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Mirrors > Home > MPE Home > Th. List > dgrcl | Structured version Visualization version GIF version |
Description: The degree of any polynomial is a nonnegative integer. (Contributed by Mario Carneiro, 22-Jul-2014.) |
Ref | Expression |
---|---|
dgrcl | ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2825 | . . 3 ⊢ (coeff‘𝐹) = (coeff‘𝐹) | |
2 | 1 | dgrval 24390 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) = sup((◡(coeff‘𝐹) “ (ℂ ∖ {0})), ℕ0, < )) |
3 | nn0ssre 11629 | . . . . 5 ⊢ ℕ0 ⊆ ℝ | |
4 | ltso 10444 | . . . . 5 ⊢ < Or ℝ | |
5 | soss 5284 | . . . . 5 ⊢ (ℕ0 ⊆ ℝ → ( < Or ℝ → < Or ℕ0)) | |
6 | 3, 4, 5 | mp2 9 | . . . 4 ⊢ < Or ℕ0 |
7 | 6 | a1i 11 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → < Or ℕ0) |
8 | 0zd 11723 | . . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 0 ∈ ℤ) | |
9 | cnvimass 5730 | . . . . 5 ⊢ (◡(coeff‘𝐹) “ (ℂ ∖ {0})) ⊆ dom (coeff‘𝐹) | |
10 | 1 | coef 24392 | . . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0})) |
11 | 9, 10 | fssdm 6298 | . . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → (◡(coeff‘𝐹) “ (ℂ ∖ {0})) ⊆ ℕ0) |
12 | 1 | dgrlem 24391 | . . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → ((coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0}) ∧ ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡(coeff‘𝐹) “ (ℂ ∖ {0}))𝑥 ≤ 𝑛)) |
13 | 12 | simprd 491 | . . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡(coeff‘𝐹) “ (ℂ ∖ {0}))𝑥 ≤ 𝑛) |
14 | nn0uz 12011 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
15 | 14 | uzsupss 12070 | . . . 4 ⊢ ((0 ∈ ℤ ∧ (◡(coeff‘𝐹) “ (ℂ ∖ {0})) ⊆ ℕ0 ∧ ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡(coeff‘𝐹) “ (ℂ ∖ {0}))𝑥 ≤ 𝑛) → ∃𝑛 ∈ ℕ0 (∀𝑥 ∈ (◡(coeff‘𝐹) “ (ℂ ∖ {0})) ¬ 𝑛 < 𝑥 ∧ ∀𝑥 ∈ ℕ0 (𝑥 < 𝑛 → ∃𝑦 ∈ (◡(coeff‘𝐹) “ (ℂ ∖ {0}))𝑥 < 𝑦))) |
16 | 8, 11, 13, 15 | syl3anc 1494 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℕ0 (∀𝑥 ∈ (◡(coeff‘𝐹) “ (ℂ ∖ {0})) ¬ 𝑛 < 𝑥 ∧ ∀𝑥 ∈ ℕ0 (𝑥 < 𝑛 → ∃𝑦 ∈ (◡(coeff‘𝐹) “ (ℂ ∖ {0}))𝑥 < 𝑦))) |
17 | 7, 16 | supcl 8639 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → sup((◡(coeff‘𝐹) “ (ℂ ∖ {0})), ℕ0, < ) ∈ ℕ0) |
18 | 2, 17 | eqeltrd 2906 | 1 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 ∈ wcel 2164 ∀wral 3117 ∃wrex 3118 ∖ cdif 3795 ∪ cun 3796 ⊆ wss 3798 {csn 4399 class class class wbr 4875 Or wor 5264 ◡ccnv 5345 “ cima 5349 ⟶wf 6123 ‘cfv 6127 supcsup 8621 ℂcc 10257 ℝcr 10258 0cc0 10259 < clt 10398 ≤ cle 10399 ℕ0cn0 11625 ℤcz 11711 Polycply 24346 coeffccoe 24348 degcdgr 24349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 ax-addf 10338 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-pm 8130 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-sup 8623 df-inf 8624 df-oi 8691 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-n0 11626 df-z 11712 df-uz 11976 df-rp 12120 df-fz 12627 df-fzo 12768 df-fl 12895 df-seq 13103 df-exp 13162 df-hash 13418 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-clim 14603 df-rlim 14604 df-sum 14801 df-0p 23843 df-ply 24350 df-coe 24352 df-dgr 24353 |
This theorem is referenced by: dgrub 24396 dgrub2 24397 dgrlb 24398 coeidlem 24399 plyco 24403 dgreq 24406 0dgr 24407 dgrnznn 24409 coefv0 24410 coeaddlem 24411 coemullem 24412 coemulhi 24416 dgreq0 24427 dgrlt 24428 dgradd2 24430 dgrmul 24432 dgrmulc 24433 dgrcolem2 24436 dgrco 24437 plycj 24439 coecj 24440 plymul0or 24442 dvply2g 24446 plydivlem3 24456 plydivlem4 24457 plydivex 24458 plydiveu 24459 plyrem 24466 fta1lem 24468 fta1 24469 quotcan 24470 vieta1lem1 24471 vieta1lem2 24472 elqaalem2 24481 elqaalem3 24482 aareccl 24487 aannenlem1 24489 aannenlem2 24490 aalioulem1 24493 aaliou2 24501 taylply2 24528 signsplypnf 31170 signsply0 31171 dgraa0p 38557 mpaaeu 38558 elaa2lem 41238 etransclem46 41285 etransclem47 41286 etransclem48 41287 |
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