signstfvn - Mathbox for Thierry Arnoux

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Theorem signstfvn 34606

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 34591	. . . 4 ⊢ {-1, 0, 1} = (Base‘𝑊)
4	1, 2	signswmnd 34594	. . . . 5 ⊢ 𝑊 ∈ Mnd
5	4	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → 𝑊 ∈ Mnd)
6		eldifi 4111	. . . . . . . . 9 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → 𝐹 ∈ Word ℝ)
7		lencl 14556	. . . . . . . . 9 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℕ₀)
8	6, 7	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (♯‘𝐹) ∈ ℕ₀)
9		eldifsn 4767	. . . . . . . . 9 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) ↔ (𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅))
10		hasheq0 14386	. . . . . . . . . . 11 ⊢ (𝐹 ∈ Word ℝ → ((♯‘𝐹) = 0 ↔ 𝐹 = ∅))
11	10	necon3bid 2977	. . . . . . . . . 10 ⊢ (𝐹 ∈ Word ℝ → ((♯‘𝐹) ≠ 0 ↔ 𝐹 ≠ ∅))
12	11	biimpar 477	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅) → (♯‘𝐹) ≠ 0)
13	9, 12	sylbi 217	. . . . . . . 8 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (♯‘𝐹) ≠ 0)
14		elnnne0 12520	. . . . . . . 8 ⊢ ((♯‘𝐹) ∈ ℕ ↔ ((♯‘𝐹) ∈ ℕ₀ ∧ (♯‘𝐹) ≠ 0))
15	8, 13, 14	sylanbrc 583	. . . . . . 7 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → (♯‘𝐹) ∈ ℕ)
16	15	adantr 480	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (♯‘𝐹) ∈ ℕ)
17		nnm1nn0 12547	. . . . . 6 ⊢ ((♯‘𝐹) ∈ ℕ → ((♯‘𝐹) − 1) ∈ ℕ₀)
18	16, 17	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((♯‘𝐹) − 1) ∈ ℕ₀)
19		nn0uz 12899	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
20	18, 19	eleqtrdi 2845	. . . 4 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((♯‘𝐹) − 1) ∈ (ℤ_≥‘0))
21		ccatws1cl 14639	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
22	21	adantr 480	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
23		wrdf 14541	. . . . . . . . 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 12619	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℤ)
26		fzoval 13682	. . . . . . . . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0..^(♯‘𝐹)) = (0...((♯‘𝐹) − 1)))
29		fzossfz 13700	. . . . . . . . . . 11 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
30	28, 29	eqsstrrdi 4009	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0...((♯‘𝐹) − 1)) ⊆ (0...(♯‘𝐹)))
31		s1cl 14625	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℝ → ⟨“𝐾”⟩ ∈ Word ℝ)
32		ccatlen 14598	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)))
33	31, 32	sylan2 593	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)))
34		s1len 14629	. . . . . . . . . . . . . 14 ⊢ (♯‘⟨“𝐾”⟩) = 1
35	34	oveq2i 7421	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)) = ((♯‘𝐹) + 1)
36	33, 35	eqtrdi 2787	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + 1))
37	36	oveq2d 7426	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))) = (0..^((♯‘𝐹) + 1)))
38	25	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (♯‘𝐹) ∈ ℤ)
39	38	peano2zd 12705	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → ((♯‘𝐹) + 1) ∈ ℤ)
40		fzoval 13682	. . . . . . . . . . . 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 12569	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℂ)
43		1cnd 11235	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ Word ℝ → 1 ∈ ℂ)
44	42, 43	pncand 11600	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ Word ℝ → (((♯‘𝐹) + 1) − 1) = (♯‘𝐹))
45	44	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (((♯‘𝐹) + 1) − 1) = (♯‘𝐹))
46	45	oveq2d 7426	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0...(((♯‘𝐹) + 1) − 1)) = (0...(♯‘𝐹)))
47	37, 41, 46	3eqtrd 2775	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))) = (0...(♯‘𝐹)))
48	30, 47	sseqtrrd 4001	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (0...((♯‘𝐹) − 1)) ⊆ (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))))
49	48	sselda 3963	. . . . . . . 8 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → 𝑖 ∈ (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))))
50	24, 49	ffvelcdmd 7080	. . . . . . 7 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) ∈ ℝ)
51	6, 50	sylanl1 680	. . . . . 6 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) ∈ ℝ)
52	51	rexrd 11290	. . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) ∈ ℝ^*)
53		sgncl 32815	. . . . 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 34592	. . . 4 ⊢ ⨣ = (+_g‘𝑊)
56		rexr 11286	. . . . . 6 ⊢ (𝐾 ∈ ℝ → 𝐾 ∈ ℝ^*)
57		sgncl 32815	. . . . . 6 ⊢ (𝐾 ∈ ℝ^* → (sgn‘𝐾) ∈ {-1, 0, 1})
58	56, 57	syl 17	. . . . 5 ⊢ (𝐾 ∈ ℝ → (sgn‘𝐾) ∈ {-1, 0, 1})
59	58	adantl 481	. . . 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 11603	. . . . . . . . . 10 ⊢ (𝐹 ∈ Word ℝ → (((♯‘𝐹) − 1) + 1) = (♯‘𝐹))
62	61	adantr 480	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (((♯‘𝐹) − 1) + 1) = (♯‘𝐹))
63	60, 62	sylan9eqr 2793	. . . . . . . 8 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → 𝑖 = (♯‘𝐹))
