ftalem4 - Metamath Proof Explorer

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Theorem ftalem4 27033

Description: Lemma for fta 27037: Closure of the auxiliary variables for ftalem5 27034. (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 4029	. . . . . 6 ⊢ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ⊆ ℕ
3		nnuz 12781	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
4	2, 3	sseqtri 3979	. . . . 5 ⊢ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ⊆ (ℤ_≥‘1)
5		fveq2 6831	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝐴‘𝑛) = (𝐴‘𝑁))
6	5	neeq1d 2988	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝐴‘𝑛) ≠ 0 ↔ (𝐴‘𝑁) ≠ 0))
7		ftalem.4	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
8	7	nnne0d 12186	. . . . . . . 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 26218	. . . . . . . . . . 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 6831	. . . . . . . . . . . 12 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = (deg‘0_𝑝))
15		dgr0 26215	. . . . . . . . . . . 12 ⊢ (deg‘0_𝑝) = 0
16	14, 15	eqtrdi 2784	. . . . . . . . . . 11 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = 0)
17	10, 16	eqtrid 2780	. . . . . . . . . 10 ⊢ (𝐹 = 0_𝑝 → 𝑁 = 0)
18	13, 17	biimtrrdi 254	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴‘𝑁) = 0 → 𝑁 = 0))
19	18	necon3d 2950	. . . . . . . 8 ⊢ (𝜑 → (𝑁 ≠ 0 → (𝐴‘𝑁) ≠ 0))
20	8, 19	mpd 15	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝑁) ≠ 0)
21	6, 7, 20	elrabd 3645	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0})
22	21	ne0d 4291	. . . . 5 ⊢ (𝜑 → {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ≠ ∅)
23		infssuzcl 12836	. . . . 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 2837	. . 3 ⊢ (𝜑 → 𝐾 ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0})
26		fveq2 6831	. . . . 5 ⊢ (𝑛 = 𝐾 → (𝐴‘𝑛) = (𝐴‘𝐾))
27	26	neeq1d 2988	. . . 4 ⊢ (𝑛 = 𝐾 → ((𝐴‘𝑛) ≠ 0 ↔ (𝐴‘𝐾) ≠ 0))
28	27	elrab 3643	. . 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 26150	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
32	9, 31	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
33		0cn 11115	. . . . . . . 8 ⊢ 0 ∈ ℂ
34		ffvelcdm 7023	. . . . . . . 8 ⊢ ((𝐹:ℂ⟶ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) ∈ ℂ)
35	32, 33, 34	sylancl 586	. . . . . . 7 ⊢ (𝜑 → (𝐹‘0) ∈ ℂ)
36	11	coef3 26184	. . . . . . . . 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 12453	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
40	37, 39	ffvelcdmd 7027	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝐾) ∈ ℂ)
41	29	simprd 495	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝐾) ≠ 0)
42	35, 40, 41	divcld 11908	. . . . . 6 ⊢ (𝜑 → ((𝐹‘0) / (𝐴‘𝐾)) ∈ ℂ)
43	42	negcld 11470	. . . . 5 ⊢ (𝜑 → -((𝐹‘0) / (𝐴‘𝐾)) ∈ ℂ)
44	38	nnrecred 12187	. . . . . 6 ⊢ (𝜑 → (1 / 𝐾) ∈ ℝ)
45	44	recnd 11151	. . . . 5 ⊢ (𝜑 → (1 / 𝐾) ∈ ℂ)
46	43, 45	cxpcld 26664	. . . 4 ⊢ (𝜑 → (-((𝐹‘0) / (𝐴‘𝐾))↑_𝑐(1 / 𝐾)) ∈ ℂ)
47	30, 46	eqeltrid 2837	. . 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 15365	. . . . 5 ⊢ (𝜑 → (abs‘(𝐹‘0)) ∈ ℝ⁺)
51		fzfid 13887	. . . . . . 7 ⊢ (𝜑 → ((𝐾 + 1)...𝑁) ∈ Fin)
52		peano2nn0 12432	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ₀ → (𝐾 + 1) ∈ ℕ₀)
53	39, 52	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾 + 1) ∈ ℕ₀)
54		elfzuz 13427	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ((𝐾 + 1)...𝑁) → 𝑘 ∈ (ℤ_≥‘(𝐾 + 1)))
55		eluznn0 12821	. . . . . . . . . . 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 7026	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
58	56, 57	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐴‘𝑘) ∈ ℂ)
59		expcl 13993	. . . . . . . . . 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 11143	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → ((𝐴‘𝑘) · (𝑇↑𝑘)) ∈ ℂ)
62	61	abscld 15353	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) ∈ ℝ)
63	51, 62	fsumrecl 15648	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) ∈ ℝ)
64	61	absge0d 15361	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → 0 ≤ (abs‘((𝐴‘𝑘) · (𝑇↑𝑘))))
65	51, 62, 64	fsumge0 15709	. . . . . 6 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))))
66	63, 65	ge0p1rpd 12970	. . . . 5 ⊢ (𝜑 → (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1) ∈ ℝ⁺)
67	50, 66	rpdivcld 12957	. . . 4 ⊢ (𝜑 → ((abs‘(𝐹‘0)) / (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1)) ∈ ℝ⁺)
68	48, 67	eqeltrid 2837	. . 3 ⊢ (𝜑 → 𝑈 ∈ ℝ⁺)
69		ftalem4.9	. . . 4 ⊢ 𝑋 = if(1 ≤ 𝑈, 1, 𝑈)
70		1rp 12900	. . . . 5 ⊢ 1 ∈ ℝ⁺
71		ifcl 4522	. . . . 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 2837	. . 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 2113 ≠ wne 2929 {crab 3396 ⊆ wss 3898 ∅c0 4282 ifcif 4476 class class class wbr 5095 ⟶wf 6485 ‘cfv 6489 (class class class)co 7355 infcinf 9336 ℂcc 11015 ℝcr 11016 0cc0 11017 1c1 11018 + caddc 11020 · cmul 11022 < clt 11157 ≤ cle 11158 -cneg 11356 / cdiv 11785 ℕcn 12136 ℕ₀cn0 12392 ℤ_≥cuz 12742 ℝ⁺crp 12896 ...cfz 13414 ↑cexp 13975 abscabs 15148 Σcsu 15600 0_𝑝c0p 25617 Polycply 26136 coeffccoe 26138 degcdgr 26139 ↑_𝑐ccxp 26511

