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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 2739 | . . 3 ⊢ (coeff‘𝐹) = (coeff‘𝐹) | |
2 | 1 | dgrval 24980 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) = sup((◡(coeff‘𝐹) “ (ℂ ∖ {0})), ℕ0, < )) |
3 | nn0ssre 11983 | . . . . 5 ⊢ ℕ0 ⊆ ℝ | |
4 | ltso 10802 | . . . . 5 ⊢ < Or ℝ | |
5 | soss 5463 | . . . . 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 12077 | . . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 0 ∈ ℤ) | |
9 | cnvimass 5924 | . . . . 5 ⊢ (◡(coeff‘𝐹) “ (ℂ ∖ {0})) ⊆ dom (coeff‘𝐹) | |
10 | 1 | coef 24982 | . . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0})) |
11 | 9, 10 | fssdm 6525 | . . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → (◡(coeff‘𝐹) “ (ℂ ∖ {0})) ⊆ ℕ0) |
12 | 1 | dgrlem 24981 | . . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → ((coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0}) ∧ ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡(coeff‘𝐹) “ (ℂ ∖ {0}))𝑥 ≤ 𝑛)) |
13 | 12 | simprd 499 | . . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡(coeff‘𝐹) “ (ℂ ∖ {0}))𝑥 ≤ 𝑛) |
14 | nn0uz 12365 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
15 | 14 | uzsupss 12425 | . . . 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 1372 | . . 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 8998 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → sup((◡(coeff‘𝐹) “ (ℂ ∖ {0})), ℕ0, < ) ∈ ℕ0) |
18 | 2, 17 | eqeltrd 2834 | 1 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∈ wcel 2114 ∀wral 3054 ∃wrex 3055 ∖ cdif 3841 ∪ cun 3842 ⊆ wss 3844 {csn 4517 class class class wbr 5031 Or wor 5442 ◡ccnv 5525 “ cima 5529 ⟶wf 6336 ‘cfv 6340 supcsup 8980 ℂcc 10616 ℝcr 10617 0cc0 10618 < clt 10756 ≤ cle 10757 ℕ0cn0 11979 ℤcz 12065 Polycply 24936 coeffccoe 24938 degcdgr 24939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-inf2 9180 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 ax-pre-sup 10696 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-se 5485 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-of 7428 df-om 7603 df-1st 7717 df-2nd 7718 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-er 8323 df-map 8442 df-pm 8443 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-sup 8982 df-inf 8983 df-oi 9050 df-card 9444 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-div 11379 df-nn 11720 df-2 11782 df-3 11783 df-n0 11980 df-z 12066 df-uz 12328 df-rp 12476 df-fz 12985 df-fzo 13128 df-fl 13256 df-seq 13464 df-exp 13525 df-hash 13786 df-cj 14551 df-re 14552 df-im 14553 df-sqrt 14687 df-abs 14688 df-clim 14938 df-rlim 14939 df-sum 15139 df-0p 24425 df-ply 24940 df-coe 24942 df-dgr 24943 |
This theorem is referenced by: dgrub 24986 dgrub2 24987 dgrlb 24988 coeidlem 24989 plyco 24993 dgreq 24996 0dgr 24997 dgrnznn 24999 coefv0 25000 coeaddlem 25001 coemullem 25002 coemulhi 25006 dgreq0 25017 dgrlt 25018 dgradd2 25020 dgrmul 25022 dgrmulc 25023 dgrcolem2 25026 dgrco 25027 plycj 25029 coecj 25030 plymul0or 25032 dvply2g 25036 plydivlem3 25046 plydivlem4 25047 plydivex 25048 plydiveu 25049 plyrem 25056 fta1lem 25058 fta1 25059 quotcan 25060 vieta1lem1 25061 vieta1lem2 25062 elqaalem2 25071 elqaalem3 25072 aareccl 25077 aannenlem1 25079 aannenlem2 25080 aalioulem1 25083 aaliou2 25091 taylply2 25118 signsplypnf 32102 signsply0 32103 dgraa0p 40569 mpaaeu 40570 elaa2lem 43339 etransclem46 43386 etransclem47 43387 etransclem48 43388 |
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