signstfvp - Mathbox for Thierry Arnoux

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Theorem signstfvp 35193

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 1210	. . . . . 6 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ 𝑖 ∈ (0...𝑁)) → 𝐹 ∈ Word ℝ)
2		s1cl 14742	. . . . . . . 8 ⊢ (𝐾 ∈ ℝ → ⟨“𝐾”⟩ ∈ Word ℝ)
3	2	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ⟨“𝐾”⟩ ∈ Word ℝ)
4	3	adantr 486	. . . . . 6 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ 𝑖 ∈ (0...𝑁)) → ⟨“𝐾”⟩ ∈ Word ℝ)
5		fzssfzo 35164	. . . . . . . 8 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → (0...𝑁) ⊆ (0..^(♯‘𝐹)))
6	5	3ad2ant3 1153	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → (0...𝑁) ⊆ (0..^(♯‘𝐹)))
7	6	sselda 3931	. . . . . 6 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ 𝑖 ∈ (0...𝑁)) → 𝑖 ∈ (0..^(♯‘𝐹)))
8		ccatval1 14715	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = (𝐹‘𝑖))
9	1, 4, 7, 8	syl3anc 1398	. . . . 5 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ 𝑖 ∈ (0...𝑁)) → ((𝐹 ++ ⟨“𝐾”⟩)‘𝑖) = (𝐹‘𝑖))
10	9	fveq2d 6887	. . . 4 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ 𝑖 ∈ (0...𝑁)) → (sgn‘((𝐹 ++ ⟨“𝐾”⟩)‘𝑖)) = (sgn‘(𝐹‘𝑖)))
11	10	mpteq2dva 5198	. . 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 7434	. 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 14757	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
14	13	3adant3 1150	. . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → (𝐹 ++ ⟨“𝐾”⟩) ∈ Word ℝ)
15		lencl 14671	. . . . . . . . 9 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℕ₀)
16	15	nn0zd 12711	. . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℤ)
17	16	uzidd 12974	. . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ (ℤ_≥‘(♯‘𝐹)))
18		peano2uz 13021	. . . . . . 7 ⊢ ((♯‘𝐹) ∈ (ℤ_≥‘(♯‘𝐹)) → ((♯‘𝐹) + 1) ∈ (ℤ_≥‘(♯‘𝐹)))
19		fzoss2 13815	. . . . . . 7 ⊢ (((♯‘𝐹) + 1) ∈ (ℤ_≥‘(♯‘𝐹)) → (0..^(♯‘𝐹)) ⊆ (0..^((♯‘𝐹) + 1)))
20	17, 18, 19	3syl 19	. . . . . 6 ⊢ (𝐹 ∈ Word ℝ → (0..^(♯‘𝐹)) ⊆ (0..^((♯‘𝐹) + 1)))
21	20	sselda 3931	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → 𝑁 ∈ (0..^((♯‘𝐹) + 1)))
22	21	3adant2 1149	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → 𝑁 ∈ (0..^((♯‘𝐹) + 1)))
23		ccatlen 14713	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“𝐾”⟩ ∈ Word ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)))
24	2, 23	sylan2 605	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)))
25	24	3adant3 1150	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)))
26		s1len 14746	. . . . . . 7 ⊢ (♯‘⟨“𝐾”⟩) = 1
27	26	oveq2i 7429	. . . . . 6 ⊢ ((♯‘𝐹) + (♯‘⟨“𝐾”⟩)) = ((♯‘𝐹) + 1)
28	25, 27	eqtrdi 2812	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → (♯‘(𝐹 ++ ⟨“𝐾”⟩)) = ((♯‘𝐹) + 1))
29	28	oveq2d 7434	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → (0..^(♯‘(𝐹 ++ ⟨“𝐾”⟩))) = (0..^((♯‘𝐹) + 1)))
30	22, 29	eleqtrrd 2864	. . 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 35186	. . 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 596	. 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 35186	. . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘𝐹)‘𝑁) = (𝑊 Σ_g (𝑖 ∈ (0...𝑁) ↦ (sgn‘(𝐹‘𝑖)))))
38	37	3adant2 1149	. 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 2806	1 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑁) = ((𝑇‘𝐹)‘𝑁))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ⊆ wss 3899 ifcif 4482 {cpr 4586 {ctp 4588 ⟨cop 4590 ↦ cmpt 5186 ‘cfv 6537 (class class class)co 7418 ∈ cmpo 7420 ℝcr 11192 0cc0 11193 1c1 11194 + caddc 11196 − cmin 11534 -cneg 11535 ℤ_≥cuz 12958 ...cfz 13632 ..^cfzo 13781 ♯chash 14467 Word cword 14651 ++ cconcat 14708 ⟨“cs1 14735 sgncsgn 15232 Σcsu 15846 ndxcnx 17364 Basecbs 17380 +_gcplusg 17421 Σ_g cgsu 17604

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 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 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-1st 7999 df-2nd 8000 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-er 8710 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-nn 12329 df-n0 12600 df-z 12687 df-uz 12959 df-fz 13633 df-fzo 13782 df-hash 14468 df-word 14652 df-concat 14709 df-s1 14736

This theorem is used by: signstfvneq0 35194 signstfvc 35196 signstfveq0 35199 signsvfn 35204

W3C validator