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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 2738 | . . 3 ⊢ (coeff‘𝐹) = (coeff‘𝐹) | |
2 | 1 | dgrval 25389 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) = sup((◡(coeff‘𝐹) “ (ℂ ∖ {0})), ℕ0, < )) |
3 | nn0ssre 12237 | . . . . 5 ⊢ ℕ0 ⊆ ℝ | |
4 | ltso 11055 | . . . . 5 ⊢ < Or ℝ | |
5 | soss 5523 | . . . . 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 12331 | . . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 0 ∈ ℤ) | |
9 | cnvimass 5989 | . . . . 5 ⊢ (◡(coeff‘𝐹) “ (ℂ ∖ {0})) ⊆ dom (coeff‘𝐹) | |
10 | 1 | coef 25391 | . . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0})) |
11 | 9, 10 | fssdm 6620 | . . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → (◡(coeff‘𝐹) “ (ℂ ∖ {0})) ⊆ ℕ0) |
12 | 1 | dgrlem 25390 | . . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → ((coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0}) ∧ ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡(coeff‘𝐹) “ (ℂ ∖ {0}))𝑥 ≤ 𝑛)) |
13 | 12 | simprd 496 | . . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡(coeff‘𝐹) “ (ℂ ∖ {0}))𝑥 ≤ 𝑛) |
14 | nn0uz 12620 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
15 | 14 | uzsupss 12680 | . . . 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 1370 | . . 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 9217 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → sup((◡(coeff‘𝐹) “ (ℂ ∖ {0})), ℕ0, < ) ∈ ℕ0) |
18 | 2, 17 | eqeltrd 2839 | 1 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 ∖ cdif 3884 ∪ cun 3885 ⊆ wss 3887 {csn 4561 class class class wbr 5074 Or wor 5502 ◡ccnv 5588 “ cima 5592 ⟶wf 6429 ‘cfv 6433 supcsup 9199 ℂcc 10869 ℝcr 10870 0cc0 10871 < clt 11009 ≤ cle 11010 ℕ0cn0 12233 ℤcz 12319 Polycply 25345 coeffccoe 25347 degcdgr 25348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-rlim 15198 df-sum 15398 df-0p 24834 df-ply 25349 df-coe 25351 df-dgr 25352 |
This theorem is referenced by: dgrub 25395 dgrub2 25396 dgrlb 25397 coeidlem 25398 plyco 25402 dgreq 25405 0dgr 25406 dgrnznn 25408 coefv0 25409 coeaddlem 25410 coemullem 25411 coemulhi 25415 dgreq0 25426 dgrlt 25427 dgradd2 25429 dgrmul 25431 dgrmulc 25432 dgrcolem2 25435 dgrco 25436 plycj 25438 coecj 25439 plymul0or 25441 dvply2g 25445 plydivlem3 25455 plydivlem4 25456 plydivex 25457 plydiveu 25458 plyrem 25465 fta1lem 25467 fta1 25468 quotcan 25469 vieta1lem1 25470 vieta1lem2 25471 elqaalem2 25480 elqaalem3 25481 aareccl 25486 aannenlem1 25488 aannenlem2 25489 aalioulem1 25492 aaliou2 25500 taylply2 25527 signsplypnf 32529 signsply0 32530 dgraa0p 40974 mpaaeu 40975 elaa2lem 43774 etransclem46 43821 etransclem47 43822 etransclem48 43823 |
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