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| Mirrors > Home > MPE Home > Th. List > ftalem4 | Structured version Visualization version GIF version | ||
| Description: Lemma for fta 27123: Closure of the auxiliary variables for ftalem5 27120. (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 4080 | . . . . . 6 ⊢ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ⊆ ℕ | |
| 3 | nnuz 12921 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
| 4 | 2, 3 | sseqtri 4032 | . . . . 5 ⊢ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ⊆ (ℤ≥‘1) |
| 5 | fveq2 6906 | . . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝐴‘𝑛) = (𝐴‘𝑁)) | |
| 6 | 5 | neeq1d 3000 | . . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝐴‘𝑛) ≠ 0 ↔ (𝐴‘𝑁) ≠ 0)) |
| 7 | ftalem.4 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
| 8 | 7 | nnne0d 12316 | . . . . . . . 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 26305 | . . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0𝑝 ↔ (𝐴‘𝑁) = 0)) |
| 13 | 9, 12 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐹 = 0𝑝 ↔ (𝐴‘𝑁) = 0)) |
| 14 | fveq2 6906 | . . . . . . . . . . . 12 ⊢ (𝐹 = 0𝑝 → (deg‘𝐹) = (deg‘0𝑝)) | |
| 15 | dgr0 26302 | . . . . . . . . . . . 12 ⊢ (deg‘0𝑝) = 0 | |
| 16 | 14, 15 | eqtrdi 2793 | . . . . . . . . . . 11 ⊢ (𝐹 = 0𝑝 → (deg‘𝐹) = 0) |
| 17 | 10, 16 | eqtrid 2789 | . . . . . . . . . 10 ⊢ (𝐹 = 0𝑝 → 𝑁 = 0) |
| 18 | 13, 17 | biimtrrdi 254 | . . . . . . . . 9 ⊢ (𝜑 → ((𝐴‘𝑁) = 0 → 𝑁 = 0)) |
| 19 | 18 | necon3d 2961 | . . . . . . . 8 ⊢ (𝜑 → (𝑁 ≠ 0 → (𝐴‘𝑁) ≠ 0)) |
| 20 | 8, 19 | mpd 15 | . . . . . . 7 ⊢ (𝜑 → (𝐴‘𝑁) ≠ 0) |
| 21 | 6, 7, 20 | elrabd 3694 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}) |
| 22 | 21 | ne0d 4342 | . . . . 5 ⊢ (𝜑 → {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0} ≠ ∅) |
| 23 | infssuzcl 12974 | . . . . 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 2845 | . . 3 ⊢ (𝜑 → 𝐾 ∈ {𝑛 ∈ ℕ ∣ (𝐴‘𝑛) ≠ 0}) |
| 26 | fveq2 6906 | . . . . 5 ⊢ (𝑛 = 𝐾 → (𝐴‘𝑛) = (𝐴‘𝐾)) | |
| 27 | 26 | neeq1d 3000 | . . . 4 ⊢ (𝑛 = 𝐾 → ((𝐴‘𝑛) ≠ 0 ↔ (𝐴‘𝐾) ≠ 0)) |
| 28 | 27 | elrab 3692 | . . 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 26237 | . . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ) | |
| 32 | 9, 31 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ) |
| 33 | 0cn 11253 | . . . . . . . 8 ⊢ 0 ∈ ℂ | |
| 34 | ffvelcdm 7101 | . . . . . . . 8 ⊢ ((𝐹:ℂ⟶ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) ∈ ℂ) | |
| 35 | 32, 33, 34 | sylancl 586 | . . . . . . 7 ⊢ (𝜑 → (𝐹‘0) ∈ ℂ) |
| 36 | 11 | coef3 26271 | . . . . . . . . 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 12587 | . . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℕ0) |
| 40 | 37, 39 | ffvelcdmd 7105 | . . . . . . 7 ⊢ (𝜑 → (𝐴‘𝐾) ∈ ℂ) |
| 41 | 29 | simprd 495 | . . . . . . 7 ⊢ (𝜑 → (𝐴‘𝐾) ≠ 0) |
| 42 | 35, 40, 41 | divcld 12043 | . . . . . 6 ⊢ (𝜑 → ((𝐹‘0) / (𝐴‘𝐾)) ∈ ℂ) |
| 43 | 42 | negcld 11607 | . . . . 5 ⊢ (𝜑 → -((𝐹‘0) / (𝐴‘𝐾)) ∈ ℂ) |
| 44 | 38 | nnrecred 12317 | . . . . . 6 ⊢ (𝜑 → (1 / 𝐾) ∈ ℝ) |
