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Mirrors > Home > MPE Home > Th. List > dgreq | Structured version Visualization version GIF version |
Description: If the highest term in a polynomial expression is nonzero, then the polynomial's degree is completely determined. (Contributed by Mario Carneiro, 24-Jul-2014.) |
Ref | Expression |
---|---|
dgreq.1 | ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) |
dgreq.2 | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
dgreq.3 | ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) |
dgreq.4 | ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) |
dgreq.5 | ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))) |
dgreq.6 | ⊢ (𝜑 → (𝐴‘𝑁) ≠ 0) |
Ref | Expression |
---|---|
dgreq | ⊢ (𝜑 → (deg‘𝐹) = 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dgreq.1 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) | |
2 | dgreq.2 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
3 | dgreq.3 | . . . 4 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) | |
4 | elfznn0 12988 | . . . 4 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0) | |
5 | ffvelrn 6841 | . . . 4 ⊢ ((𝐴:ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) | |
6 | 3, 4, 5 | syl2an 595 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐴‘𝑘) ∈ ℂ) |
7 | dgreq.5 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))) | |
8 | 1, 2, 6, 7 | dgrle 24760 | . 2 ⊢ (𝜑 → (deg‘𝐹) ≤ 𝑁) |
9 | dgreq.4 | . . . . . 6 ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) | |
10 | 1, 2, 3, 9, 7 | coeeq 24744 | . . . . 5 ⊢ (𝜑 → (coeff‘𝐹) = 𝐴) |
11 | 10 | fveq1d 6665 | . . . 4 ⊢ (𝜑 → ((coeff‘𝐹)‘𝑁) = (𝐴‘𝑁)) |
12 | dgreq.6 | . . . 4 ⊢ (𝜑 → (𝐴‘𝑁) ≠ 0) | |
13 | 11, 12 | eqnetrd 3080 | . . 3 ⊢ (𝜑 → ((coeff‘𝐹)‘𝑁) ≠ 0) |
14 | eqid 2818 | . . . 4 ⊢ (coeff‘𝐹) = (coeff‘𝐹) | |
15 | eqid 2818 | . . . 4 ⊢ (deg‘𝐹) = (deg‘𝐹) | |
16 | 14, 15 | dgrub 24751 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑁 ∈ ℕ0 ∧ ((coeff‘𝐹)‘𝑁) ≠ 0) → 𝑁 ≤ (deg‘𝐹)) |
17 | 1, 2, 13, 16 | syl3anc 1363 | . 2 ⊢ (𝜑 → 𝑁 ≤ (deg‘𝐹)) |
18 | dgrcl 24750 | . . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0) | |
19 | 1, 18 | syl 17 | . . . 4 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ0) |
20 | 19 | nn0red 11944 | . . 3 ⊢ (𝜑 → (deg‘𝐹) ∈ ℝ) |
21 | 2 | nn0red 11944 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
22 | 20, 21 | letri3d 10770 | . 2 ⊢ (𝜑 → ((deg‘𝐹) = 𝑁 ↔ ((deg‘𝐹) ≤ 𝑁 ∧ 𝑁 ≤ (deg‘𝐹)))) |
23 | 8, 17, 22 | mpbir2and 709 | 1 ⊢ (𝜑 → (deg‘𝐹) = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 {csn 4557 class class class wbr 5057 ↦ cmpt 5137 “ cima 5551 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 + caddc 10528 · cmul 10530 ≤ cle 10664 ℕ0cn0 11885 ℤ≥cuz 12231 ...cfz 12880 ↑cexp 13417 Σcsu 15030 Polycply 24701 coeffccoe 24703 degcdgr 24704 |
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 |
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-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-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 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-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12881 df-fzo 13022 df-fl 13150 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-rlim 14834 df-sum 15031 df-0p 24198 df-ply 24705 df-coe 24707 df-dgr 24708 |
This theorem is referenced by: coe1termlem 24775 basellem2 25586 |
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