signsvtn - Mathbox for Thierry Arnoux

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Theorem signsvtn 32463

Description: Adding a letter of a different sign as the highest coefficient changes the sign. (Contributed by Thierry Arnoux, 12-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))
signsvf.e	⊢ (𝜑 → 𝐸 ∈ (Word ℝ ∖ {∅}))
signsvf.0	⊢ (𝜑 → (𝐸‘0) ≠ 0)
signsvf.f	⊢ (𝜑 → 𝐹 = (𝐸 ++ ⟨“𝐴”⟩))
signsvf.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
signsvf.n	⊢ 𝑁 = (♯‘𝐸)
signsvt.b	⊢ 𝐵 = ((𝑇‘𝐸)‘(𝑁 − 1))

**Assertion**
Ref	Expression
signsvtn	⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → ((𝑉‘𝐹) − (𝑉‘𝐸)) = 1)

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

**Proof of Theorem signsvtn**
Step	Hyp	Ref	Expression
1		signsvf.f	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝐸 ++ ⟨“𝐴”⟩))
2	1	fveq2d 6760	. . . . 5 ⊢ (𝜑 → (𝑉‘𝐹) = (𝑉‘(𝐸 ++ ⟨“𝐴”⟩)))
3		signsvf.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (Word ℝ ∖ {∅}))
4		signsvf.0	. . . . . 6 ⊢ (𝜑 → (𝐸‘0) ≠ 0)
5		signsvf.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
6		signsv.p	. . . . . . 7 ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
7		signsv.w	. . . . . . 7 ⊢ 𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+_g‘ndx), ⨣ ⟩}
8		signsv.t	. . . . . . 7 ⊢ 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(♯‘𝑓)) ↦ (𝑊 Σ_g (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓‘𝑖))))))
9		signsv.v	. . . . . . 7 ⊢ 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(♯‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0))
10	6, 7, 8, 9	signsvfn 32461	. . . . . 6 ⊢ (((𝐸 ∈ (Word ℝ ∖ {∅}) ∧ (𝐸‘0) ≠ 0) ∧ 𝐴 ∈ ℝ) → (𝑉‘(𝐸 ++ ⟨“𝐴”⟩)) = ((𝑉‘𝐸) + if((((𝑇‘𝐸)‘((♯‘𝐸) − 1)) · 𝐴) < 0, 1, 0)))
11	3, 4, 5, 10	syl21anc 834	. . . . 5 ⊢ (𝜑 → (𝑉‘(𝐸 ++ ⟨“𝐴”⟩)) = ((𝑉‘𝐸) + if((((𝑇‘𝐸)‘((♯‘𝐸) − 1)) · 𝐴) < 0, 1, 0)))
12	2, 11	eqtrd 2778	. . . 4 ⊢ (𝜑 → (𝑉‘𝐹) = ((𝑉‘𝐸) + if((((𝑇‘𝐸)‘((♯‘𝐸) − 1)) · 𝐴) < 0, 1, 0)))
13	12	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝑉‘𝐹) = ((𝑉‘𝐸) + if((((𝑇‘𝐸)‘((♯‘𝐸) − 1)) · 𝐴) < 0, 1, 0)))
14		signsvt.b	. . . . . . . 8 ⊢ 𝐵 = ((𝑇‘𝐸)‘(𝑁 − 1))
15		signsvf.n	. . . . . . . . . 10 ⊢ 𝑁 = (♯‘𝐸)
16	15	oveq1i 7265	. . . . . . . . 9 ⊢ (𝑁 − 1) = ((♯‘𝐸) − 1)
17	16	fveq2i 6759	. . . . . . . 8 ⊢ ((𝑇‘𝐸)‘(𝑁 − 1)) = ((𝑇‘𝐸)‘((♯‘𝐸) − 1))
18	14, 17	eqtri 2766	. . . . . . 7 ⊢ 𝐵 = ((𝑇‘𝐸)‘((♯‘𝐸) − 1))
19	18	oveq1i 7265	. . . . . 6 ⊢ (𝐵 · 𝐴) = (((𝑇‘𝐸)‘((♯‘𝐸) − 1)) · 𝐴)
20	3	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝐸 ∈ (Word ℝ ∖ {∅}))
21	20	eldifad 3895	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝐸 ∈ Word ℝ)
22	6, 7, 8, 9	signstf 32445	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ Word ℝ → (𝑇‘𝐸) ∈ Word ℝ)
23		wrdf 14150	. . . . . . . . . . . 12 ⊢ ((𝑇‘𝐸) ∈ Word ℝ → (𝑇‘𝐸):(0..^(♯‘(𝑇‘𝐸)))⟶ℝ)
24	21, 22, 23	3syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝑇‘𝐸):(0..^(♯‘(𝑇‘𝐸)))⟶ℝ)
25		eldifsn 4717	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∈ (Word ℝ ∖ {∅}) ↔ (𝐸 ∈ Word ℝ ∧ 𝐸 ≠ ∅))
26	3, 25	sylib 217	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸 ∈ Word ℝ ∧ 𝐸 ≠ ∅))
27	26	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝐸 ∈ Word ℝ ∧ 𝐸 ≠ ∅))
28		lennncl 14165	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ Word ℝ ∧ 𝐸 ≠ ∅) → (♯‘𝐸) ∈ ℕ)
29		fzo0end 13407	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐸) ∈ ℕ → ((♯‘𝐸) − 1) ∈ (0..^(♯‘𝐸)))
30	27, 28, 29	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → ((♯‘𝐸) − 1) ∈ (0..^(♯‘𝐸)))
31	6, 7, 8, 9	signstlen 32446	. . . . . . . . . . . . . 14 ⊢ (𝐸 ∈ Word ℝ → (♯‘(𝑇‘𝐸)) = (♯‘𝐸))
32	21, 31	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (♯‘(𝑇‘𝐸)) = (♯‘𝐸))