| 45 | 44 | recnd 11289 | . . . . 5 ⊢ (𝜑 → (1 / 𝐾) ∈ ℂ) |
| 46 | 43, 45 | cxpcld 26750 | . . . 4 ⊢ (𝜑 → (-((𝐹‘0) / (𝐴‘𝐾))↑𝑐(1 / 𝐾)) ∈ ℂ) |
| 47 | 30, 46 | eqeltrid 2845 | . . 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 15487 | . . . . 5 ⊢ (𝜑 → (abs‘(𝐹‘0)) ∈ ℝ+) |
| 51 | fzfid 14014 | . . . . . . 7 ⊢ (𝜑 → ((𝐾 + 1)...𝑁) ∈ Fin) | |
| 52 | peano2nn0 12566 | . . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ0 → (𝐾 + 1) ∈ ℕ0) | |
| 53 | 39, 52 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐾 + 1) ∈ ℕ0) |
| 54 | elfzuz 13560 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ ((𝐾 + 1)...𝑁) → 𝑘 ∈ (ℤ≥‘(𝐾 + 1))) | |
| 55 | eluznn0 12959 | . . . . . . . . . . 11 ⊢ (((𝐾 + 1) ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘(𝐾 + 1))) → 𝑘 ∈ ℕ0) | |
| 56 | 53, 54, 55 | syl2an 596 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → 𝑘 ∈ ℕ0) |
| 57 | 37 | ffvelcdmda 7104 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
| 58 | 56, 57 | syldan 591 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐴‘𝑘) ∈ ℂ) |
| 59 | expcl 14120 | . . . . . . . . . 10 ⊢ ((𝑇 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑇↑𝑘) ∈ ℂ) | |
| 60 | 47, 56, 59 | syl2an2r 685 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝑇↑𝑘) ∈ ℂ) |
| 61 | 58, 60 | mulcld 11281 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → ((𝐴‘𝑘) · (𝑇↑𝑘)) ∈ ℂ) |
| 62 | 61 | abscld 15475 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) ∈ ℝ) |
| 63 | 51, 62 | fsumrecl 15770 | . . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) ∈ ℝ) |
| 64 | 61 | absge0d 15483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → 0 ≤ (abs‘((𝐴‘𝑘) · (𝑇↑𝑘)))) |
| 65 | 51, 62, 64 | fsumge0 15831 | . . . . . 6 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘)))) |
| 66 | 63, 65 | ge0p1rpd 13107 | . . . . 5 ⊢ (𝜑 → (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1) ∈ ℝ+) |
| 67 | 50, 66 | rpdivcld 13094 | . . . 4 ⊢ (𝜑 → ((abs‘(𝐹‘0)) / (Σ𝑘 ∈ ((𝐾 + 1)...𝑁)(abs‘((𝐴‘𝑘) · (𝑇↑𝑘))) + 1)) ∈ ℝ+) |
| 68 | 48, 67 | eqeltrid 2845 | . . 3 ⊢ (𝜑 → 𝑈 ∈ ℝ+) |
| 69 | ftalem4.9 | . . . 4 ⊢ 𝑋 = if(1 ≤ 𝑈, 1, 𝑈) | |
| 70 | 1rp 13038 | . . . . 5 ⊢ 1 ∈ ℝ+ | |
| 71 | ifcl 4571 | . . . . 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 2845 | . . 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 1540 ∈ wcel 2108 ≠ wne 2940 {crab 3436 ⊆ wss 3951 ∅c0 4333 ifcif 4525 class class class wbr 5143 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 infcinf 9481 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 ≤ cle 11296 -cneg 11493 / cdiv 11920 ℕcn 12266 ℕ0cn0 12526 ℤ≥cuz 12878 ℝ+crp 13034 ...cfz 13547 ↑cexp 14102 abscabs 15273 Σcsu 15722 0𝑝c0p 25704 Polycply 26223 coeffccoe 26225 degcdgr 26226 ↑𝑐ccxp 26597 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-0p 25705 df-limc 25901 df-dv 25902 df-ply 26227 df-coe 26229 df-dgr 26230 df-log 26598 df-cxp 26599 |
| This theorem is referenced by: ftalem5 27120 |
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