signshf - Mathbox for Thierry Arnoux

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Theorem signshf 30995

Description: 𝐻, corresponding to the word 𝐹 multiplied by (𝑥 − 𝐶), as a function. (Contributed by Thierry Arnoux, 29-Sep-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))
signs.h	⊢ 𝐻 = ((⟨“0”⟩ ++ 𝐹) ∘_𝑓 − ((𝐹 ++ ⟨“0”⟩)∘_𝑓/𝑐 · 𝐶))

**Assertion**
Ref	Expression
signshf	⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → 𝐻:(0..^((♯‘𝐹) + 1))⟶ℝ)

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

Proof of Theorem signshf Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		resubcl 10557	. . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 − 𝑦) ∈ ℝ)
2	1	adantl 473	. . 3 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 − 𝑦) ∈ ℝ)
3		0red 10253	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → 0 ∈ ℝ)
4	3	s1cld 13593	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → ⟨“0”⟩ ∈ Word ℝ)
5		simpl 474	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → 𝐹 ∈ Word ℝ)
6		ccatcl 13566	. . . . . 6 ⊢ ((⟨“0”⟩ ∈ Word ℝ ∧ 𝐹 ∈ Word ℝ) → (⟨“0”⟩ ++ 𝐹) ∈ Word ℝ)
7	4, 5, 6	syl2anc 696	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (⟨“0”⟩ ++ 𝐹) ∈ Word ℝ)
8		wrdf 13516	. . . . 5 ⊢ ((⟨“0”⟩ ++ 𝐹) ∈ Word ℝ → (⟨“0”⟩ ++ 𝐹):(0..^(♯‘(⟨“0”⟩ ++ 𝐹)))⟶ℝ)
9	7, 8	syl 17	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (⟨“0”⟩ ++ 𝐹):(0..^(♯‘(⟨“0”⟩ ++ 𝐹)))⟶ℝ)
10		ccatlen 13567	. . . . . . . . 9 ⊢ ((⟨“0”⟩ ∈ Word ℝ ∧ 𝐹 ∈ Word ℝ) → (♯‘(⟨“0”⟩ ++ 𝐹)) = ((♯‘⟨“0”⟩) + (♯‘𝐹)))
11	4, 5, 10	syl2anc 696	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (♯‘(⟨“0”⟩ ++ 𝐹)) = ((♯‘⟨“0”⟩) + (♯‘𝐹)))
12		s1len 13596	. . . . . . . . 9 ⊢ (♯‘⟨“0”⟩) = 1
13	12	oveq1i 6824	. . . . . . . 8 ⊢ ((♯‘⟨“0”⟩) + (♯‘𝐹)) = (1 + (♯‘𝐹))
14	11, 13	syl6eq 2810	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (♯‘(⟨“0”⟩ ++ 𝐹)) = (1 + (♯‘𝐹)))
15		1cnd 10268	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → 1 ∈ ℂ)
16		wrdfin 13529	. . . . . . . . . 10 ⊢ (𝐹 ∈ Word ℝ → 𝐹 ∈ Fin)
17		hashcl 13359	. . . . . . . . . 10 ⊢ (𝐹 ∈ Fin → (♯‘𝐹) ∈ ℕ₀)
18	5, 16, 17	3syl 18	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (♯‘𝐹) ∈ ℕ₀)
19	18	nn0cnd 11565	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (♯‘𝐹) ∈ ℂ)
20	15, 19	addcomd 10450	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (1 + (♯‘𝐹)) = ((♯‘𝐹) + 1))
21	14, 20	eqtrd 2794	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (♯‘(⟨“0”⟩ ++ 𝐹)) = ((♯‘𝐹) + 1))
22	21	oveq2d 6830	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (0..^(♯‘(⟨“0”⟩ ++ 𝐹))) = (0..^((♯‘𝐹) + 1)))
23	22	feq2d 6192	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → ((⟨“0”⟩ ++ 𝐹):(0..^(♯‘(⟨“0”⟩ ++ 𝐹)))⟶ℝ ↔ (⟨“0”⟩ ++ 𝐹):(0..^((♯‘𝐹) + 1))⟶ℝ))
24	9, 23	mpbid 222	. . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (⟨“0”⟩ ++ 𝐹):(0..^((♯‘𝐹) + 1))⟶ℝ)
25		remulcl 10233	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ)
26	25	adantl 473	. . . 4 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
27		ccatcl 13566	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“0”⟩ ∈ Word ℝ) → (𝐹 ++ ⟨“0”⟩) ∈ Word ℝ)
