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Mirrors > Home > MPE Home > Th. List > ftalem4 | Structured version Visualization version GIF version |
Description: Lemma for fta 25584: Closure of the auxiliary variables for ftalem5 25581. (Contributed by Mario Carneiro, 20-Sep-2014.) (Revised by AV, 28-Sep-2020.) |
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) |
Ref | Expression |
---|---|
ftalem4 | ⊢ (𝜑 → ((𝐾 ∈ ℕ ∧ (𝐴‘𝐾) ≠ 0) ∧ (𝑇 ∈ ℂ ∧ 𝑈 ∈ ℝ+ ∧ 𝑋 ∈ ℝ+))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ftalem4.6 | . . . 4 ⊢ 𝐾 = inf({𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}, ℝ, < ) | |
2 | ssrab2 4053 | . . . . . 6 ⊢ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ⊆ ℕ | |
3 | nnuz 12269 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
4 | 2, 3 | sseqtri 4000 | . . . . 5 ⊢ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ⊆ (ℤ≥‘1) |
5 | fveq2 6663 | . . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝐴‘𝑛) = (𝐴‘𝑁)) | |
6 | 5 | neeq1d 3072 | . . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝐴‘𝑛) ≠ 0 ↔ (𝐴‘𝑁) ≠ 0)) |
7 | ftalem.4 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
8 | 7 | nnne0d 11675 | . . . . . . . 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 24782 | . . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0𝑝 ↔ (𝐴‘𝑁) = 0)) |
13 | 9, 12 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐹 = 0𝑝 ↔ (𝐴‘𝑁) = 0)) |
14 | fveq2 6663 | . . . . . . . . . . . 12 ⊢ (𝐹 = 0𝑝 → (deg‘𝐹) = (deg‘0𝑝)) | |
15 | dgr0 24779 | . . . . . . . . . . . 12 ⊢ (deg‘0𝑝) = 0 | |
16 | 14, 15 | syl6eq 2869 | . . . . . . . . . . 11 ⊢ (𝐹 = 0𝑝 → (deg‘𝐹) = 0) |
17 | 10, 16 | syl5eq 2865 | . . . . . . . . . 10 ⊢ (𝐹 = 0𝑝 → 𝑁 = 0) |
18 | 13, 17 | syl6bir 255 | . . . . . . . . 9 ⊢ (𝜑 → ((𝐴‘𝑁) = 0 → 𝑁 = 0)) |
19 | 18 | necon3d 3034 | . . . . . . . 8 ⊢ (𝜑 → (𝑁 ≠ 0 → (𝐴‘𝑁) ≠ 0)) |
20 | 8, 19 | mpd 15 | . . . . . . 7 ⊢ (𝜑 → (𝐴‘𝑁) ≠ 0) |
21 | 6, 7, 20 | elrabd 3679 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}) |
22 | 21 | ne0d 4298 | . . . . 5 ⊢ (𝜑 → {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ≠ ∅) |
23 | infssuzcl 12320 | . . . . 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 2914 | . . 3 ⊢ (𝜑 → 𝐾 ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}) |
26 | fveq2 6663 | . . . . 5 ⊢ (𝑛 = 𝐾 → (𝐴‘𝑛) = (𝐴‘𝐾)) | |
27 | 26 | neeq1d 3072 | . . . 4 ⊢ (𝑛 = 𝐾 → ((𝐴‘𝑛) ≠ 0 ↔ (𝐴‘𝐾) ≠ 0)) |
28 | 27 | elrab 3677 | . . 3 ⊢ (𝐾 ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ↔ (𝐾 ∈ ℕ ∧ (𝐴‘𝐾) ≠ 0)) |
29 | 25, 28 | sylib 219 | . 2 ⊢ (𝜑 → (𝐾 ∈ ℕ ∧ (𝐴‘𝐾) ≠ 0)) |
30 | ftalem4.7 | . . . 4 ⊢ 𝑇 = (-((𝐹‘0) / (𝐴‘𝐾))↑𝑐(1 / 𝐾)) | |
31 | plyf 24715 | . . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ) | |
32 | 9, 31 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ) |
33 | 0cn 10621 | . . . . . . . 8 ⊢ 0 ∈ ℂ | |
34 | ffvelrn 6841 | . . . . . . . 8 ⊢ ((𝐹:ℂ⟶ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) ∈ ℂ) | |
35 | 32, 33, 34 | sylancl 586 | . . . . . . 7 ⊢ (𝜑 → (𝐹‘0) ∈ ℂ) |
36 | 11 | coef3 24749 | . . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ) |
37 | 9, 36 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) |
38 | 29 | simpld 495 | . . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℕ) |
39 | 38 | nnnn0d 11943 | . . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℕ0) |
40 | 37, 39 | ffvelrnd 6844 | . . . . . . 7 ⊢ (𝜑 → (𝐴‘𝐾) ∈ ℂ) |
41 | 29 | simprd 496 | . . . . . . 7 ⊢ (𝜑 → (𝐴‘𝐾) ≠ 0) |
42 | 35, 40, 41 | divcld 11404 | . . . . . 6 ⊢ (𝜑 → ((𝐹‘0) / (𝐴‘𝐾)) ∈ ℂ) |
43 | 42 | negcld 10972 | . . . . 5 ⊢ (𝜑 → -((𝐹‘0) / (𝐴‘𝐾)) ∈ ℂ) |
44 | 38 | nnrecred 11676 | . . . . . 6 ⊢ (𝜑 → (1 / 𝐾) ∈ ℝ) |
