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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 2798 | . . 3 ⊢ (coeff‘𝐹) = (coeff‘𝐹) | |
2 | 1 | dgrval 24825 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) = sup((◡(coeff‘𝐹) “ (ℂ ∖ {0})), ℕ0, < )) |
3 | nn0ssre 11889 | . . . . 5 ⊢ ℕ0 ⊆ ℝ | |
4 | ltso 10710 | . . . . 5 ⊢ < Or ℝ | |
5 | soss 5457 | . . . . 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 11981 | . . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 0 ∈ ℤ) | |
9 | cnvimass 5916 | . . . . 5 ⊢ (◡(coeff‘𝐹) “ (ℂ ∖ {0})) ⊆ dom (coeff‘𝐹) | |
10 | 1 | coef 24827 | . . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0})) |
11 | 9, 10 | fssdm 6504 | . . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → (◡(coeff‘𝐹) “ (ℂ ∖ {0})) ⊆ ℕ0) |
12 | 1 | dgrlem 24826 | . . . . 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 12268 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
15 | 14 | uzsupss 12328 | . . . 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 1368 | . . 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 8906 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → sup((◡(coeff‘𝐹) “ (ℂ ∖ {0})), ℕ0, < ) ∈ ℕ0) |
18 | 2, 17 | eqeltrd 2890 | 1 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 ∖ cdif 3878 ∪ cun 3879 ⊆ wss 3881 {csn 4525 class class class wbr 5030 Or wor 5437 ◡ccnv 5518 “ cima 5522 ⟶wf 6320 ‘cfv 6324 supcsup 8888 ℂcc 10524 ℝcr 10525 0cc0 10526 < clt 10664 ≤ cle 10665 ℕ0cn0 11885 ℤcz 11969 Polycply 24781 coeffccoe 24783 degcdgr 24784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-sum 15035 df-0p 24274 df-ply 24785 df-coe 24787 df-dgr 24788 |
This theorem is referenced by: dgrub 24831 dgrub2 24832 dgrlb 24833 coeidlem 24834 plyco 24838 dgreq 24841 0dgr 24842 dgrnznn 24844 coefv0 24845 coeaddlem 24846 coemullem 24847 coemulhi 24851 dgreq0 24862 dgrlt 24863 dgradd2 24865 dgrmul 24867 dgrmulc 24868 dgrcolem2 24871 dgrco 24872 plycj 24874 coecj 24875 plymul0or 24877 dvply2g 24881 plydivlem3 24891 plydivlem4 24892 plydivex 24893 plydiveu 24894 plyrem 24901 fta1lem 24903 fta1 24904 quotcan 24905 vieta1lem1 24906 vieta1lem2 24907 elqaalem2 24916 elqaalem3 24917 aareccl 24922 aannenlem1 24924 aannenlem2 24925 aalioulem1 24928 aaliou2 24936 taylply2 24963 signsplypnf 31930 signsply0 31931 dgraa0p 40093 mpaaeu 40094 elaa2lem 42875 etransclem46 42922 etransclem47 42923 etransclem48 42924 |
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