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Mirrors > Home > MPE Home > Th. List > ftalem4 | Structured version Visualization version GIF version |
Description: Lemma for fta 25665: Closure of the auxiliary variables for ftalem5 25662. (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 4007 | . . . . . 6 ⊢ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ⊆ ℕ | |
3 | nnuz 12269 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
4 | 2, 3 | sseqtri 3951 | . . . . 5 ⊢ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ⊆ (ℤ≥‘1) |
5 | fveq2 6645 | . . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝐴‘𝑛) = (𝐴‘𝑁)) | |
6 | 5 | neeq1d 3046 | . . . . . . 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 24862 | . . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0𝑝 ↔ (𝐴‘𝑁) = 0)) |
13 | 9, 12 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐹 = 0𝑝 ↔ (𝐴‘𝑁) = 0)) |
14 | fveq2 6645 | . . . . . . . . . . . 12 ⊢ (𝐹 = 0𝑝 → (deg‘𝐹) = (deg‘0𝑝)) | |
15 | dgr0 24859 | . . . . . . . . . . . 12 ⊢ (deg‘0𝑝) = 0 | |
16 | 14, 15 | eqtrdi 2849 | . . . . . . . . . . 11 ⊢ (𝐹 = 0𝑝 → (deg‘𝐹) = 0) |
17 | 10, 16 | syl5eq 2845 | . . . . . . . . . 10 ⊢ (𝐹 = 0𝑝 → 𝑁 = 0) |
18 | 13, 17 | syl6bir 257 | . . . . . . . . 9 ⊢ (𝜑 → ((𝐴‘𝑁) = 0 → 𝑁 = 0)) |
19 | 18 | necon3d 3008 | . . . . . . . 8 ⊢ (𝜑 → (𝑁 ≠ 0 → (𝐴‘𝑁) ≠ 0)) |
20 | 8, 19 | mpd 15 | . . . . . . 7 ⊢ (𝜑 → (𝐴‘𝑁) ≠ 0) |
21 | 6, 7, 20 | elrabd 3630 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}) |
22 | 21 | ne0d 4251 | . . . . 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 590 | . . . 4 ⊢ (𝜑 → inf({𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}) |
25 | 1, 24 | eqeltrid 2894 | . . 3 ⊢ (𝜑 → 𝐾 ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}) |
26 | fveq2 6645 | . . . . 5 ⊢ (𝑛 = 𝐾 → (𝐴‘𝑛) = (𝐴‘𝐾)) | |
27 | 26 | neeq1d 3046 | . . . 4 ⊢ (𝑛 = 𝐾 → ((𝐴‘𝑛) ≠ 0 ↔ (𝐴‘𝐾) ≠ 0)) |
28 | 27 | elrab 3628 | . . 3 ⊢ (𝐾 ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ↔ (𝐾 ∈ ℕ ∧ (𝐴‘𝐾) ≠ 0)) |
29 | 25, 28 | sylib 221 | . 2 ⊢ (𝜑 → (𝐾 ∈ ℕ ∧ (𝐴‘𝐾) ≠ 0)) |
30 | ftalem4.7 | . . . 4 ⊢ 𝑇 = (-((𝐹‘0) / (𝐴‘𝐾))↑𝑐(1 / 𝐾)) | |
31 | plyf 24795 | . . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ) | |
32 | 9, 31 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ) |
33 | 0cn 10622 | . . . . . . . 8 ⊢ 0 ∈ ℂ | |
34 | ffvelrn 6826 | . . . . . . . 8 ⊢ ((𝐹:ℂ⟶ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) ∈ ℂ) | |
35 | 32, 33, 34 | sylancl 589 | . . . . . . 7 ⊢ (𝜑 → (𝐹‘0) ∈ ℂ) |
36 | 11 | coef3 24829 | . . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ) |
37 | 9, 36 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) |
38 | 29 | simpld 498 | . . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℕ) |
39 | 38 | nnnn0d 11943 | . . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℕ0) |
40 | 37, 39 | ffvelrnd 6829 | . . . . . . 7 ⊢ (𝜑 → (𝐴‘𝐾) ∈ ℂ) |
41 | 29 | simprd 499 | . . . . . . 7 ⊢ (𝜑 → (𝐴‘𝐾) ≠ 0) |