45 | 44 | recnd 10657 | . . . . 5 ⊢ (𝜑 → (1 / 𝐾) ∈ ℂ) |
46 | 43, 45 | cxpcld 25218 | . . . 4 ⊢ (𝜑 → (-((𝐹‘0) / (𝐴‘𝐾))↑𝑐(1 / 𝐾)) ∈ ℂ) |
47 | 30, 46 | eqeltrid 2914 | . . 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 14796 | . . . . 5 ⊢ (𝜑 → (abs‘(𝐹‘0)) ∈ ℝ+) |
51 | fzfid 13329 | . . . . . . 7 ⊢ (𝜑 → ((𝐾 + 1)...𝑁) ∈ Fin) | |
52 | peano2nn0 11925 | . . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ0 → (𝐾 + 1) ∈ ℕ0) | |
53 | 39, 52 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐾 + 1) ∈ ℕ0) |
54 | elfzuz 12892 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ ((𝐾 + 1)...𝑁) → 𝑘 ∈ (ℤ≥‘(𝐾 + 1))) | |
55 | eluznn0 12305 | . . . . . . . . . . 11 ⊢ (((𝐾 + 1) ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘(𝐾 + 1))) → 𝑘 ∈ ℕ0) | |
56 | 53, 54, 55 | syl2an 595 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → 𝑘 ∈ ℕ0) |
57 | 37 | ffvelrnda 6843 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
58 | 56, 57 | syldan 591 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐴‘𝑘) ∈ ℂ) |
59 | expcl 13435 | . . . . . . . . . 10 ⊢ ((𝑇 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑇↑𝑘) ∈ ℂ) | |
60 | 47, 56, 59 | syl2an2r 681 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝑇↑𝑘) ∈ ℂ) |
61 | 58, 60 | mulcld 10649 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → ((𝐴‘𝑘) · (𝑇↑𝑘)) ∈ ℂ) |
62 | 61 | abscld 14784 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) ∈ ℝ) |
63 | 51, 62 | fsumrecl 15079 | . . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) ∈ ℝ) |
64 | 61 | absge0d 14792 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → 0 ≤ (abs‘((𝐴‘𝑘) · (𝑇↑𝑘)))) |
65 | 51, 62, 64 | fsumge0 15138 | . . . . . 6 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘)))) |
66 | 63, 65 | ge0p1rpd 12449 | . . . . 5 ⊢ (𝜑 → (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1) ∈ ℝ+) |
67 | 50, 66 | rpdivcld 12436 | . . . 4 ⊢ (𝜑 → ((abs‘(𝐹‘0)) / (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1)) ∈ ℝ+) |
68 | 48, 67 | eqeltrid 2914 | . . 3 ⊢ (𝜑 → 𝑈 ∈ ℝ+) |
69 | ftalem4.9 | . . . 4 ⊢ 𝑋 = if(1 ≤ 𝑈, 1, 𝑈) | |
70 | 1rp 12381 | . . . . 5 ⊢ 1 ∈ ℝ+ | |
71 | ifcl 4507 | . . . . 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 2914 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ+) |
74 | 47, 68, 73 | 3jca 1120 | . 2 ⊢ (𝜑 → (𝑇 ∈ ℂ ∧ 𝑈 ∈ ℝ+ ∧ 𝑋 ∈ ℝ+)) |
75 | 29, 74 | jca 512 | 1 ⊢ (𝜑 → ((𝐾 ∈ ℕ ∧ (𝐴‘𝐾) ≠ 0) ∧ (𝑇 ∈ ℂ ∧ 𝑈 ∈ ℝ+ ∧ 𝑋 ∈ ℝ+))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 {crab 3139 ⊆ wss 3933 ∅c0 4288 ifcif 4463 class class class wbr 5057 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 infcinf 8893 ℂcc 10523 ℝcr 10524 0cc0 10525 1c1 10526 + caddc 10528 · cmul 10530 < clt 10663 ≤ cle 10664 -cneg 10859 / cdiv 11285 ℕcn 11626 ℕ0cn0 11885 ℤ≥cuz 12231 ℝ+crp 12377 ...cfz 12880 ↑cexp 13417 abscabs 14581 Σcsu 15030 0𝑝c0p 24197 Polycply 24701 coeffccoe 24703 degcdgr 24704 ↑𝑐ccxp 25066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-fac 13622 df-bc 13651 df-hash 13679 df-shft 14414 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-limsup 14816 df-clim 14833 df-rlim 14834 df-sum 15031 df-ef 15409 df-sin 15411 df-cos 15412 df-pi 15414 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-fbas 20470 df-fg 20471 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-lp 21672 df-perf 21673 df-cn 21763 df-cnp 21764 df-haus 21851 df-tx 22098 df-hmeo 22291 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-xms 22857 df-ms 22858 df-tms 22859 df-cncf 23413 df-0p 24198 df-limc 24391 df-dv 24392 df-ply 24705 df-coe 24707 df-dgr 24708 df-log 25067 df-cxp 25068 |
This theorem is referenced by: ftalem5 25581 |
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