28	4, 27	syldan 488	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (𝐹 ++ ⟨“0”⟩) ∈ Word ℝ)
29		wrdf 13516	. . . . . 6 ⊢ ((𝐹 ++ ⟨“0”⟩) ∈ Word ℝ → (𝐹 ++ ⟨“0”⟩):(0..^(♯‘(𝐹 ++ ⟨“0”⟩)))⟶ℝ)
30	28, 29	syl 17	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (𝐹 ++ ⟨“0”⟩):(0..^(♯‘(𝐹 ++ ⟨“0”⟩)))⟶ℝ)
31		ccatlen 13567	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“0”⟩ ∈ Word ℝ) → (♯‘(𝐹 ++ ⟨“0”⟩)) = ((♯‘𝐹) + (♯‘⟨“0”⟩)))
32	4, 31	syldan 488	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (♯‘(𝐹 ++ ⟨“0”⟩)) = ((♯‘𝐹) + (♯‘⟨“0”⟩)))
33	12	oveq2i 6825	. . . . . . . 8 ⊢ ((♯‘𝐹) + (♯‘⟨“0”⟩)) = ((♯‘𝐹) + 1)
34	32, 33	syl6eq 2810	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (♯‘(𝐹 ++ ⟨“0”⟩)) = ((♯‘𝐹) + 1))
35	34	oveq2d 6830	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (0..^(♯‘(𝐹 ++ ⟨“0”⟩))) = (0..^((♯‘𝐹) + 1)))
36	35	feq2d 6192	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → ((𝐹 ++ ⟨“0”⟩):(0..^(♯‘(𝐹 ++ ⟨“0”⟩)))⟶ℝ ↔ (𝐹 ++ ⟨“0”⟩):(0..^((♯‘𝐹) + 1))⟶ℝ))
37	30, 36	mpbid 222	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (𝐹 ++ ⟨“0”⟩):(0..^((♯‘𝐹) + 1))⟶ℝ)
38		ovexd 6844	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (0..^((♯‘𝐹) + 1)) ∈ V)
39		simpr 479	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℝ⁺)
40	39	rpred 12085	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℝ)
41	26, 37, 38, 40	ofcf 30495	. . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → ((𝐹 ++ ⟨“0”⟩)∘_𝑓/𝑐 · 𝐶):(0..^((♯‘𝐹) + 1))⟶ℝ)
42		inidm 3965	. . 3 ⊢ ((0..^((♯‘𝐹) + 1)) ∩ (0..^((♯‘𝐹) + 1))) = (0..^((♯‘𝐹) + 1))
43	2, 24, 41, 38, 38, 42	off 7078	. 2 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → ((⟨“0”⟩ ++ 𝐹) ∘_𝑓 − ((𝐹 ++ ⟨“0”⟩)∘_𝑓/𝑐 · 𝐶)):(0..^((♯‘𝐹) + 1))⟶ℝ)
44		signs.h	. . 3 ⊢ 𝐻 = ((⟨“0”⟩ ++ 𝐹) ∘_𝑓 − ((𝐹 ++ ⟨“0”⟩)∘_𝑓/𝑐 · 𝐶))
45	44	feq1i 6197	. 2 ⊢ (𝐻:(0..^((♯‘𝐹) + 1))⟶ℝ ↔ ((⟨“0”⟩ ++ 𝐹) ∘_𝑓 − ((𝐹 ++ ⟨“0”⟩)∘_𝑓/𝑐 · 𝐶)):(0..^((♯‘𝐹) + 1))⟶ℝ)
46	43, 45	sylibr 224	1 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → 𝐻:(0..^((♯‘𝐹) + 1))⟶ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 Vcvv 3340 ifcif 4230 {cpr 4323 {ctp 4325 ⟨cop 4327 ↦ cmpt 4881 ⟶wf 6045 ‘cfv 6049 (class class class)co 6814 ↦ cmpt2 6816 ∘_𝑓 cof 7061 Fincfn 8123 ℝcr 10147 0cc0 10148 1c1 10149 + caddc 10151 · cmul 10153 − cmin 10478 -cneg 10479 ℕ₀cn0 11504 ℝ⁺crp 12045 ...cfz 12539 ..^cfzo 12679 ♯chash 13331 Word cword 13497 ++ cconcat 13499 ⟨“cs1 13500 sgncsgn 14045 Σcsu 14635 ndxcnx 16076 Basecbs 16079 +_gcplusg 16163 Σ_g cgsu 16323 ∘_𝑓/𝑐cofc 30487

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-of 7063 df-om 7232 df-1st 7334 df-2nd 7335 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-oadd 7734 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-card 8975 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-n0 11505 df-z 11590 df-uz 11900 df-rp 12046 df-fz 12540 df-fzo 12680 df-hash 13332 df-word 13505 df-concat 13507 df-s1 13508 df-ofc 30488

This theorem is referenced by: signshwrd 30996 signshlen 30997

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