42 | 35, 40, 41 | divcld 11405 | . . . . . 6 ⊢ (𝜑 → ((𝐹‘0) / (𝐴‘𝐾)) ∈ ℂ) |
43 | 42 | negcld 10973 | . . . . 5 ⊢ (𝜑 → -((𝐹‘0) / (𝐴‘𝐾)) ∈ ℂ) |
44 | 38 | nnrecred 11676 | . . . . . 6 ⊢ (𝜑 → (1 / 𝐾) ∈ ℝ) |
45 | 44 | recnd 10658 | . . . . 5 ⊢ (𝜑 → (1 / 𝐾) ∈ ℂ) |
46 | 43, 45 | cxpcld 25299 | . . . 4 ⊢ (𝜑 → (-((𝐹‘0) / (𝐴‘𝐾))↑𝑐(1 / 𝐾)) ∈ ℂ) |
47 | 30, 46 | eqeltrid 2894 | . . 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 14800 | . . . . 5 ⊢ (𝜑 → (abs‘(𝐹‘0)) ∈ ℝ+) |
51 | fzfid 13336 | . . . . . . 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 12898 | . . . . . . . . . . 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 598 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → 𝑘 ∈ ℕ0) |
57 | 37 | ffvelrnda 6828 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
58 | 56, 57 | syldan 594 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐴‘𝑘) ∈ ℂ) |
59 | expcl 13443 | . . . . . . . . . 10 ⊢ ((𝑇 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑇↑𝑘) ∈ ℂ) | |
60 | 47, 56, 59 | syl2an2r 684 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝑇↑𝑘) ∈ ℂ) |
61 | 58, 60 | mulcld 10650 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → ((𝐴‘𝑘) · (𝑇↑𝑘)) ∈ ℂ) |
62 | 61 | abscld 14788 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) ∈ ℝ) |
63 | 51, 62 | fsumrecl 15083 | . . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) ∈ ℝ) |
64 | 61 | absge0d 14796 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → 0 ≤ (abs‘((𝐴‘𝑘) · (𝑇↑𝑘)))) |
65 | 51, 62, 64 | fsumge0 15142 | . . . . . 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 2894 | . . 3 ⊢ (𝜑 → 𝑈 ∈ ℝ+) |
69 | ftalem4.9 | . . . 4 ⊢ 𝑋 = if(1 ≤ 𝑈, 1, 𝑈) | |
70 | 1rp 12381 | . . . . 5 ⊢ 1 ∈ ℝ+ | |
71 | ifcl 4469 | . . . . 5 ⊢ ((1 ∈ ℝ+ ∧ 𝑈 ∈ ℝ+) → if(1 ≤ 𝑈, 1, 𝑈) ∈ ℝ+) | |
72 | 70, 68, 71 | sylancr 590 | . . . 4 ⊢ (𝜑 → if(1 ≤ 𝑈, 1, 𝑈) ∈ ℝ+) |
73 | 69, 72 | eqeltrid 2894 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ+) |
74 | 47, 68, 73 | 3jca 1125 | . 2 ⊢ (𝜑 → (𝑇 ∈ ℂ ∧ 𝑈 ∈ ℝ+ ∧ 𝑋 ∈ ℝ+)) |
75 | 29, 74 | jca 515 | 1 ⊢ (𝜑 → ((𝐾 ∈ ℕ ∧ (𝐴‘𝐾) ≠ 0) ∧ (𝑇 ∈ ℂ ∧ 𝑈 ∈ ℝ+ ∧ 𝑋 ∈ ℝ+))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 {crab 3110 ⊆ wss 3881 ∅c0 4243 ifcif 4425 class class class wbr 5030 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 infcinf 8889 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 < clt 10664 ≤ cle 10665 -cneg 10860 / cdiv 11286 ℕcn 11625 ℕ0cn0 11885 ℤ≥cuz 12231 ℝ+crp 12377 ...cfz 12885 ↑cexp 13425 abscabs 14585 Σcsu 15034 0𝑝c0p 24273 Polycply 24781 coeffccoe 24783 degcdgr 24784 ↑𝑐ccxp 25147 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 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 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-sin 15415 df-cos 15416 df-pi 15418 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-0p 24274 df-limc 24469 df-dv 24470 df-ply 24785 df-coe 24787 df-dgr 24788 df-log 25148 df-cxp 25149 |
This theorem is referenced by: ftalem5 25662 |
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