ftalem4 - Metamath Proof Explorer

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Theorem ftalem4 27054

Description: Lemma for fta 27058: Closure of the auxiliary variables for ftalem5 27055. (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 4034	. . . . . 6 ⊢ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ⊆ ℕ
3		nnuz 12802	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
4	2, 3	sseqtri 3984	. . . . 5 ⊢ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ⊆ (ℤ_≥‘1)
5		fveq2 6842	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝐴‘𝑛) = (𝐴‘𝑁))
6	5	neeq1d 2992	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝐴‘𝑛) ≠ 0 ↔ (𝐴‘𝑁) ≠ 0))
7		ftalem.4	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
8	7	nnne0d 12207	. . . . . . . 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 26239	. . . . . . . . . . 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 6842	. . . . . . . . . . . 12 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = (deg‘0_𝑝))
15		dgr0 26236	. . . . . . . . . . . 12 ⊢ (deg‘0_𝑝) = 0
16	14, 15	eqtrdi 2788	. . . . . . . . . . 11 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = 0)
17	10, 16	eqtrid 2784	. . . . . . . . . 10 ⊢ (𝐹 = 0_𝑝 → 𝑁 = 0)
18	13, 17	biimtrrdi 254	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴‘𝑁) = 0 → 𝑁 = 0))
19	18	necon3d 2954	. . . . . . . 8 ⊢ (𝜑 → (𝑁 ≠ 0 → (𝐴‘𝑁) ≠ 0))
20	8, 19	mpd 15	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝑁) ≠ 0)
21	6, 7, 20	elrabd 3650	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0})
22	21	ne0d 4296	. . . . 5 ⊢ (𝜑 → {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ≠ ∅)
23		infssuzcl 12857	. . . . 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 2841	. . 3 ⊢ (𝜑 → 𝐾 ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0})
26		fveq2 6842	. . . . 5 ⊢ (𝑛 = 𝐾 → (𝐴‘𝑛) = (𝐴‘𝐾))
27	26	neeq1d 2992	. . . 4 ⊢ (𝑛 = 𝐾 → ((𝐴‘𝑛) ≠ 0 ↔ (𝐴‘𝐾) ≠ 0))
28	27	elrab 3648	. . 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 26171	. . . . . . . . 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 7035	. . . . . . . 8 ⊢ ((𝐹:ℂ⟶ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) ∈ ℂ)
35	32, 33, 34	sylancl 587	. . . . . . 7 ⊢ (𝜑 → (𝐹‘0) ∈ ℂ)
36	11	coef3 26205	. . . . . . . . 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 12474	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
40	37, 39	ffvelcdmd 7039	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝐾) ∈ ℂ)
41	29	simprd 495	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝐾) ≠ 0)
42	35, 40, 41	divcld 11929	. . . . . 6 ⊢ (𝜑 → ((𝐹‘0) / (𝐴‘𝐾)) ∈ ℂ)
43	42	negcld 11491	. . . . 5 ⊢ (𝜑 → -((𝐹‘0) / (𝐴‘𝐾)) ∈ ℂ)
44	38	nnrecred 12208	. . . . . 6 ⊢ (𝜑 → (1 / 𝐾) ∈ ℝ)
45	44	recnd 11172	. . . . 5 ⊢ (𝜑 → (1 / 𝐾) ∈ ℂ)
46	43, 45	cxpcld 26685	. . . 4 ⊢ (𝜑 → (-((𝐹‘0) / (𝐴‘𝐾))↑_𝑐(1 / 𝐾)) ∈ ℂ)
47	30, 46	eqeltrid 2841	. . 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 15386	. . . . 5 ⊢ (𝜑 → (abs‘(𝐹‘0)) ∈ ℝ⁺)
51		fzfid 13908	. . . . . . 7 ⊢ (𝜑 → ((𝐾 + 1)...𝑁) ∈ Fin)
52		peano2nn0 12453	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ₀ → (𝐾 + 1) ∈ ℕ₀)
53	39, 52	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾 + 1) ∈ ℕ₀)
54		elfzuz 13448	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ((𝐾 + 1)...𝑁) → 𝑘 ∈ (ℤ_≥‘(𝐾 + 1)))
55		eluznn0 12842	. . . . . . . . . . 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 7038	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
58	56, 57	syldan 592	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐴‘𝑘) ∈ ℂ)
59		expcl 14014	. . . . . . . . . 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 11164	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → ((𝐴‘𝑘) · (𝑇↑𝑘)) ∈ ℂ)
62	61	abscld 15374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) ∈ ℝ)
63	51, 62	fsumrecl 15669	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) ∈ ℝ)
64	61	absge0d 15382	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → 0 ≤ (abs‘((𝐴‘𝑘) · (𝑇↑𝑘))))
65	51, 62, 64	fsumge0 15730	. . . . . 6 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))))
66	63, 65	ge0p1rpd 12991	. . . . 5 ⊢ (𝜑 → (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1) ∈ ℝ⁺)
67	50, 66	rpdivcld 12978	. . . 4 ⊢ (𝜑 → ((abs‘(𝐹‘0)) / (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1)) ∈ ℝ⁺)
68	48, 67	eqeltrid 2841	. . 3 ⊢ (𝜑 → 𝑈 ∈ ℝ⁺)
69		ftalem4.9	. . . 4 ⊢ 𝑋 = if(1 ≤ 𝑈, 1, 𝑈)
70		1rp 12921	. . . . 5 ⊢ 1 ∈ ℝ⁺
71		ifcl 4527	. . . . 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 2841	. . 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 2933 {crab 3401 ⊆ wss 3903 ∅c0 4287 ifcif 4481 class class class wbr 5100 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 infcinf 9356 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 < clt 11178 ≤ cle 11179 -cneg 11377 / cdiv 11806 ℕcn 12157 ℕ₀cn0 12413 ℤ_≥cuz 12763 ℝ⁺crp 12917 ...cfz 13435 ↑cexp 13996 abscabs 15169 Σcsu 15621 0_𝑝c0p 25638 Polycply 26157 coeffccoe 26159 degcdgr 26160 ↑_𝑐ccxp 26532

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 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ioc 13278 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997 df-fac 14209 df-bc 14238 df-hash 14266 df-shft 15002 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-limsup 15406 df-clim 15423 df-rlim 15424 df-sum 15622 df-ef 16002 df-sin 16004 df-cos 16005 df-pi 16007 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-mulg 19010 df-cntz 19258 df-cmn 19723 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-lp 23092 df-perf 23093 df-cn 23183 df-cnp 23184 df-haus 23271 df-tx 23518 df-hmeo 23711 df-fil 23802 df-fm 23894 df-flim 23895 df-flf 23896 df-xms 24276 df-ms 24277 df-tms 24278 df-cncf 24839 df-0p 25639 df-limc 25835 df-dv 25836 df-ply 26161 df-coe 26163 df-dgr 26164 df-log 26533 df-cxp 26534

This theorem is referenced by: ftalem5 27055

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