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Mirrors > Home > MPE Home > Th. List > ftalem4 | Structured version Visualization version GIF version |
Description: Lemma for fta 24606: Closure of the auxiliary variables for ftalem5 24603. (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 3650 | . . . . . 6 ⊢ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ⊆ ℕ | |
3 | nnuz 11599 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
4 | 2, 3 | sseqtri 3600 | . . . . 5 ⊢ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ⊆ (ℤ≥‘1) |
5 | ftalem.4 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
6 | 5 | nnne0d 10942 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ≠ 0) |
7 | ftalem.3 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) | |
8 | ftalem.2 | . . . . . . . . . . . 12 ⊢ 𝑁 = (deg‘𝐹) | |
9 | ftalem.1 | . . . . . . . . . . . 12 ⊢ 𝐴 = (coeff‘𝐹) | |
10 | 8, 9 | dgreq0 23825 | . . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0𝑝 ↔ (𝐴‘𝑁) = 0)) |
11 | 7, 10 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐹 = 0𝑝 ↔ (𝐴‘𝑁) = 0)) |
12 | fveq2 6103 | . . . . . . . . . . . 12 ⊢ (𝐹 = 0𝑝 → (deg‘𝐹) = (deg‘0𝑝)) | |
13 | dgr0 23822 | . . . . . . . . . . . 12 ⊢ (deg‘0𝑝) = 0 | |
14 | 12, 13 | syl6eq 2660 | . . . . . . . . . . 11 ⊢ (𝐹 = 0𝑝 → (deg‘𝐹) = 0) |
15 | 8, 14 | syl5eq 2656 | . . . . . . . . . 10 ⊢ (𝐹 = 0𝑝 → 𝑁 = 0) |
16 | 11, 15 | syl6bir 243 | . . . . . . . . 9 ⊢ (𝜑 → ((𝐴‘𝑁) = 0 → 𝑁 = 0)) |
17 | 16 | necon3d 2803 | . . . . . . . 8 ⊢ (𝜑 → (𝑁 ≠ 0 → (𝐴‘𝑁) ≠ 0)) |
18 | 6, 17 | mpd 15 | . . . . . . 7 ⊢ (𝜑 → (𝐴‘𝑁) ≠ 0) |
19 | fveq2 6103 | . . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝐴‘𝑛) = (𝐴‘𝑁)) | |
20 | 19 | neeq1d 2841 | . . . . . . . 8 ⊢ (𝑛 = 𝑁 → ((𝐴‘𝑛) ≠ 0 ↔ (𝐴‘𝑁) ≠ 0)) |
21 | 20 | elrab 3331 | . . . . . . 7 ⊢ (𝑁 ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ↔ (𝑁 ∈ ℕ ∧ (𝐴‘𝑁) ≠ 0)) |
22 | 5, 18, 21 | sylanbrc 695 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}) |
23 | ne0i 3880 | . . . . . 6 ⊢ (𝑁 ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} → {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ≠ ∅) | |
24 | 22, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ≠ ∅) |
25 | infssuzcl 11648 | . . . . 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}) | |
26 | 4, 24, 25 | sylancr 694 | . . . 4 ⊢ (𝜑 → inf({𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}) |
27 | 1, 26 | syl5eqel 2692 | . . 3 ⊢ (𝜑 → 𝐾 ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}) |
28 | fveq2 6103 | . . . . 5 ⊢ (𝑛 = 𝐾 → (𝐴‘𝑛) = (𝐴‘𝐾)) | |
29 | 28 | neeq1d 2841 | . . . 4 ⊢ (𝑛 = 𝐾 → ((𝐴‘𝑛) ≠ 0 ↔ (𝐴‘𝐾) ≠ 0)) |
30 | 29 | elrab 3331 | . . 3 ⊢ (𝐾 ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ↔ (𝐾 ∈ ℕ ∧ (𝐴‘𝐾) ≠ 0)) |
31 | 27, 30 | sylib 207 | . 2 ⊢ (𝜑 → (𝐾 ∈ ℕ ∧ (𝐴‘𝐾) ≠ 0)) |
32 | ftalem4.7 | . . . 4 ⊢ 𝑇 = (-((𝐹‘0) / (𝐴‘𝐾))↑𝑐(1 / 𝐾)) | |
33 | plyf 23758 | . . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ) | |
34 | 7, 33 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ) |
35 | 0cn 9911 | . . . . . . . 8 ⊢ 0 ∈ ℂ | |
36 | ffvelrn 6265 | . . . . . . . 8 ⊢ ((𝐹:ℂ⟶ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) ∈ ℂ) | |
37 | 34, 35, 36 | sylancl 693 | . . . . . . 7 ⊢ (𝜑 → (𝐹‘0) ∈ ℂ) |
38 | 9 | coef3 23792 | . . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ) |
39 | 7, 38 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) |
40 | 31 | simpld 474 | . . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℕ) |
41 | 40 | nnnn0d 11228 | . . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℕ0) |
42 | 39, 41 | ffvelrnd 6268 | . . . . . . 7 ⊢ (𝜑 → (𝐴‘𝐾) ∈ ℂ) |
43 | 31 | simprd 478 | . . . . . . 7 ⊢ (𝜑 → (𝐴‘𝐾) ≠ 0) |
