signstfvp - Mathbox for Thierry Arnoux

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Theorem signstfvp 34539

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
signstfvp	⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑁) = ((𝑇‘𝐹)‘𝑁))

Distinct variable groups: 𝑎,𝑏, ⨣ 𝑓,𝑖,𝑛,𝐹 𝑓,𝐾,𝑖,𝑛 𝑓,𝑊,𝑖,𝑛 𝑖,𝑁,𝑛 Allowed substitution hints: ⨣ (𝑓,𝑖,𝑗,𝑛) 𝑇(𝑓,𝑖,𝑗,𝑛,𝑎,𝑏) 𝐹(𝑗,𝑎,𝑏) 𝐾(𝑗,𝑎,𝑏) 𝑁(𝑓,𝑗,𝑎,𝑏) 𝑉(𝑓,𝑖,𝑗,𝑛,𝑎,𝑏) 𝑊(𝑗,𝑎,𝑏)

**Proof of Theorem signstfvp**
Step	Hyp	Ref	Expression
1		simpl1 1192	. . . . . 6 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ 𝑖 ∈ (0...𝑁)) → 𝐹 ∈ Word ℝ)
2		s1cl 14509	. . . . . . . 8 ⊢ (𝐾 ∈ ℝ → ⟨“𝐾”⟩ ∈ Word ℝ)
3	2	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ⟨“𝐾”⟩ ∈ Word ℝ)
4	3	adantr 480	. . . . . 6 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ 𝑖 ∈ (0...𝑁)) → ⟨“𝐾”⟩ ∈ Word ℝ)
5		fzssfzo 34507	. . . . . . . 8 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → (0...𝑁) ⊆ (0..^(♯‘𝐹)))
6	5	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → (0...𝑁) ⊆ (0..^(♯‘𝐹)))
7	6	sselda 3935	. . . . . 6 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ 𝑖 ∈ (0...𝑁)) → 𝑖 ∈ (0..^(♯‘𝐹)))
8		ccatval1 14484	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = (𝐹‘𝑖))
9	1, 4, 7, 8	syl3anc 1373	. . . . 5 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ 𝑖 ∈ (0...𝑁)) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = (𝐹‘𝑖))
10	9	fveq2d 6826	. . . 4 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ 𝑖 ∈ (0...𝑁)) → (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)) = (sgn‘(𝐹‘𝑖)))
11	10	mpteq2dva 5185	. . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → (𝑖 ∈ (0...𝑁) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖))) = (𝑖 ∈ (0...𝑁) ↦ (sgn‘(𝐹‘𝑖))))
12	11	oveq2d 7365	. 2 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → (𝑊 Σ_g (𝑖 ∈ (0...𝑁) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)))) = (𝑊 Σ_g (𝑖 ∈ (0...𝑁) ↦ (sgn‘(𝐹‘𝑖)))))
13		ccatws1cl 14523	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
14	13	3adant3 1132	. . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
15		lencl 14440	. . . . . . . . 9 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℕ₀)
16	15	nn0zd 12497	. . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℤ)
17	16	uzidd 12751	. . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ (ℤ_≥‘(♯‘𝐹)))
18		peano2uz 12802	. . . . . . 7 ⊢ ((♯‘𝐹) ∈ (ℤ_≥‘(♯‘𝐹)) → ((♯‘𝐹) + 1) ∈ (ℤ_≥‘(♯‘𝐹)))
19		fzoss2 13590	. . . . . . 7 ⊢ (((♯‘𝐹) + 1) ∈ (ℤ_≥‘(♯‘𝐹)) → (0..^(♯‘𝐹)) ⊆ (0..^((♯‘𝐹) + 1)))
20	17, 18, 19	3syl 18	. . . . . 6 ⊢ (𝐹 ∈ Word ℝ → (0..^(♯‘𝐹)) ⊆ (0..^((♯‘𝐹) + 1)))
21	20	sselda 3935	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → 𝑁 ∈ (0..^((♯‘𝐹) + 1)))
22	21	3adant2 1131	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → 𝑁 ∈ (0..^((♯‘𝐹) + 1)))
23		ccatlen 14482	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)))
24	2, 23	sylan2 593	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)))
25	24	3adant3 1132	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)))
26		s1len 14513	. . . . . . 7 ⊢ (♯‘⟨“𝐾”⟩) = 1
27	26	oveq2i 7360	. . . . . 6 ⊢ ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)) = ((♯‘𝐹) + 1)
28	25, 27	eqtrdi 2780	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + 1))
29	28	oveq2d 7365	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))) = (0..^((♯‘𝐹) + 1)))
30	22, 29	eleqtrrd 2831	. . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → 𝑁 ∈ (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))))
31		signsv.p	. . . 4 ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
32		signsv.w	. . . 4 ⊢ 𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+_g‘ndx), ⨣ ⟩}
33		signsv.t	. . . 4 ⊢ 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(♯‘𝑓)) ↦ (𝑊 Σ_g (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓‘𝑖))))))
34		signsv.v	. . . 4 ⊢ 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(♯‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0))
35	31, 32, 33, 34	signstfval 34532	. . 3 ⊢ (((𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩)))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑁) = (𝑊 Σ_g (𝑖 ∈ (0...𝑁) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)))))
36	14, 30, 35	syl2anc 584	. 2 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑁) = (𝑊 Σ_g (𝑖 ∈ (0...𝑁) ↦ (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)))))
37	31, 32, 33, 34	signstfval 34532	. . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘𝐹)‘𝑁) = (𝑊 Σ_g (𝑖 ∈ (0...𝑁) ↦ (sgn‘(𝐹‘𝑖)))))
38	37	3adant2 1131	. 2 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘𝐹)‘𝑁) = (𝑊 Σ_g (𝑖 ∈ (0...𝑁) ↦ (sgn‘(𝐹‘𝑖)))))
39	12, 36, 38	3eqtr4d 2774	1 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑁) = ((𝑇‘𝐹)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ⊆ wss 3903 ifcif 4476 {cpr 4579 {ctp 4581 ⟨cop 4583 ↦ cmpt 5173 ‘cfv 6482 (class class class)co 7349 ∈ cmpo 7351 ℝcr 11008 0cc0 11009 1c1 11010 + caddc 11012 − cmin 11347 -cneg 11348 ℤ_≥cuz 12735 ...cfz 13410 ..^cfzo 13557 ♯chash 14237 Word cword 14420 ++ cconcat 14477 ⟨“cs1 14502 sgncsgn 14993 Σcsu 15593 ndxcnx 17104 Basecbs 17120 +_gcplusg 17161 Σ_g cgsu 17344

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 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-n0 12385 df-z 12472 df-uz 12736 df-fz 13411 df-fzo 13558 df-hash 14238 df-word 14421 df-concat 14478 df-s1 14503

This theorem is referenced by: signstfvneq0 34540 signstfvc 34542 signstfveq0 34545 signsvfn 34550

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