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Theorem dgrlb 24753
Description: If all the coefficients above 𝑀 are zero, then the degree of 𝐹 is at most 𝑀. (Contributed by Mario Carneiro, 22-Jul-2014.)
Hypotheses
Ref Expression
dgrub.1 𝐴 = (coeff‘𝐹)
dgrub.2 𝑁 = (deg‘𝐹)
Assertion
Ref Expression
dgrlb ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝑁𝑀)

Proof of Theorem dgrlb
Dummy variables 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dgrub.2 . . . . 5 𝑁 = (deg‘𝐹)
2 dgrcl 24750 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
31, 2eqeltrid 2914 . . . 4 (𝐹 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
433ad2ant1 1125 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝑁 ∈ ℕ0)
54nn0red 11944 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝑁 ∈ ℝ)
6 simp2 1129 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝑀 ∈ ℕ0)
76nn0red 11944 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝑀 ∈ ℝ)
8 dgrub.1 . . . . . . . . . . . . 13 𝐴 = (coeff‘𝐹)
98dgrlem 24746 . . . . . . . . . . . 12 (𝐹 ∈ (Poly‘𝑆) → (𝐴:ℕ0⟶(𝑆 ∪ {0}) ∧ ∃𝑛 ∈ ℤ ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛))
109simpld 495 . . . . . . . . . . 11 (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
11103ad2ant1 1125 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
12 ffn 6507 . . . . . . . . . 10 (𝐴:ℕ0⟶(𝑆 ∪ {0}) → 𝐴 Fn ℕ0)
13 elpreima 6820 . . . . . . . . . 10 (𝐴 Fn ℕ0 → (𝑦 ∈ (𝐴 “ (ℂ ∖ {0})) ↔ (𝑦 ∈ ℕ0 ∧ (𝐴𝑦) ∈ (ℂ ∖ {0}))))
1411, 12, 133syl 18 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → (𝑦 ∈ (𝐴 “ (ℂ ∖ {0})) ↔ (𝑦 ∈ ℕ0 ∧ (𝐴𝑦) ∈ (ℂ ∖ {0}))))
1514biimpa 477 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → (𝑦 ∈ ℕ0 ∧ (𝐴𝑦) ∈ (ℂ ∖ {0})))
1615simpld 495 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → 𝑦 ∈ ℕ0)
1716nn0red 11944 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → 𝑦 ∈ ℝ)
187adantr 481 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → 𝑀 ∈ ℝ)
19 eldifsni 4714 . . . . . . . 8 ((𝐴𝑦) ∈ (ℂ ∖ {0}) → (𝐴𝑦) ≠ 0)
2015, 19simpl2im 504 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → (𝐴𝑦) ≠ 0)
21 simp3 1130 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})
228coef3 24749 . . . . . . . . . . . 12 (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
23223ad2ant1 1125 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝐴:ℕ0⟶ℂ)
24 plyco0 24709 . . . . . . . . . . 11 ((𝑀 ∈ ℕ0𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ‘(𝑀 + 1))) = {0} ↔ ∀𝑦 ∈ ℕ0 ((𝐴𝑦) ≠ 0 → 𝑦𝑀)))
256, 23, 24syl2anc 584 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → ((𝐴 “ (ℤ‘(𝑀 + 1))) = {0} ↔ ∀𝑦 ∈ ℕ0 ((𝐴𝑦) ≠ 0 → 𝑦𝑀)))
2621, 25mpbid 233 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → ∀𝑦 ∈ ℕ0 ((𝐴𝑦) ≠ 0 → 𝑦𝑀))
2726r19.21bi 3205 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ ℕ0) → ((𝐴𝑦) ≠ 0 → 𝑦𝑀))
2816, 27syldan 591 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → ((𝐴𝑦) ≠ 0 → 𝑦𝑀))
2920, 28mpd 15 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → 𝑦𝑀)
3017, 18, 29lensymd 10779 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → ¬ 𝑀 < 𝑦)
3130ralrimiva 3179 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → ∀𝑦 ∈ (𝐴 “ (ℂ ∖ {0})) ¬ 𝑀 < 𝑦)
32 nn0ssre 11889 . . . . . . 7 0 ⊆ ℝ
33 ltso 10709 . . . . . . 7 < Or ℝ
34 soss 5486 . . . . . . 7 (ℕ0 ⊆ ℝ → ( < Or ℝ → < Or ℕ0))
3532, 33, 34mp2 9 . . . . . 6 < Or ℕ0
3635a1i 11 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → < Or ℕ0)
37 0zd 11981 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → 0 ∈ ℤ)
38 cnvimass 5942 . . . . . . . 8 (𝐴 “ (ℂ ∖ {0})) ⊆ dom 𝐴
3938, 10fssdm 6523 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℂ ∖ {0})) ⊆ ℕ0)
409simprd 496 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛)
41 nn0uz 12268 . . . . . . . 8 0 = (ℤ‘0)
4241uzsupss 12328 . . . . . . 7 ((0 ∈ ℤ ∧ (𝐴 “ (ℂ ∖ {0})) ⊆ ℕ0 ∧ ∃𝑛 ∈ ℤ ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛) → ∃𝑛 ∈ ℕ0 (∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0})) ¬ 𝑛 < 𝑥 ∧ ∀𝑥 ∈ ℕ0 (𝑥 < 𝑛 → ∃𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥 < 𝑦)))
4337, 39, 40, 42syl3anc 1363 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℕ0 (∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0})) ¬ 𝑛 < 𝑥 ∧ ∀𝑥 ∈ ℕ0 (𝑥 < 𝑛 → ∃𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥 < 𝑦)))
44433ad2ant1 1125 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → ∃𝑛 ∈ ℕ0 (∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0})) ¬ 𝑛 < 𝑥 ∧ ∀𝑥 ∈ ℕ0 (𝑥 < 𝑛 → ∃𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥 < 𝑦)))
4536, 44supnub 8914 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → ((𝑀 ∈ ℕ0 ∧ ∀𝑦 ∈ (𝐴 “ (ℂ ∖ {0})) ¬ 𝑀 < 𝑦) → ¬ 𝑀 < sup((𝐴 “ (ℂ ∖ {0})), ℕ0, < )))
466, 31, 45mp2and 695 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → ¬ 𝑀 < sup((𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
478dgrval 24745 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) = sup((𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
481, 47syl5eq 2865 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → 𝑁 = sup((𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
49483ad2ant1 1125 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝑁 = sup((𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
5049breq2d 5069 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → (𝑀 < 𝑁𝑀 < sup((𝐴 “ (ℂ ∖ {0})), ℕ0, < )))
5146, 50mtbird 326 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → ¬ 𝑀 < 𝑁)
525, 7, 51nltled 10778 1 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝑁𝑀)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  cdif 3930  cun 3931  wss 3933  {csn 4557   class class class wbr 5057   Or wor 5466  ccnv 5547  cima 5551   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  supcsup 8892  cc 10523  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   < clt 10663  cle 10664  0cn0 11885  cz 11969  cuz 12231  Polycply 24701  coeffccoe 24703  degcdgr 24704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603  ax-addf 10604
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-fzo 13022  df-fl 13150  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-rlim 14834  df-sum 15031  df-0p 24198  df-ply 24705  df-coe 24707  df-dgr 24708
This theorem is referenced by:  coeidlem  24754  dgrle  24760  dgreq0  24782
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