd 11268	. . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) ∈ ℝ^*)
53		sgncl 33835	. . . . 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 33864	. . . 4 ⊢ ⨣ = (+_g‘𝑊)
56		rexr 11264	. . . . . 6 ⊢ (𝐾 ∈ ℝ → 𝐾 ∈ ℝ^*)
57		sgncl 33835	. . . . . 6 ⊢ (𝐾 ∈ ℝ^* → (sgn‘𝐾) ∈ {-1, 0, 1})
58	56, 57	syl 17	. . . . 5 ⊢ (𝐾 ∈ ℝ → (sgn‘𝐾) ∈ {-1, 0, 1})
59	58	adantl 480	. . . 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 11579	. . . . . . . . . 10 ⊢ (𝐹 ∈ Word ℝ → (((♯‘𝐹) − 1) + 1) = (♯‘𝐹))
62	61	adantr 479	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (((♯‘𝐹) − 1) + 1) = (♯‘𝐹))
63	60, 62	sylan9eqr 2792	. . . . . . . 8 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → 𝑖 = (♯‘𝐹))
64	63	fveq2d 6894	. . . . . . 7 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = ((𝐹 ++ ⟨“𝐾”⟩)‘(♯‘𝐹)))
65		ccatws1ls 14587	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → ((𝐹 ++ ⟨“𝐾”⟩)‘(♯‘𝐹)) = 𝐾)
66	65	adantr 479	. . . . . . 7 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘(♯‘𝐹)) = 𝐾)
67	64, 66	eqtrd 2770	. . . . . 6 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = 𝐾)
68	6, 67	sylanl1 676	. . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = 𝐾)
69	68	fveq2d 6894	. . . 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 33850	. . 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 479	. . . . . 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 5242	. . . 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 763	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → 𝐹 ∈ Word ℝ)
77	31	ad2antlr 723	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ⟨“𝐾”⟩ ∈ Word ℝ)
78	28	eleq2d 2817	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (𝑖 ∈ (0..^(♯‘𝐹)) ↔ 𝑖 ∈ (0...((♯‘𝐹) − 1))))
79	78	biimpar 476	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → 𝑖 ∈ (0..^(♯‘𝐹)))
80		ccatval1 14531	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = (𝐹‘𝑖))
81	76, 77, 79, 80	syl3anc 1369	. . . . . . . 8 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = (𝐹‘𝑖))
82	81	fveq2d 6894	. . . . . . 7 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)) = (sgn‘(𝐹‘𝑖)))
83	82	mpteq2dva 5247	. . . . . 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 578	. . . . 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 2778	. 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 862	. . . . . . 7 ⊢ ((♯‘𝐹) ∈ (0..^(♯‘𝐹)) ∨ (♯‘𝐹) = (♯‘𝐹))
90	7, 19	eleqtrdi 2841	. . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ (ℤ_≥‘0))
91		fzosplitsni 13747	. . . . . . . 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 257	. . . . . 6 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ (0..^((♯‘𝐹) + 1)))
94	93	adantr 479	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (♯‘𝐹) ∈ (0..^((♯‘𝐹) + 1)))
95	94, 37	eleqtrrd 2834	. . . 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 33873	. . . 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 582	. . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) = (𝑊 Σ_g (𝑖 ∈ (0...(♯‘𝐹)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)))))
100	6, 99	sylan 578	. 2 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) = (𝑊 Σ_g (𝑖 ∈ (0...(♯‘𝐹)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)))))
101		fzo0end 13728	. . . . . 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 33873	. . . . 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 582	. . . 4 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) = (𝑊 Σ_g (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖)))))
105	104	adantr 479	. . 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 2780	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 205 ∧ wa 394 ∨ wo 843 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∖ cdif 3944 ∅c0 4321 ifcif 4527 {csn 4627 {cpr 4629 {ctp 4631 ⟨cop 4633 ↦ cmpt 5230 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ∈ cmpo 7413 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 ℝ^*cxr 11251 − cmin 11448 -cneg 11449 ℕcn 12216 ℕ₀cn0 12476 ℤcz 12562 ℤ_≥cuz 12826 ...cfz 13488 ..^cfzo 13631 ♯chash 14294 Word cword 14468 ++ cconcat 14524 ⟨“cs1 14549 sgncsgn 15037 Σcsu 15636 ndxcnx 17130 Basecbs 17148 +_gcplusg 17201 Σ_g cgsu 17390 Mndcmnd 18659

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632 df-seq 13971 df-hash 14295 df-word 14469 df-concat 14525 df-s1 14550 df-sgn 15038 df-struct 17084 df-slot 17119 df-ndx 17131 df-base 17149 df-plusg 17214 df-0g 17391 df-gsum 17392 df-mgm 18565 df-sgrp 18644 df-mnd 18660

This theorem is referenced by: signsvtn0 33879 signstfvneq0 33881 signstfveq0 33886 signsvfn 33891

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