ftalem4 - Metamath Proof Explorer

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Theorem ftalem4 27133

Description: Lemma for fta 27137: Closure of the auxiliary variables for ftalem5 27134. (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 4089	. . . . . 6 ⊢ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ⊆ ℕ
3		nnuz 12918	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
4	2, 3	sseqtri 4031	. . . . 5 ⊢ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ⊆ (ℤ_≥‘1)
5		fveq2 6906	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝐴‘𝑛) = (𝐴‘𝑁))
6	5	neeq1d 2997	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝐴‘𝑛) ≠ 0 ↔ (𝐴‘𝑁) ≠ 0))
7		ftalem.4	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
8	7	nnne0d 12313	. . . . . . . 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 26319	. . . . . . . . . . 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 6906	. . . . . . . . . . . 12 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = (deg‘0_𝑝))
15		dgr0 26316	. . . . . . . . . . . 12 ⊢ (deg‘0_𝑝) = 0
16	14, 15	eqtrdi 2790	. . . . . . . . . . 11 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = 0)
17	10, 16	eqtrid 2786	. . . . . . . . . 10 ⊢ (𝐹 = 0_𝑝 → 𝑁 = 0)
18	13, 17	biimtrrdi 254	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴‘𝑁) = 0 → 𝑁 = 0))
19	18	necon3d 2958	. . . . . . . 8 ⊢ (𝜑 → (𝑁 ≠ 0 → (𝐴‘𝑁) ≠ 0))
20	8, 19	mpd 15	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝑁) ≠ 0)
21	6, 7, 20	elrabd 3696	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0})
22	21	ne0d 4347	. . . . 5 ⊢ (𝜑 → {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ≠ ∅)
23		infssuzcl 12971	. . . . 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 2842	. . 3 ⊢ (𝜑 → 𝐾 ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0})
26		fveq2 6906	. . . . 5 ⊢ (𝑛 = 𝐾 → (𝐴‘𝑛) = (𝐴‘𝐾))
27	26	neeq1d 2997	. . . 4 ⊢ (𝑛 = 𝐾 → ((𝐴‘𝑛) ≠ 0 ↔ (𝐴‘𝐾) ≠ 0))
28	27	elrab 3694	. . 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 26251	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
32	9, 31	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
33		0cn 11250	. . . . . . . 8 ⊢ 0 ∈ ℂ
34		ffvelcdm 7100	. . . . . . . 8 ⊢ ((𝐹:ℂ⟶ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) ∈ ℂ)
35	32, 33, 34	sylancl 586	. . . . . . 7 ⊢ (𝜑 → (𝐹‘0) ∈ ℂ)
36	11	coef3 26285	. . . . . . . . 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 12584	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
40	37, 39	ffvelcdmd 7104	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝐾) ∈ ℂ)
41	29	simprd 495	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝐾) ≠ 0)
42	35, 40, 41	divcld 12040	. . . . . 6 ⊢ (𝜑 → ((𝐹‘0) / (𝐴‘𝐾)) ∈ ℂ)
43	42	negcld 11604	. . . . 5 ⊢ (𝜑 → -((𝐹‘0) / (𝐴‘𝐾)) ∈ ℂ)
44	38	nnrecred 12314	. . . . . 6 ⊢ (𝜑 → (1 / 𝐾) ∈ ℝ)
45	44	recnd 11286	. . . . 5 ⊢ (𝜑 → (1 / 𝐾) ∈ ℂ)
46	43, 45	cxpcld 26764	. . . 4 ⊢ (𝜑 → (-((𝐹‘0) / (𝐴‘𝐾))↑_𝑐(1 / 𝐾)) ∈ ℂ)
47	30, 46	eqeltrid 2842	. . 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 15483	. . . . 5 ⊢ (𝜑 → (abs‘(𝐹‘0)) ∈ ℝ⁺)
51		fzfid 14010	. . . . . . 7 ⊢ (𝜑 → ((𝐾 + 1)...𝑁) ∈ Fin)
52		peano2nn0 12563	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ₀ → (𝐾 + 1) ∈ ℕ₀)
53	39, 52	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾 + 1) ∈ ℕ₀)
54		elfzuz 13556	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ((𝐾 + 1)...𝑁) → 𝑘 ∈ (ℤ_≥‘(𝐾 + 1)))
55		eluznn0 12956	. . . . . . . . . . 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 7103	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
58	56, 57	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐴‘𝑘) ∈ ℂ)
59		expcl 14116	. . . . . . . . . 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 11278	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → ((𝐴‘𝑘) · (𝑇↑𝑘)) ∈ ℂ)
62	61	abscld 15471	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) ∈ ℝ)
63	51, 62	fsumrecl 15766	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) ∈ ℝ)
64	61	absge0d 15479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → 0 ≤ (abs‘((𝐴‘𝑘) · (𝑇↑𝑘))))
65	51, 62, 64	fsumge0 15827	. . . . . 6 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))))
66	63, 65	ge0p1rpd 13104	. . . . 5 ⊢ (𝜑 → (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1) ∈ ℝ⁺)
67	50, 66	rpdivcld 13091	. . . 4 ⊢ (𝜑 → ((abs‘(𝐹‘0)) / (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1)) ∈ ℝ⁺)
68	48, 67	eqeltrid 2842	. . 3 ⊢ (𝜑 → 𝑈 ∈ ℝ⁺)
69		ftalem4.9	. . . 4 ⊢ 𝑋 = if(1 ≤ 𝑈, 1, 𝑈)
70		1rp 13035	. . . . 5 ⊢ 1 ∈ ℝ⁺
71		ifcl 4575	. . . . 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 2842	. . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
74	47, 68, 73	3jca 1127	. 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 1536 ∈ wcel 2105 ≠ wne 2937 {crab 3432 ⊆ wss 3962 ∅c0 4338 ifcif 4530 class class class wbr 5147 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 infcinf 9478 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 < clt 11292 ≤ cle 11293 -cneg 11490 / cdiv 11917 ℕcn 12263 ℕ₀cn0 12523 ℤ_≥cuz 12875 ℝ⁺crp 13031 ...cfz 13543 ↑cexp 14098 abscabs 15269 Σcsu 15718 0_𝑝c0p 25717 Polycply 26237 coeffccoe 26239 degcdgr 26240 ↑_𝑐ccxp 26611

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ioc 13388 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-shft 15102 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-sum 15719 df-ef 16099 df-sin 16101 df-cos 16102 df-pi 16104 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-perf 23160 df-cn 23250 df-cnp 23251 df-haus 23338 df-tx 23585 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-xms 24345 df-ms 24346 df-tms 24347 df-cncf 24917 df-0p 25718 df-limc 25915 df-dv 25916 df-ply 26241 df-coe 26243 df-dgr 26244 df-log 26612 df-cxp 26613

This theorem is referenced by: ftalem5 27134

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