signshf - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > signshf		Structured version Visualization version GIF version

Theorem signshf 31471

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 10804	. . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 − 𝑦) ∈ ℝ)
2	1	adantl 482	. . 3 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 − 𝑦) ∈ ℝ)
3		0red 10497	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → 0 ∈ ℝ)
4	3	s1cld 13805	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → ⟨“0”⟩ ∈ Word ℝ)
5		simpl 483	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → 𝐹 ∈ Word ℝ)
6		ccatcl 13776	. . . . . 6 ⊢ ((⟨“0”⟩ ∈ Word ℝ ∧ 𝐹 ∈ Word ℝ) → (⟨“0”⟩ ++ 𝐹) ∈ Word ℝ)
7	4, 5, 6	syl2anc 584	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (⟨“0”⟩ ++ 𝐹) ∈ Word ℝ)
8		wrdf 13716	. . . . 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 13777	. . . . . . . . 9 ⊢ ((⟨“0”⟩ ∈ Word ℝ ∧ 𝐹 ∈ Word ℝ) → (♯‘(⟨“0”⟩ ++ 𝐹)) = ((♯‘⟨“0”⟩) + (♯‘𝐹)))
11	4, 5, 10	syl2anc 584	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (♯‘(⟨“0”⟩ ++ 𝐹)) = ((♯‘⟨“0”⟩) + (♯‘𝐹)))
12		s1len 13808	. . . . . . . . 9 ⊢ (♯‘⟨“0”⟩) = 1
13	12	oveq1i 7033	. . . . . . . 8 ⊢ ((♯‘⟨“0”⟩) + (♯‘𝐹)) = (1 + (♯‘𝐹))
14	11, 13	syl6eq 2849	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (♯‘(⟨“0”⟩ ++ 𝐹)) = (1 + (♯‘𝐹)))
15		1cnd 10489	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → 1 ∈ ℂ)
16		wrdfin 13732	. . . . . . . . . 10 ⊢ (𝐹 ∈ Word ℝ → 𝐹 ∈ Fin)
17		hashcl 13571	. . . . . . . . . 10 ⊢ (𝐹 ∈ Fin → (♯‘𝐹) ∈ ℕ₀)
18	5, 16, 17	3syl 18	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (♯‘𝐹) ∈ ℕ₀)
19	18	nn0cnd 11811	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (♯‘𝐹) ∈ ℂ)
20	15, 19	addcomd 10695	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (1 + (♯‘𝐹)) = ((♯‘𝐹) + 1))
21	14, 20	eqtrd 2833	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (♯‘(⟨“0”⟩ ++ 𝐹)) = ((♯‘𝐹) + 1))
22	21	oveq2d 7039	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (0..^(♯‘(⟨“0”⟩ ++ 𝐹))) = (0..^((♯‘𝐹) + 1)))
23	22	feq2d 6375	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → ((⟨“0”⟩ ++ 𝐹):(0..^(♯‘(⟨“0”⟩ ++ 𝐹)))⟶ℝ ↔ (⟨“0”⟩ ++ 𝐹):(0..^((♯‘𝐹) + 1))⟶ℝ))
24	9, 23	mpbid 233	. . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (⟨“0”⟩ ++ 𝐹):(0..^((♯‘𝐹) + 1))⟶ℝ)
25		remulcl 10475	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ)
26	25	adantl 482	. . . 4 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
27		ccatcl 13776	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“0”⟩ ∈ Word ℝ) → (𝐹 ++ ⟨“0”⟩) ∈ Word ℝ)
28	4, 27	syldan 591	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (𝐹 ++ ⟨“0”⟩) ∈ Word ℝ)
29		wrdf 13716	. . . . . 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 13777	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ ⟨“0”⟩ ∈ Word ℝ) → (♯‘(𝐹 ++ ⟨“0”⟩)) = ((♯‘𝐹) + (♯‘⟨“0”⟩)))
32	4, 31	syldan 591	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (♯‘(𝐹 ++ ⟨“0”⟩)) = ((♯‘𝐹) + (♯‘⟨“0”⟩)))
33	12	oveq2i 7034	. . . . . . . 8 ⊢ ((♯‘𝐹) + (♯‘⟨“0”⟩)) = ((♯‘𝐹) + 1)
34	32, 33	syl6eq 2849	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (♯‘(𝐹 ++ ⟨“0”⟩)) = ((♯‘𝐹) + 1))
35	34	oveq2d 7039	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (0..^(♯‘(𝐹 ++ ⟨“0”⟩))) = (0..^((♯‘𝐹) + 1)))
36	35	feq2d 6375	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → ((𝐹 ++ ⟨“0”⟩):(0..^(♯‘(𝐹 ++ ⟨“0”⟩)))⟶ℝ ↔ (𝐹 ++ ⟨“0”⟩):(0..^((♯‘𝐹) + 1))⟶ℝ))
37	30, 36	mpbid 233	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (𝐹 ++ ⟨“0”⟩):(0..^((♯‘𝐹) + 1))⟶ℝ)
38		ovexd 7057	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → (0..^((♯‘𝐹) + 1)) ∈ V)
39		simpr 485	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℝ⁺)
40	39	rpred 12285	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℝ)
41	26, 37, 38, 40	ofcf 30975	. . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → ((𝐹 ++ ⟨“0”⟩)∘_𝑓/𝑐 · 𝐶):(0..^((♯‘𝐹) + 1))⟶ℝ)
42		inidm 4121	. . 3 ⊢ ((0..^((♯‘𝐹) + 1)) ∩ (0..^((♯‘𝐹) + 1))) = (0..^((♯‘𝐹) + 1))
43	2, 24, 41, 38, 38, 42	off 7289	. 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 6380	. 2 ⊢ (𝐻:(0..^((♯‘𝐹) + 1))⟶ℝ ↔ ((⟨“0”⟩ ++ 𝐹) ∘_𝑓 − ((𝐹 ++ ⟨“0”⟩)∘_𝑓/𝑐 · 𝐶)):(0..^((♯‘𝐹) + 1))⟶ℝ)
46	43, 45	sylibr 235	1 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ⁺) → 𝐻:(0..^((♯‘𝐹) + 1))⟶ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1525 ∈ wcel 2083 ≠ wne 2986 Vcvv 3440 ifcif 4387 {cpr 4480 {ctp 4482 ⟨cop 4484 ↦ cmpt 5047 ⟶wf 6228 ‘cfv 6232 (class class class)co 7023 ∈ cmpo 7025 ∘_𝑓 cof 7272 Fincfn 8364 ℝcr 10389 0cc0 10390 1c1 10391 + caddc 10393 · cmul 10395 − cmin 10723 -cneg 10724 ℕ₀cn0 11751 ℝ⁺crp 12243 ...cfz 12746 ..^cfzo 12887 ♯chash 13544 Word cword 13711 ++ cconcat 13772 ⟨“cs1 13797 sgncsgn 14283 Σcsu 14880 ndxcnx 16313 Basecbs 16316 +_gcplusg 16398 Σ_g cgsu 16547 ∘_𝑓/𝑐cofc 30967

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-of 7274 df-om 7444 df-1st 7552 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-oadd 7964 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-card 9221 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-n0 11752 df-z 11836 df-uz 12098 df-rp 12244 df-fz 12747 df-fzo 12888 df-hash 13545 df-word 13712 df-concat 13773 df-s1 13798 df-ofc 30968

This theorem is referenced by: signshwrd 31472 signshlen 31473

W3C validator