64	63	fveq2d 6885	. . . . . . 7 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = ((𝐹 ++ ⟨“𝐾”⟩)‘(♯‘𝐹)))
65		ccatws1ls 14656	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → ((𝐹 ++ ⟨“𝐾”⟩)‘(♯‘𝐹)) = 𝐾)
66	65	adantr 480	. . . . . . 7 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘(♯‘𝐹)) = 𝐾)
67	64, 66	eqtrd 2771	. . . . . 6 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = 𝐾)
68	6, 67	sylanl1 680	. . . . 5 ⊢ (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((♯‘𝐹) − 1) + 1)) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = 𝐾)
69	68	fveq2d 6885	. . . 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 34578	. . 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 480	. . . . . 6 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (((♯‘𝐹) − 1) + 1) = (♯‘𝐹))
73	72	oveq2d 7426	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (0...(((♯‘𝐹) − 1) + 1)) = (0...(♯‘𝐹)))
74	73	mpteq1d 5215	. . . 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 7426	. . 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 766	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → 𝐹 ∈ Word ℝ)
77	31	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ⟨“𝐾”⟩ ∈ Word ℝ)
78	28	eleq2d 2821	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (𝑖 ∈ (0..^(♯‘𝐹)) ↔ 𝑖 ∈ (0...((♯‘𝐹) − 1))))
79	78	biimpar 477	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → 𝑖 ∈ (0..^(♯‘𝐹)))
80		ccatval1 14600	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = (𝐹‘𝑖))
81	76, 77, 79, 80	syl3anc 1373	. . . . . . . 8 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = (𝐹‘𝑖))
82	81	fveq2d 6885	. . . . . . 7 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((♯‘𝐹) − 1))) → (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)) = (sgn‘(𝐹‘𝑖)))
83	82	mpteq2dva 5219	. . . . . 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 580	. . . . 5 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖))) = (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖))))
85	84	oveq2d 7426	. . . 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 7425	. . 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 2779	. 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 2736	. . . . . . . 8 ⊢ (♯‘𝐹) = (♯‘𝐹)
89	88	olci 866	. . . . . . 7 ⊢ ((♯‘𝐹) ∈ (0..^(♯‘𝐹)) ∨ (♯‘𝐹) = (♯‘𝐹))
90	7, 19	eleqtrdi 2845	. . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ (ℤ_≥‘0))
91		fzosplitsni 13799	. . . . . . . 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 258	. . . . . 6 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ (0..^((♯‘𝐹) + 1)))
94	93	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (♯‘𝐹) ∈ (0..^((♯‘𝐹) + 1)))
95	94, 37	eleqtrrd 2838	. . . 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 34601	. . . 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 584	. . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) = (𝑊 Σ_g (𝑖 ∈ (0...(♯‘𝐹)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)))))
100	6, 99	sylan 580	. 2 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(♯‘𝐹)) = (𝑊 Σ_g (𝑖 ∈ (0...(♯‘𝐹)) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)))))
101		fzo0end 13779	. . . . . 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 34601	. . . . 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 584	. . . 4 ⊢ (𝐹 ∈ (Word ℝ ∖ {∅}) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) = (𝑊 Σ_g (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖)))))
105	104	adantr 480	. . 3 ⊢ ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝑇‘𝐹)‘((♯‘𝐹) − 1)) = (𝑊 Σ_g (𝑖 ∈ (0...((♯‘𝐹) − 1)) ↦ (sgn‘(𝐹‘𝑖)))))
106	105	oveq1d 7425	. 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 2781	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 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 ∖ cdif 3928 ∅c0 4313 ifcif 4505 {csn 4606 {cpr 4608 {ctp 4610 ⟨cop 4612 ↦ cmpt 5206 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ∈ cmpo 7412 ℝcr 11133 0cc0 11134 1c1 11135 + caddc 11137 ℝ^*cxr 11273 − cmin 11471 -cneg 11472 ℕcn 12245 ℕ₀cn0 12506 ℤcz 12593 ℤ_≥cuz 12857 ...cfz 13529 ..^cfzo 13676 ♯chash 14353 Word cword 14536 ++ cconcat 14593 ⟨“cs1 14618 sgncsgn 15110 Σcsu 15707 ndxcnx 17217 Basecbs 17233 +_gcplusg 17276 Σ_g cgsu 17459 Mndcmnd 18717

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 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-n0 12507 df-z 12594 df-uz 12858 df-fz 13530 df-fzo 13677 df-seq 14025 df-hash 14354 df-word 14537 df-concat 14594 df-s1 14619 df-sgn 15111 df-struct 17171 df-slot 17206 df-ndx 17218 df-base 17234 df-plusg 17289 df-0g 17460 df-gsum 17461 df-mgm 18623 df-sgrp 18702 df-mnd 18718

This theorem is referenced by: signsvtn0 34607 signstfvneq0 34609 signstfveq0 34614 signsvfn 34619

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