ftalem4 - Metamath Proof Explorer

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Theorem ftalem4 27008

Description: Lemma for fta 27012: Closure of the auxiliary variables for ftalem5 27009. (Contributed by Mario Carneiro, 20-Sep-2014.) (Revised by AV, 28-Sep-2020.)

**Hypotheses**
Ref	Expression
ftalem.1	⊢ 𝐴 = (coeff‘𝐹)
ftalem.2	⊢ 𝑁 = (deg‘𝐹)
ftalem.3	⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
ftalem.4	⊢ (𝜑 → 𝑁 ∈ ℕ)
ftalem4.5	⊢ (𝜑 → (𝐹‘0) ≠ 0)
ftalem4.6	⊢ 𝐾 = inf({𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}, ℝ, < )
ftalem4.7	⊢ 𝑇 = (-((𝐹‘0) / (𝐴‘𝐾))↑_𝑐(1 / 𝐾))
ftalem4.8	⊢ 𝑈 = ((abs‘(𝐹‘0)) / (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1))
ftalem4.9	⊢ 𝑋 = if(1 ≤ 𝑈, 1, 𝑈)

**Assertion**
Ref	Expression
ftalem4	⊢ (𝜑 → ((𝐾 ∈ ℕ ∧ (𝐴‘𝐾) ≠ 0) ∧ (𝑇 ∈ ℂ ∧ 𝑈 ∈ ℝ⁺ ∧ 𝑋 ∈ ℝ⁺)))

Distinct variable groups: 𝑘,𝑛,𝐴 𝑘,𝐾,𝑛 𝑘,𝑁,𝑛 𝑘,𝐹,𝑛 𝜑,𝑘 𝑆,𝑘 𝑇,𝑘 𝑘,𝑋,𝑛 Allowed substitution hints: 𝜑(𝑛) 𝑆(𝑛) 𝑇(𝑛) 𝑈(𝑘,𝑛)