33	32	oveq2d 7271	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (0..^(♯‘(𝑇‘𝐸))) = (0..^(♯‘𝐸)))
34	30, 33	eleqtrrd 2842	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → ((♯‘𝐸) − 1) ∈ (0..^(♯‘(𝑇‘𝐸))))
35	24, 34	ffvelrnd 6944	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → ((𝑇‘𝐸)‘((♯‘𝐸) − 1)) ∈ ℝ)
36	18, 35	eqeltrid 2843	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝐵 ∈ ℝ)
37	36	recnd 10934	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝐵 ∈ ℂ)
38	5	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝐴 ∈ ℝ)
39	38	recnd 10934	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝐴 ∈ ℂ)
40	37, 39	mulcomd 10927	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝐵 · 𝐴) = (𝐴 · 𝐵))
41		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝐴 · 𝐵) < 0)
42	40, 41	eqbrtrd 5092	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝐵 · 𝐴) < 0)
43	19, 42	eqbrtrrid 5106	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (((𝑇‘𝐸)‘((♯‘𝐸) − 1)) · 𝐴) < 0)
44	43	iftrued 4464	. . . 4 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → if((((𝑇‘𝐸)‘((♯‘𝐸) − 1)) · 𝐴) < 0, 1, 0) = 1)
45	44	oveq2d 7271	. . 3 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → ((𝑉‘𝐸) + if((((𝑇‘𝐸)‘((♯‘𝐸) − 1)) · 𝐴) < 0, 1, 0)) = ((𝑉‘𝐸) + 1))
46	13, 45	eqtr2d 2779	. 2 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → ((𝑉‘𝐸) + 1) = (𝑉‘𝐹))
47	6, 7, 8, 9	signsvvf 32458	. . . . . 6 ⊢ 𝑉:Word ℝ⟶ℕ₀
48	47	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝑉:Word ℝ⟶ℕ₀)
49	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝐹 = (𝐸 ++ ⟨“𝐴”⟩))
50	38	s1cld 14236	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → ⟨“𝐴”⟩ ∈ Word ℝ)
51		ccatcl 14205	. . . . . . 7 ⊢ ((𝐸 ∈ Word ℝ ∧ ⟨“𝐴”⟩ ∈ Word ℝ) → (𝐸 ++ ⟨“𝐴”⟩) ∈ Word ℝ)
52	21, 50, 51	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝐸 ++ ⟨“𝐴”⟩) ∈ Word ℝ)
53	49, 52	eqeltrd 2839	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝐹 ∈ Word ℝ)
54	48, 53	ffvelrnd 6944	. . . 4 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝑉‘𝐹) ∈ ℕ₀)
55	54	nn0cnd 12225	. . 3 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝑉‘𝐹) ∈ ℂ)
56	48, 21	ffvelrnd 6944	. . . 4 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝑉‘𝐸) ∈ ℕ₀)
57	56	nn0cnd 12225	. . 3 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝑉‘𝐸) ∈ ℂ)
58		1cnd 10901	. . 3 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 1 ∈ ℂ)
59	55, 57, 58	subaddd 11280	. 2 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (((𝑉‘𝐹) − (𝑉‘𝐸)) = 1 ↔ ((𝑉‘𝐸) + 1) = (𝑉‘𝐹)))
60	46, 59	mpbird 256	1 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → ((𝑉‘𝐹) − (𝑉‘𝐸)) = 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∖ cdif 3880 ∅c0 4253 ifcif 4456 {csn 4558 {cpr 4560 {ctp 4562 ⟨cop 4564 class class class wbr 5070 ↦ cmpt 5153 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ∈ cmpo 7257 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 < clt 10940 − cmin 11135 -cneg 11136 ℕcn 11903 ℕ₀cn0 12163 ...cfz 13168 ..^cfzo 13311 ♯chash 13972 Word cword 14145 ++ cconcat 14201 ⟨“cs1 14228 sgncsgn 14725 Σcsu 15325 ndxcnx 16822 Basecbs 16840 +_gcplusg 16888 Σ_g cgsu 17068

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-xnn0 12236 df-z 12250 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-hash 13973 df-word 14146 df-lsw 14194 df-concat 14202 df-s1 14229 df-substr 14282 df-pfx 14312 df-sgn 14726 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-sum 15326 df-struct 16776 df-slot 16811 df-ndx 16823 df-base 16841 df-plusg 16901 df-0g 17069 df-gsum 17070 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mulg 18616 df-cntz 18838

This theorem is referenced by: signsvfnn 32465

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