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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 2821 | . . 3 ⊢ (coeff‘𝐹) = (coeff‘𝐹) | |
2 | 1 | dgrval 24818 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) = sup((◡(coeff‘𝐹) “ (ℂ ∖ {0})), ℕ0, < )) |
3 | nn0ssre 11902 | . . . . 5 ⊢ ℕ0 ⊆ ℝ | |
4 | ltso 10721 | . . . . 5 ⊢ < Or ℝ | |
5 | soss 5493 | . . . . 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 11994 | . . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 0 ∈ ℤ) | |
9 | cnvimass 5949 | . . . . 5 ⊢ (◡(coeff‘𝐹) “ (ℂ ∖ {0})) ⊆ dom (coeff‘𝐹) | |
10 | 1 | coef 24820 | . . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0})) |
11 | 9, 10 | fssdm 6530 | . . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → (◡(coeff‘𝐹) “ (ℂ ∖ {0})) ⊆ ℕ0) |
12 | 1 | dgrlem 24819 | . . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → ((coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0}) ∧ ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡(coeff‘𝐹) “ (ℂ ∖ {0}))𝑥 ≤ 𝑛)) |
13 | 12 | simprd 498 | . . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡(coeff‘𝐹) “ (ℂ ∖ {0}))𝑥 ≤ 𝑛) |
14 | nn0uz 12281 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
15 | 14 | uzsupss 12341 | . . . 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 1367 | . . 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 8922 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → sup((◡(coeff‘𝐹) “ (ℂ ∖ {0})), ℕ0, < ) ∈ ℕ0) |
18 | 2, 17 | eqeltrd 2913 | 1 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 ∖ cdif 3933 ∪ cun 3934 ⊆ wss 3936 {csn 4567 class class class wbr 5066 Or wor 5473 ◡ccnv 5554 “ cima 5558 ⟶wf 6351 ‘cfv 6355 supcsup 8904 ℂcc 10535 ℝcr 10536 0cc0 10537 < clt 10675 ≤ cle 10676 ℕ0cn0 11898 ℤcz 11982 Polycply 24774 coeffccoe 24776 degcdgr 24777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-fl 13163 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-rlim 14846 df-sum 15043 df-0p 24271 df-ply 24778 df-coe 24780 df-dgr 24781 |
This theorem is referenced by: dgrub 24824 dgrub2 24825 dgrlb 24826 coeidlem 24827 plyco 24831 dgreq 24834 0dgr 24835 dgrnznn 24837 coefv0 24838 coeaddlem 24839 coemullem 24840 coemulhi 24844 dgreq0 24855 dgrlt 24856 dgradd2 24858 dgrmul 24860 dgrmulc 24861 dgrcolem2 24864 dgrco 24865 plycj 24867 coecj 24868 plymul0or 24870 dvply2g 24874 plydivlem3 24884 plydivlem4 24885 plydivex 24886 plydiveu 24887 plyrem 24894 fta1lem 24896 fta1 24897 quotcan 24898 vieta1lem1 24899 vieta1lem2 24900 elqaalem2 24909 elqaalem3 24910 aareccl 24915 aannenlem1 24917 aannenlem2 24918 aalioulem1 24921 aaliou2 24929 taylply2 24956 signsplypnf 31820 signsply0 31821 dgraa0p 39769 mpaaeu 39770 elaa2lem 42538 etransclem46 42585 etransclem47 42586 etransclem48 42587 |
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