signsvtn - Mathbox for Thierry Arnoux

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Theorem signsvtn 34599

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 6910	. . . . 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 34597	. . . . . 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 838	. . . . 5 ⊢ (𝜑 → (𝑉‘(𝐸 ++ ⟨“𝐴”⟩)) = ((𝑉‘𝐸) + if((((𝑇‘𝐸)‘((♯‘𝐸) − 1)) · 𝐴) < 0, 1, 0)))
12	2, 11	eqtrd 2777	. . . 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 7441	. . . . . . . . 9 ⊢ (𝑁 − 1) = ((♯‘𝐸) − 1)
17	16	fveq2i 6909	. . . . . . . 8 ⊢ ((𝑇‘𝐸)‘(𝑁 − 1)) = ((𝑇‘𝐸)‘((♯‘𝐸) − 1))
18	14, 17	eqtri 2765	. . . . . . 7 ⊢ 𝐵 = ((𝑇‘𝐸)‘((♯‘𝐸) − 1))
19	18	oveq1i 7441	. . . . . 6 ⊢ (𝐵 · 𝐴) = (((𝑇‘𝐸)‘((♯‘𝐸) − 1)) · 𝐴)
20	3	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝐸 ∈ (Word ℝ ∖ {∅}))
21	20	eldifad 3963	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝐸 ∈ Word ℝ)
22	6, 7, 8, 9	signstf 34581	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ Word ℝ → (𝑇‘𝐸) ∈ Word ℝ)
23		wrdf 14557	. . . . . . . . . . . 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 4786	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∈ (Word ℝ ∖ {∅}) ↔ (𝐸 ∈ Word ℝ ∧ 𝐸 ≠ ∅))
26	3, 25	sylib 218	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸 ∈ Word ℝ ∧ 𝐸 ≠ ∅))
27	26	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝐸 ∈ Word ℝ ∧ 𝐸 ≠ ∅))
28		lennncl 14572	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ Word ℝ ∧ 𝐸 ≠ ∅) → (♯‘𝐸) ∈ ℕ)
29		fzo0end 13797	. . . . . . . . . . . . 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 34582	. . . . . . . . . . . . . 14 ⊢ (𝐸 ∈ Word ℝ → (♯‘(𝑇‘𝐸)) = (♯‘𝐸))
32	21, 31	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (♯‘(𝑇‘𝐸)) = (♯‘𝐸))
33	32	oveq2d 7447	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (0..^(♯‘(𝑇‘𝐸))) = (0..^(♯‘𝐸)))
34	30, 33	eleqtrrd 2844	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → ((♯‘𝐸) − 1) ∈ (0..^(♯‘(𝑇‘𝐸))))
35	24, 34	ffvelcdmd 7105	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → ((𝑇‘𝐸)‘((♯‘𝐸) − 1)) ∈ ℝ)
36	18, 35	eqeltrid 2845	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝐵 ∈ ℝ)
37	36	recnd 11289	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝐵 ∈ ℂ)
38	5	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝐴 ∈ ℝ)
39	38	recnd 11289	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝐴 ∈ ℂ)
40	37, 39	mulcomd 11282	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝐵 · 𝐴) = (𝐴 · 𝐵))
41		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝐴 · 𝐵) < 0)
42	40, 41	eqbrtrd 5165	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝐵 · 𝐴) < 0)
43	19, 42	eqbrtrrid 5179	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (((𝑇‘𝐸)‘((♯‘𝐸) − 1)) · 𝐴) < 0)
44	43	iftrued 4533	. . . 4 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → if((((𝑇‘𝐸)‘((♯‘𝐸) − 1)) · 𝐴) < 0, 1, 0) = 1)
45	44	oveq2d 7447	. . 3 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → ((𝑉‘𝐸) + if((((𝑇‘𝐸)‘((♯‘𝐸) − 1)) · 𝐴) < 0, 1, 0)) = ((𝑉‘𝐸) + 1))
46	13, 45	eqtr2d 2778	. 2 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → ((𝑉‘𝐸) + 1) = (𝑉‘𝐹))
47	6, 7, 8, 9	signsvvf 34594	. . . . . 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 14641	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → ⟨“𝐴”⟩ ∈ Word ℝ)
51		ccatcl 14612	. . . . . . 7 ⊢ ((𝐸 ∈ Word ℝ ∧ ⟨“𝐴”⟩ ∈ Word ℝ) → (𝐸 ++ ⟨“𝐴”⟩) ∈ Word ℝ)
52	21, 50, 51	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝐸 ++ ⟨“𝐴”⟩) ∈ Word ℝ)
53	49, 52	eqeltrd 2841	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 𝐹 ∈ Word ℝ)
54	48, 53	ffvelcdmd 7105	. . . 4 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝑉‘𝐹) ∈ ℕ₀)
55	54	nn0cnd 12589	. . 3 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝑉‘𝐹) ∈ ℂ)
56	48, 21	ffvelcdmd 7105	. . . 4 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝑉‘𝐸) ∈ ℕ₀)
57	56	nn0cnd 12589	. . 3 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (𝑉‘𝐸) ∈ ℂ)
58		1cnd 11256	. . 3 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → 1 ∈ ℂ)
59	55, 57, 58	subaddd 11638	. 2 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → (((𝑉‘𝐹) − (𝑉‘𝐸)) = 1 ↔ ((𝑉‘𝐸) + 1) = (𝑉‘𝐹)))
60	46, 59	mpbird 257	1 ⊢ ((𝜑 ∧ (𝐴 · 𝐵) < 0) → ((𝑉‘𝐹) − (𝑉‘𝐸)) = 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 ∅c0 4333 ifcif 4525 {csn 4626 {cpr 4628 {ctp 4630 ⟨cop 4632 class class class wbr 5143 ↦ cmpt 5225 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 − cmin 11492 -cneg 11493 ℕcn 12266 ℕ₀cn0 12526 ...cfz 13547 ..^cfzo 13694 ♯chash 14369 Word cword 14552 ++ cconcat 14608 ⟨“cs1 14633 sgncsgn 15125 Σcsu 15722 ndxcnx 17230 Basecbs 17247 +_gcplusg 17297 Σ_g cgsu 17485

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-word 14553 df-lsw 14601 df-concat 14609 df-s1 14634 df-substr 14679 df-pfx 14709 df-sgn 15126 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-0g 17486 df-gsum 17487 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mulg 19086 df-cntz 19335

This theorem is referenced by: signsvfnn 34601

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