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-inf2 9542 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-pre-sup 11095 ax-addf 11096

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 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-om 7806 df-1st 7930 df-2nd 7931 df-supp 8100 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-er 8631 df-map 8761 df-pm 8762 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9257 df-fi 9306 df-sup 9337 df-inf 9338 df-oi 9407 df-card 9843 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-div 11786 df-nn 12137 df-2 12199 df-3 12200 df-4 12201 df-5 12202 df-6 12203 df-7 12204 df-8 12205 df-9 12206 df-n0 12393 df-z 12480 df-dec 12599 df-uz 12743 df-q 12853 df-rp 12897 df-xneg 13017 df-xadd 13018 df-xmul 13019 df-ioo 13256 df-ioc 13257 df-ico 13258 df-icc 13259 df-fz 13415 df-fzo 13562 df-fl 13703 df-mod 13781 df-seq 13916 df-exp 13976 df-fac 14188 df-bc 14217 df-hash 14245 df-shft 14981 df-cj 15013 df-re 15014 df-im 15015 df-sqrt 15149 df-abs 15150 df-limsup 15385 df-clim 15402 df-rlim 15403 df-sum 15601 df-ef 15981 df-sin 15983 df-cos 15984 df-pi 15986 df-struct 17065 df-sets 17082 df-slot 17100 df-ndx 17112 df-base 17128 df-ress 17149 df-plusg 17181 df-mulr 17182 df-starv 17183 df-sca 17184 df-vsca 17185 df-ip 17186 df-tset 17187 df-ple 17188 df-ds 17190 df-unif 17191 df-hom 17192 df-cco 17193 df-rest 17333 df-topn 17334 df-0g 17352 df-gsum 17353 df-topgen 17354 df-pt 17355 df-prds 17358 df-xrs 17414 df-qtop 17419 df-imas 17420 df-xps 17422 df-mre 17496 df-mrc 17497 df-acs 17499 df-mgm 18556 df-sgrp 18635 df-mnd 18651 df-submnd 18700 df-mulg 18989 df-cntz 19237 df-cmn 19702 df-psmet 21292 df-xmet 21293 df-met 21294 df-bl 21295 df-mopn 21296 df-fbas 21297 df-fg 21298 df-cnfld 21301 df-top 22829 df-topon 22846 df-topsp 22868 df-bases 22881 df-cld 22954 df-ntr 22955 df-cls 22956 df-nei 23033 df-lp 23071 df-perf 23072 df-cn 23162 df-cnp 23163 df-haus 23250 df-tx 23497 df-hmeo 23690 df-fil 23781 df-fm 23873 df-flim 23874 df-flf 23875 df-xms 24255 df-ms 24256 df-tms 24257 df-cncf 24818 df-0p 25618 df-limc 25814 df-dv 25815 df-ply 26140 df-coe 26142 df-dgr 26143 df-log 26512 df-cxp 26513

This theorem is referenced by: ftalem5 27034

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