signstfvp - Mathbox for Thierry Arnoux

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Theorem signstfvp 34569

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 14574	. . . . . . . 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 34537	. . . . . . . 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 3949	. . . . . 6 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ 𝑖 ∈ (0...𝑁)) → 𝑖 ∈ (0..^(♯‘𝐹)))
8		ccatval1 14549	. . . . . 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 6865	. . . 4 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ 𝑖 ∈ (0...𝑁)) → (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)) = (sgn‘(𝐹‘𝑖)))
11	10	mpteq2dva 5203	. . 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 7406	. 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 14588	. . . 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 14505	. . . . . . . . 9 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℕ₀)
16	15	nn0zd 12562	. . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℤ)
17	16	uzidd 12816	. . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ (ℤ_≥‘(♯‘𝐹)))
18		peano2uz 12867	. . . . . . 7 ⊢ ((♯‘𝐹) ∈ (ℤ_≥‘(♯‘𝐹)) → ((♯‘𝐹) + 1) ∈ (ℤ_≥‘(♯‘𝐹)))
19		fzoss2 13655	. . . . . . 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 3949	. . . . 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 14547	. . . . . . . 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 14578	. . . . . . 7 ⊢ (♯‘⟨“𝐾”⟩) = 1
27	26	oveq2i 7401	. . . . . 6 ⊢ ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)) = ((♯‘𝐹) + 1)
28	25, 27	eqtrdi 2781	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + 1))
29	28	oveq2d 7406	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))) = (0..^((♯‘𝐹) + 1)))
30	22, 29	eleqtrrd 2832	. . 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 34562	. . 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 34562	. . 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 2775	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 2926 ⊆ wss 3917 ifcif 4491 {cpr 4594 {ctp 4596 ⟨cop 4598 ↦ cmpt 5191 ‘cfv 6514 (class class class)co 7390 ∈ cmpo 7392 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 − cmin 11412 -cneg 11413 ℤ_≥cuz 12800 ...cfz 13475 ..^cfzo 13622 ♯chash 14302 Word cword 14485 ++ cconcat 14542 ⟨“cs1 14567 sgncsgn 15059 Σcsu 15659 ndxcnx 17170 Basecbs 17186 +_gcplusg 17227 Σ_g cgsu 17410

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476 df-fzo 13623 df-hash 14303 df-word 14486 df-concat 14543 df-s1 14568

This theorem is referenced by: signstfvneq0 34570 signstfvc 34572 signstfveq0 34575 signsvfn 34580

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