ftalem4 - Metamath Proof Explorer

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Theorem ftalem4 27039

Description: Lemma for fta 27043: Closure of the auxiliary variables for ftalem5 27040. (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 4020	. . . . . 6 ⊢ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ⊆ ℕ
3		nnuz 12827	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
4	2, 3	sseqtri 3970	. . . . 5 ⊢ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ⊆ (ℤ_≥‘1)
5		fveq2 6840	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝐴‘𝑛) = (𝐴‘𝑁))
6	5	neeq1d 2991	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝐴‘𝑛) ≠ 0 ↔ (𝐴‘𝑁) ≠ 0))
7		ftalem.4	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
8	7	nnne0d 12227	. . . . . . . 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 26230	. . . . . . . . . . 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 6840	. . . . . . . . . . . 12 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = (deg‘0_𝑝))
15		dgr0 26227	. . . . . . . . . . . 12 ⊢ (deg‘0_𝑝) = 0
16	14, 15	eqtrdi 2787	. . . . . . . . . . 11 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = 0)
17	10, 16	eqtrid 2783	. . . . . . . . . 10 ⊢ (𝐹 = 0_𝑝 → 𝑁 = 0)
18	13, 17	biimtrrdi 254	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴‘𝑁) = 0 → 𝑁 = 0))
19	18	necon3d 2953	. . . . . . . 8 ⊢ (𝜑 → (𝑁 ≠ 0 → (𝐴‘𝑁) ≠ 0))
20	8, 19	mpd 15	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝑁) ≠ 0)
21	6, 7, 20	elrabd 3636	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0})
22	21	ne0d 4282	. . . . 5 ⊢ (𝜑 → {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ≠ ∅)
23		infssuzcl 12882	. . . . 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 588	. . . 4 ⊢ (𝜑 → inf({𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0})
25	1, 24	eqeltrid 2840	. . 3 ⊢ (𝜑 → 𝐾 ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0})
26		fveq2 6840	. . . . 5 ⊢ (𝑛 = 𝐾 → (𝐴‘𝑛) = (𝐴‘𝐾))
27	26	neeq1d 2991	. . . 4 ⊢ (𝑛 = 𝐾 → ((𝐴‘𝑛) ≠ 0 ↔ (𝐴‘𝐾) ≠ 0))
28	27	elrab 3634	. . 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 26163	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
32	9, 31	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
33		0cn 11136	. . . . . . . 8 ⊢ 0 ∈ ℂ
34		ffvelcdm 7033	. . . . . . . 8 ⊢ ((𝐹:ℂ⟶ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) ∈ ℂ)
35	32, 33, 34	sylancl 587	. . . . . . 7 ⊢ (𝜑 → (𝐹‘0) ∈ ℂ)
36	11	coef3 26197	. . . . . . . . 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 12498	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
40	37, 39	ffvelcdmd 7037	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝐾) ∈ ℂ)
41	29	simprd 495	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝐾) ≠ 0)
42	35, 40, 41	divcld 11931	. . . . . 6 ⊢ (𝜑 → ((𝐹‘0) / (𝐴‘𝐾)) ∈ ℂ)
43	42	negcld 11492	. . . . 5 ⊢ (𝜑 → -((𝐹‘0) / (𝐴‘𝐾)) ∈ ℂ)
44	38	nnrecred 12228	. . . . . 6 ⊢ (𝜑 → (1 / 𝐾) ∈ ℝ)
45	44	recnd 11173	. . . . 5 ⊢ (𝜑 → (1 / 𝐾) ∈ ℂ)
46	43, 45	cxpcld 26672	. . . 4 ⊢ (𝜑 → (-((𝐹‘0) / (𝐴‘𝐾))↑_𝑐(1 / 𝐾)) ∈ ℂ)
47	30, 46	eqeltrid 2840	. . 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 15413	. . . . 5 ⊢ (𝜑 → (abs‘(𝐹‘0)) ∈ ℝ⁺)
51		fzfid 13935	. . . . . . 7 ⊢ (𝜑 → ((𝐾 + 1)...𝑁) ∈ Fin)
52		peano2nn0 12477	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ₀ → (𝐾 + 1) ∈ ℕ₀)
53	39, 52	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾 + 1) ∈ ℕ₀)
54		elfzuz 13474	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ((𝐾 + 1)...𝑁) → 𝑘 ∈ (ℤ_≥‘(𝐾 + 1)))
55		eluznn0 12867	. . . . . . . . . . 11 ⊢ (((𝐾 + 1) ∈ ℕ₀ ∧ 𝑘 ∈ (ℤ_≥‘(𝐾 + 1))) → 𝑘 ∈ ℕ₀)
56	53, 54, 55	syl2an 597	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → 𝑘 ∈ ℕ₀)
57	37	ffvelcdmda 7036	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
58	56, 57	syldan 592	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐴‘𝑘) ∈ ℂ)
59		expcl 14041	. . . . . . . . . 10 ⊢ ((𝑇 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑇↑𝑘) ∈ ℂ)
60	47, 56, 59	syl2an2r 686	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝑇↑𝑘) ∈ ℂ)
61	58, 60	mulcld 11165	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → ((𝐴‘𝑘) · (𝑇↑𝑘)) ∈ ℂ)
62	61	abscld 15401	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) ∈ ℝ)
63	51, 62	fsumrecl 15696	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) ∈ ℝ)
64	61	absge0d 15409	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → 0 ≤ (abs‘((𝐴‘𝑘) · (𝑇↑𝑘))))
65	51, 62, 64	fsumge0 15758	. . . . . 6 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))))
66	63, 65	ge0p1rpd 13016	. . . . 5 ⊢ (𝜑 → (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1) ∈ ℝ⁺)
67	50, 66	rpdivcld 13003	. . . 4 ⊢ (𝜑 → ((abs‘(𝐹‘0)) / (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1)) ∈ ℝ⁺)
68	48, 67	eqeltrid 2840	. . 3 ⊢ (𝜑 → 𝑈 ∈ ℝ⁺)
69		ftalem4.9	. . . 4 ⊢ 𝑋 = if(1 ≤ 𝑈, 1, 𝑈)
70		1rp 12946	. . . . 5 ⊢ 1 ∈ ℝ⁺
71		ifcl 4512	. . . . 5 ⊢ ((1 ∈ ℝ⁺ ∧ 𝑈 ∈ ℝ⁺) → if(1 ≤ 𝑈, 1, 𝑈) ∈ ℝ⁺)
72	70, 68, 71	sylancr 588	. . . 4 ⊢ (𝜑 → if(1 ≤ 𝑈, 1, 𝑈) ∈ ℝ⁺)
73	69, 72	eqeltrid 2840	. . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
74	47, 68, 73	3jca 1129	. 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 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 {crab 3389 ⊆ wss 3889 ∅c0 4273 ifcif 4466 class class class wbr 5085 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 infcinf 9354 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 < clt 11179 ≤ cle 11180 -cneg 11378 / cdiv 11807 ℕcn 12174 ℕ₀cn0 12437 ℤ_≥cuz 12788 ℝ⁺crp 12942 ...cfz 13461 ↑cexp 14023 abscabs 15196 Σcsu 15648 0_𝑝c0p 25636 Polycply 26149 coeffccoe 26151 degcdgr 26152 ↑_𝑐ccxp 26519

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-ef 16032 df-sin 16034 df-cos 16035 df-pi 16037 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-0p 25637 df-limc 25833 df-dv 25834 df-ply 26153 df-coe 26155 df-dgr 26156 df-log 26520 df-cxp 26521

This theorem is referenced by: ftalem5 27040

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