**Proof of Theorem ftalem4**
Step	Hyp	Ref	Expression
1		ftalem4.6	. . . 4 ⊢ 𝐾 = inf({𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}, ℝ, < )
2		ssrab2 4025	. . . . . 6 ⊢ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ⊆ ℕ
3		nnuz 12770	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
4	2, 3	sseqtri 3978	. . . . 5 ⊢ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ⊆ (ℤ_≥‘1)
5		fveq2 6817	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝐴‘𝑛) = (𝐴‘𝑁))
6	5	neeq1d 2987	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝐴‘𝑛) ≠ 0 ↔ (𝐴‘𝑁) ≠ 0))
7		ftalem.4	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
8	7	nnne0d 12170	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ≠ 0)
9		ftalem.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
10		ftalem.2	. . . . . . . . . . . 12 ⊢ 𝑁 = (deg‘𝐹)
11		ftalem.1	. . . . . . . . . . . 12 ⊢ 𝐴 = (coeff‘𝐹)
12	10, 11	dgreq0 26193	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0_𝑝 ↔ (𝐴‘𝑁) = 0))
13	9, 12	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 = 0_𝑝 ↔ (𝐴‘𝑁) = 0))
14		fveq2 6817	. . . . . . . . . . . 12 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = (deg‘0_𝑝))
15		dgr0 26190	. . . . . . . . . . . 12 ⊢ (deg‘0_𝑝) = 0
16	14, 15	eqtrdi 2782	. . . . . . . . . . 11 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = 0)
17	10, 16	eqtrid 2778	. . . . . . . . . 10 ⊢ (𝐹 = 0_𝑝 → 𝑁 = 0)
18	13, 17	biimtrrdi 254	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴‘𝑁) = 0 → 𝑁 = 0))
19	18	necon3d 2949	. . . . . . . 8 ⊢ (𝜑 → (𝑁 ≠ 0 → (𝐴‘𝑁) ≠ 0))
20	8, 19	mpd 15	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝑁) ≠ 0)
21	6, 7, 20	elrabd 3644	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0})
22	21	ne0d 4287	. . . . 5 ⊢ (𝜑 → {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ≠ ∅)
23		infssuzcl 12825	. . . . 5 ⊢ (({𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ⊆ (ℤ_≥‘1) ∧ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ≠ ∅) → inf({𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0})
24	4, 22, 23	sylancr 587	. . . 4 ⊢ (𝜑 → inf({𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0})
25	1, 24	eqeltrid 2835	. . 3 ⊢ (𝜑 → 𝐾 ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0})
26		fveq2 6817	. . . . 5 ⊢ (𝑛 = 𝐾 → (𝐴‘𝑛) = (𝐴‘𝐾))
27	26	neeq1d 2987	. . . 4 ⊢ (𝑛 = 𝐾 → ((𝐴‘𝑛) ≠ 0 ↔ (𝐴‘𝐾) ≠ 0))
28	27	elrab 3642	. . 3 ⊢ (𝐾 ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ↔ (𝐾 ∈ ℕ ∧ (𝐴‘𝐾) ≠ 0))
29	25, 28	sylib 218	. 2 ⊢ (𝜑 → (𝐾 ∈ ℕ ∧ (𝐴‘𝐾) ≠ 0))
30		ftalem4.7	. . . 4 ⊢ 𝑇 = (-((𝐹‘0) / (𝐴‘𝐾))↑_𝑐(1 / 𝐾))
31		plyf 26125	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
32	9, 31	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
33		0cn 11099	. . . . . . . 8 ⊢ 0 ∈ ℂ
34		ffvelcdm 7009	. . . . . . . 8 ⊢ ((𝐹:ℂ⟶ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) ∈ ℂ)
35	32, 33, 34	sylancl 586	. . . . . . 7 ⊢ (𝜑 → (𝐹‘0) ∈ ℂ)
36	11	coef3 26159	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ₀⟶ℂ)
37	9, 36	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
38	29	simpld 494	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℕ)
39	38	nnnn0d 12437	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
40	37, 39	ffvelcdmd 7013	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝐾) ∈ ℂ)
41	29	simprd 495	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝐾) ≠ 0)
42	35, 40, 41	divcld 11892	. . . . . 6 ⊢ (𝜑 → ((𝐹‘0) / (𝐴‘𝐾)) ∈ ℂ)
43	42	negcld 11454	. . . . 5 ⊢ (𝜑 → -((𝐹‘0) / (𝐴‘𝐾)) ∈ ℂ)
44	38	nnrecred 12171	. . . . . 6 ⊢ (𝜑 → (1 / 𝐾) ∈ ℝ)
45	44	recnd 11135	. . . . 5 ⊢ (𝜑 → (1 / 𝐾) ∈ ℂ)
46	43, 45	cxpcld 26639	. . . 4 ⊢ (𝜑 → (-((𝐹‘0) / (𝐴‘𝐾))↑_𝑐(1 / 𝐾)) ∈ ℂ)
47	30, 46	eqeltrid 2835	. . 3 ⊢ (𝜑 → 𝑇 ∈ ℂ)
48		ftalem4.8	. . . 4 ⊢ 𝑈 = ((abs‘(𝐹‘0)) / (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1))
49		ftalem4.5	. . . . . 6 ⊢ (𝜑 → (𝐹‘0) ≠ 0)
50	35, 49	absrpcld 15353	. . . . 5 ⊢ (𝜑 → (abs‘(𝐹‘0)) ∈ ℝ⁺)
51		fzfid 13875	. . . . . . 7 ⊢ (𝜑 → ((𝐾 + 1)...𝑁) ∈ Fin)
52		peano2nn0 12416	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ₀ → (𝐾 + 1) ∈ ℕ₀)
53	39, 52	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾 + 1) ∈ ℕ₀)
54		elfzuz 13415	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ((𝐾 + 1)...𝑁) → 𝑘 ∈ (ℤ_≥‘(𝐾 + 1)))
55		eluznn0 12810	. . . . . . . . . . 11 ⊢ (((𝐾 + 1) ∈ ℕ₀ ∧ 𝑘 ∈ (ℤ_≥‘(𝐾 + 1))) → 𝑘 ∈ ℕ₀)
56	53, 54, 55	syl2an 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → 𝑘 ∈ ℕ₀)
57	37	ffvelcdmda 7012	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
58	56, 57	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐴‘𝑘) ∈ ℂ)
59		expcl 13981	. . . . . . . . . 10 ⊢ ((𝑇 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑇↑𝑘) ∈ ℂ)
60	47, 56, 59	syl2an2r 685	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝑇↑𝑘) ∈ ℂ)
61	58, 60	mulcld 11127	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → ((𝐴‘𝑘) · (𝑇↑𝑘)) ∈ ℂ)
62	61	abscld 15341	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) ∈ ℝ)
63	51, 62	fsumrecl 15636	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) ∈ ℝ)
64	61	absge0d 15349	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → 0 ≤ (abs‘((𝐴‘𝑘) · (𝑇↑𝑘))))
65	51, 62, 64	fsumge0 15697	. . . . . 6 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))))
66	63, 65	ge0p1rpd 12959	. . . . 5 ⊢ (𝜑 → (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1) ∈ ℝ⁺)
67	50, 66	rpdivcld 12946	. . . 4 ⊢ (𝜑 → ((abs‘(𝐹‘0)) / (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1)) ∈ ℝ⁺)
68	48, 67	eqeltrid 2835	. . 3 ⊢ (𝜑 → 𝑈 ∈ ℝ⁺)
69		ftalem4.9	. . . 4 ⊢ 𝑋 = if(1 ≤ 𝑈, 1, 𝑈)
70		1rp 12889	. . . . 5 ⊢ 1 ∈ ℝ⁺
71		ifcl 4516	. . . . 5 ⊢ ((1 ∈ ℝ⁺ ∧ 𝑈 ∈ ℝ⁺) → if(1 ≤ 𝑈, 1, 𝑈) ∈ ℝ⁺)
72	70, 68, 71	sylancr 587	. . . 4 ⊢ (𝜑 → if(1 ≤ 𝑈, 1, 𝑈) ∈ ℝ⁺)
73	69, 72	eqeltrid 2835	. . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
74	47, 68, 73	3jca 1128	. 2 ⊢ (𝜑 → (𝑇 ∈ ℂ ∧ 𝑈 ∈ ℝ⁺ ∧ 𝑋 ∈ ℝ⁺))
75	29, 74	jca 511	1 ⊢ (𝜑 → ((𝐾 ∈ ℕ ∧ (𝐴‘𝐾) ≠ 0) ∧ (𝑇 ∈ ℂ ∧ 𝑈 ∈ ℝ⁺ ∧ 𝑋 ∈ ℝ⁺)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 {crab 3395 ⊆ wss 3897 ∅c0 4278 ifcif 4470 class class class wbr 5086 ⟶wf 6472 ‘cfv 6476 (class class class)co 7341 infcinf 9320 ℂcc 10999 ℝcr 11000 0cc0 11001 1c1 11002 + caddc 11004 · cmul 11006 < clt 11141 ≤ cle 11142 -cneg 11340 / cdiv 11769 ℕcn 12120 ℕ₀cn0 12376 ℤ_≥cuz 12727 ℝ⁺crp 12885 ...cfz 13402 ↑cexp 13963 abscabs 15136 Σcsu 15588 0_𝑝c0p 25592 Polycply 26111 coeffccoe 26113 degcdgr 26114 ↑_𝑐ccxp 26486