44 | 37, 42, 43 | divcld 10680 | . . . . . 6 ⊢ (𝜑 → ((𝐹‘0) / (𝐴‘𝐾)) ∈ ℂ) |
45 | 44 | negcld 10258 | . . . . 5 ⊢ (𝜑 → -((𝐹‘0) / (𝐴‘𝐾)) ∈ ℂ) |
46 | 40 | nnrecred 10943 | . . . . . 6 ⊢ (𝜑 → (1 / 𝐾) ∈ ℝ) |
47 | 46 | recnd 9947 | . . . . 5 ⊢ (𝜑 → (1 / 𝐾) ∈ ℂ) |
48 | 45, 47 | cxpcld 24254 | . . . 4 ⊢ (𝜑 → (-((𝐹‘0) / (𝐴‘𝐾))↑𝑐(1 / 𝐾)) ∈ ℂ) |
49 | 32, 48 | syl5eqel 2692 | . . 3 ⊢ (𝜑 → 𝑇 ∈ ℂ) |
50 | ftalem4.8 | . . . 4 ⊢ 𝑈 = ((abs‘(𝐹‘0)) / (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1)) | |
51 | ftalem4.5 | . . . . . 6 ⊢ (𝜑 → (𝐹‘0) ≠ 0) | |
52 | 37, 51 | absrpcld 14035 | . . . . 5 ⊢ (𝜑 → (abs‘(𝐹‘0)) ∈ ℝ+) |
53 | fzfid 12634 | . . . . . . 7 ⊢ (𝜑 → ((𝐾 + 1)...𝑁) ∈ Fin) | |
54 | peano2nn0 11210 | . . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ0 → (𝐾 + 1) ∈ ℕ0) | |
55 | 41, 54 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐾 + 1) ∈ ℕ0) |
56 | elfzuz 12209 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ ((𝐾 + 1)...𝑁) → 𝑘 ∈ (ℤ≥‘(𝐾 + 1))) | |
57 | eluznn0 11633 | . . . . . . . . . . 11 ⊢ (((𝐾 + 1) ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘(𝐾 + 1))) → 𝑘 ∈ ℕ0) | |
58 | 55, 56, 57 | syl2an 493 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → 𝑘 ∈ ℕ0) |
59 | 39 | ffvelrnda 6267 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
60 | 58, 59 | syldan 486 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐴‘𝑘) ∈ ℂ) |
61 | expcl 12740 | . . . . . . . . . . 11 ⊢ ((𝑇 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑇↑𝑘) ∈ ℂ) | |
62 | 49, 61 | sylan 487 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑇↑𝑘) ∈ ℂ) |
63 | 58, 62 | syldan 486 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝑇↑𝑘) ∈ ℂ) |
64 | 60, 63 | mulcld 9939 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → ((𝐴‘𝑘) · (𝑇↑𝑘)) ∈ ℂ) |
65 | 64 | abscld 14023 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) ∈ ℝ) |
66 | 53, 65 | fsumrecl 14312 | . . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) ∈ ℝ) |
67 | 64 | absge0d 14031 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → 0 ≤ (abs‘((𝐴‘𝑘) · (𝑇↑𝑘)))) |
68 | 53, 65, 67 | fsumge0 14368 | . . . . . 6 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘)))) |
69 | 66, 68 | ge0p1rpd 11778 | . . . . 5 ⊢ (𝜑 → (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1) ∈ ℝ+) |
70 | 52, 69 | rpdivcld 11765 | . . . 4 ⊢ (𝜑 → ((abs‘(𝐹‘0)) / (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1)) ∈ ℝ+) |
71 | 50, 70 | syl5eqel 2692 | . . 3 ⊢ (𝜑 → 𝑈 ∈ ℝ+) |
72 | ftalem4.9 | . . . 4 ⊢ 𝑋 = if(1 ≤ 𝑈, 1, 𝑈) | |
73 | 1rp 11712 | . . . . 5 ⊢ 1 ∈ ℝ+ | |
74 | ifcl 4080 | . . . . 5 ⊢ ((1 ∈ ℝ+ ∧ 𝑈 ∈ ℝ+) → if(1 ≤ 𝑈, 1, 𝑈) ∈ ℝ+) | |
75 | 73, 71, 74 | sylancr 694 | . . . 4 ⊢ (𝜑 → if(1 ≤ 𝑈, 1, 𝑈) ∈ ℝ+) |
76 | 72, 75 | syl5eqel 2692 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ+) |
77 | 49, 71, 76 | 3jca 1235 | . 2 ⊢ (𝜑 → (𝑇 ∈ ℂ ∧ 𝑈 ∈ ℝ+ ∧ 𝑋 ∈ ℝ+)) |
78 | 31, 77 | jca 553 | 1 ⊢ (𝜑 → ((𝐾 ∈ ℕ ∧ (𝐴‘𝐾) ≠ 0) ∧ (𝑇 ∈ ℂ ∧ 𝑈 ∈ ℝ+ ∧ 𝑋 ∈ ℝ+))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {crab 2900 ⊆ wss 3540 ∅c0 3874 ifcif 4036 class class class wbr 4583 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 infcinf 8230 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 -cneg 10146 / cdiv 10563 ℕcn 10897 ℕ0cn0 11169 ℤ≥cuz 11563 ℝ+crp 11708 ...cfz 12197 ↑cexp 12722 abscabs 13822 Σcsu 14264 0𝑝c0p 23242 Polycply 23744 coeffccoe 23746 degcdgr 23747 ↑𝑐ccxp 24106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-0p 23243 df-limc 23436 df-dv 23437 df-ply 23748 df-coe 23750 df-dgr 23751 df-log 24107 df-cxp 24108 |
This theorem is referenced by: ftalem5 24603 |
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