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Theorem dgrlb 24837
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 24834 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
31, 2eqeltrid 2897 . . . 4 (𝐹 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
433ad2ant1 1130 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝑁 ∈ ℕ0)
54nn0red 11948 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝑁 ∈ ℝ)
6 simp2 1134 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝑀 ∈ ℕ0)
76nn0red 11948 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝑀 ∈ ℝ)
8 dgrub.1 . . . . . . . . . . . . 13 𝐴 = (coeff‘𝐹)
98dgrlem 24830 . . . . . . . . . . . 12 (𝐹 ∈ (Poly‘𝑆) → (𝐴:ℕ0⟶(𝑆 ∪ {0}) ∧ ∃𝑛 ∈ ℤ ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛))
109simpld 498 . . . . . . . . . . 11 (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
11103ad2ant1 1130 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
12 ffn 6491 . . . . . . . . . 10 (𝐴:ℕ0⟶(𝑆 ∪ {0}) → 𝐴 Fn ℕ0)
13 elpreima 6809 . . . . . . . . . 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 480 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → (𝑦 ∈ ℕ0 ∧ (𝐴𝑦) ∈ (ℂ ∖ {0})))
1615simpld 498 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → 𝑦 ∈ ℕ0)
1716nn0red 11948 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → 𝑦 ∈ ℝ)
187adantr 484 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → 𝑀 ∈ ℝ)
19 eldifsni 4686 . . . . . . . 8 ((𝐴𝑦) ∈ (ℂ ∖ {0}) → (𝐴𝑦) ≠ 0)
2015, 19simpl2im 507 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → (𝐴𝑦) ≠ 0)
21 simp3 1135 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})
228coef3 24833 . . . . . . . . . . . 12 (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
23223ad2ant1 1130 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝐴:ℕ0⟶ℂ)
24 plyco0 24793 . . . . . . . . . . 11 ((𝑀 ∈ ℕ0𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ‘(𝑀 + 1))) = {0} ↔ ∀𝑦 ∈ ℕ0 ((𝐴𝑦) ≠ 0 → 𝑦𝑀)))
256, 23, 24syl2anc 587 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → ((𝐴 “ (ℤ‘(𝑀 + 1))) = {0} ↔ ∀𝑦 ∈ ℕ0 ((𝐴𝑦) ≠ 0 → 𝑦𝑀)))
2621, 25mpbid 235 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → ∀𝑦 ∈ ℕ0 ((𝐴𝑦) ≠ 0 → 𝑦𝑀))
2726r19.21bi 3176 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ ℕ0) → ((𝐴𝑦) ≠ 0 → 𝑦𝑀))
2816, 27syldan 594 . . . . . . 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 10784 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → ¬ 𝑀 < 𝑦)
3130ralrimiva 3152 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → ∀𝑦 ∈ (𝐴 “ (ℂ ∖ {0})) ¬ 𝑀 < 𝑦)
32 nn0ssre 11893 . . . . . . 7 0 ⊆ ℝ
33 ltso 10714 . . . . . . 7 < Or ℝ
34 soss 5461 . . . . . . 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 11985 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → 0 ∈ ℤ)
38 cnvimass 5920 . . . . . . . 8 (𝐴 “ (ℂ ∖ {0})) ⊆ dom 𝐴
3938, 10fssdm 6508 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℂ ∖ {0})) ⊆ ℕ0)
409simprd 499 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛)
41 nn0uz 12272 . . . . . . . 8 0 = (ℤ‘0)
4241uzsupss 12332 . . . . . . 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 1368 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℕ0 (∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0})) ¬ 𝑛 < 𝑥 ∧ ∀𝑥 ∈ ℕ0 (𝑥 < 𝑛 → ∃𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥 < 𝑦)))
44433ad2ant1 1130 . . . . 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 698 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → ¬ 𝑀 < sup((𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
478dgrval 24829 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) = sup((𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
481, 47syl5eq 2848 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → 𝑁 = sup((𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
49483ad2ant1 1130 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝑁 = sup((𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
5049breq2d 5045 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → (𝑀 < 𝑁𝑀 < sup((𝐴 “ (ℂ ∖ {0})), ℕ0, < )))
5146, 50mtbird 328 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → ¬ 𝑀 < 𝑁)
525, 7, 51nltled 10783 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wne 2990  wral 3109  wrex 3110  cdif 3881  cun 3882  wss 3884  {csn 4528   class class class wbr 5033   Or wor 5441  ccnv 5522  cima 5526   Fn wfn 6323  wf 6324  cfv 6328  (class class class)co 7139  supcsup 8892  cc 10528  cr 10529  0cc0 10530  1c1 10531   + caddc 10533   < clt 10668  cle 10669  0cn0 11889  cz 11973  cuz 12235  Polycply 24785  coeffccoe 24787  degcdgr 24788
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608  ax-addf 10609
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12382  df-fz 12890  df-fzo 13033  df-fl 13161  df-seq 13369  df-exp 13430  df-hash 13691  df-cj 14454  df-re 14455  df-im 14456  df-sqrt 14590  df-abs 14591  df-clim 14841  df-rlim 14842  df-sum 15039  df-0p 24278  df-ply 24789  df-coe 24791  df-dgr 24792
This theorem is referenced by:  coeidlem  24838  dgrle  24844  dgreq0  24866
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