signsvtn - Mathbox for Thierry Arnoux

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Theorem signsvtn 31856

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 6676	. . . . 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 31854	. . . . . 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 835	. . . . 5 ⊢ (𝜑 → (𝑉‘(𝐸 ++ ⟨“𝐴”⟩)) = ((𝑉‘𝐸) + if((((𝑇‘𝐸)‘((♯‘𝐸) − 1)) · 𝐴) < 0, 1, 0)))
12	2, 11	eqtrd 2858	. . . 4 ⊢ (𝜑 → (𝑉‘𝐹) = ((𝑉‘𝐸) + if((((𝑇‘𝐸)‘((♯‘𝐸) − 1)) · 𝐴) < 0, 1, 0)))
13	12	adantr 483	. . 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 7168	. . . . . . . . 9 ⊢ (𝑁 − 1) = ((♯‘𝐸) − 1)
17	16	fveq2i 6675	. . . . . . . 8 ⊢ ((𝑇‘𝐸)‘(𝑁 − 1)) = ((𝑇‘𝐸)‘((♯‘𝐸) − 1))
18	14, 17	eqtri 2846	. . . . . . 7 ⊢ 𝐵 = ((𝑇‘𝐸)‘((♯‘𝐸) − 1))
19	18	oveq1i 7168	. . . . . 6 ⊢ (𝐵 · 𝐴) = (((𝑇‘𝐸)‘((♯‘𝐸) − 1)) · 𝐴)
20	3	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝐸 ∈ (Word ℝ ∖ {∅}))
21	20	eldifad 3950	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝐸 ∈ Word ℝ)
22	6, 7, 8, 9	signstf 31838	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ Word ℝ → (𝑇‘𝐸) ∈ Word ℝ)
23		wrdf 13869	. . . . . . . . . . . 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 4721	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∈ (Word ℝ ∖ {∅}) ↔ (𝐸 ∈ Word ℝ ∧ 𝐸 ≠ ∅))
26	3, 25	sylib 220	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸 ∈ Word ℝ ∧ 𝐸 ≠ ∅))
27	26	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝐸 ∈ Word ℝ ∧ 𝐸 ≠ ∅))
28		lennncl 13886	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ Word ℝ ∧ 𝐸 ≠ ∅) → (♯‘𝐸) ∈ ℕ)
29		fzo0end 13132	. . . . . . . . . . . . 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 31839	. . . . . . . . . . . . . 14 ⊢ (𝐸 ∈ Word ℝ → (♯‘(𝑇‘𝐸)) = (♯‘𝐸))
32	21, 31	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (♯‘(𝑇‘𝐸)) = (♯‘𝐸))
33	32	oveq2d 7174	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (0..^(♯‘(𝑇‘𝐸))) = (0..^(♯‘𝐸)))
34	30, 33	eleqtrrd 2918	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → ((♯‘𝐸) − 1) ∈ (0..^(♯‘(𝑇‘𝐸))))
35	24, 34	ffvelrnd 6854	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → ((𝑇‘𝐸)‘((♯‘𝐸) − 1)) ∈ ℝ)
36	18, 35	eqeltrid 2919	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝐵 ∈ ℝ)
37	36	recnd 10671	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝐵 ∈ ℂ)
38	5	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝐴 ∈ ℝ)
39	38	recnd 10671	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝐴 ∈ ℂ)
40	37, 39	mulcomd 10664	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝐵 · 𝐴) = (𝐴 · 𝐵))
41		simpr 487	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝐴 · 𝐵) < 0)
42	40, 41	eqbrtrd 5090	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝐵 · 𝐴) < 0)
43	19, 42	eqbrtrrid 5104	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (((𝑇‘𝐸)‘((♯‘𝐸) − 1)) · 𝐴) < 0)
44	43	iftrued 4477	. . . 4 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → if((((𝑇‘𝐸)‘((♯‘𝐸) − 1)) · 𝐴) < 0, 1, 0) = 1)
45	44	oveq2d 7174	. . 3 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → ((𝑉‘𝐸) + if((((𝑇‘𝐸)‘((♯‘𝐸) − 1)) · 𝐴) < 0, 1, 0)) = ((𝑉‘𝐸) + 1))
46	13, 45	eqtr2d 2859	. 2 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → ((𝑉‘𝐸) + 1) = (𝑉‘𝐹))
47	6, 7, 8, 9	signsvvf 31851	. . . . . 6 ⊢ 𝑉:Word ℝ⟶ℕ₀
48	47	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝑉:Word ℝ⟶ℕ₀)
49	1	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝐹 = (𝐸 ++ ⟨“𝐴”⟩))
50	38	s1cld 13959	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → ⟨“𝐴”⟩ ∈ Word ℝ)
51		ccatcl 13928	. . . . . . 7 ⊢ ((𝐸 ∈ Word ℝ ∧ ⟨“𝐴”⟩ ∈ Word ℝ) → (𝐸 ++ ⟨“𝐴”⟩) ∈ Word ℝ)
52	21, 50, 51	syl2anc 586	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝐸 ++ ⟨“𝐴”⟩) ∈ Word ℝ)
53	49, 52	eqeltrd 2915	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝐹 ∈ Word ℝ)
54	48, 53	ffvelrnd 6854	. . . 4 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝑉‘𝐹) ∈ ℕ₀)
55	54	nn0cnd 11960	. . 3 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝑉‘𝐹) ∈ ℂ)
56	48, 21	ffvelrnd 6854	. . . 4 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝑉‘𝐸) ∈ ℕ₀)
57	56	nn0cnd 11960	. . 3 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝑉‘𝐸) ∈ ℂ)
58		1cnd 10638	. . 3 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 1 ∈ ℂ)
59	55, 57, 58	subaddd 11017	. 2 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (((𝑉‘𝐹) − (𝑉‘𝐸)) = 1 ↔ ((𝑉‘𝐸) + 1) = (𝑉‘𝐹)))
60	46, 59	mpbird 259	1 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → ((𝑉‘𝐹) − (𝑉‘𝐸)) = 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∖ cdif 3935 ∅c0 4293 ifcif 4469 {csn 4569 {cpr 4571 {ctp 4573 ⟨cop 4575 class class class wbr 5068 ↦ cmpt 5148 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 < clt 10677 − cmin 10872 -cneg 10873 ℕcn 11640 ℕ₀cn0 11900 ...cfz 12895 ..^cfzo 13036 ♯chash 13693 Word cword 13864 ++ cconcat 13924 ⟨“cs1 13951 sgncsgn 14447 Σcsu 15044 ndxcnx 16482 Basecbs 16485 +_gcplusg 16567 Σ_g cgsu 16716

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-word 13865 df-lsw 13917 df-concat 13925 df-s1 13952 df-substr 14005 df-pfx 14035 df-sgn 14448 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-plusg 16580 df-0g 16717 df-gsum 16718 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mulg 18227 df-cntz 18449

This theorem is referenced by: signsvfnn 31858

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