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-inf2 9526 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-pre-sup 11079 ax-addf 11080

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-iin 4939 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-se 5565 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-isom 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-er 8617 df-map 8747 df-pm 8748 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-fi 9290 df-sup 9321 df-inf 9322 df-oi 9391 df-card 9827 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-div 11770 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-7 12188 df-8 12189 df-9 12190 df-n0 12377 df-z 12464 df-dec 12584 df-uz 12728 df-q 12842 df-rp 12886 df-xneg 13006 df-xadd 13007 df-xmul 13008 df-ioo 13244 df-ioc 13245 df-ico 13246 df-icc 13247 df-fz 13403 df-fzo 13550 df-fl 13691 df-mod 13769 df-seq 13904 df-exp 13964 df-fac 14176 df-bc 14205 df-hash 14233 df-shft 14969 df-cj 15001 df-re 15002 df-im 15003 df-sqrt 15137 df-abs 15138 df-limsup 15373 df-clim 15390 df-rlim 15391 df-sum 15589 df-ef 15969 df-sin 15971 df-cos 15972 df-pi 15974 df-struct 17053 df-sets 17070 df-slot 17088 df-ndx 17100 df-base 17116 df-ress 17137 df-plusg 17169 df-mulr 17170 df-starv 17171 df-sca 17172 df-vsca 17173 df-ip 17174 df-tset 17175 df-ple 17176 df-ds 17178 df-unif 17179 df-hom 17180 df-cco 17181 df-rest 17321 df-topn 17322 df-0g 17340 df-gsum 17341 df-topgen 17342 df-pt 17343 df-prds 17346 df-xrs 17401 df-qtop 17406 df-imas 17407 df-xps 17409 df-mre 17483 df-mrc 17484 df-acs 17486 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19224 df-cmn 19689 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22804 df-topon 22821 df-topsp 22843 df-bases 22856 df-cld 22929 df-ntr 22930 df-cls 22931 df-nei 23008 df-lp 23046 df-perf 23047 df-cn 23137 df-cnp 23138 df-haus 23225 df-tx 23472 df-hmeo 23665 df-fil 23756 df-fm 23848 df-flim 23849 df-flf 23850 df-xms 24230 df-ms 24231 df-tms 24232 df-cncf 24793 df-0p 25593 df-limc 25789 df-dv 25790 df-ply 26115 df-coe 26117 df-dgr 26118 df-log 26487 df-cxp 26488

This theorem is referenced by: ftalem5 27009

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