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Theorem dgrlb 24183
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 simp2 1131 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝑀 ∈ ℕ0)
2 dgrub.1 . . . . . . . . . . . . . 14 𝐴 = (coeff‘𝐹)
32dgrlem 24176 . . . . . . . . . . . . 13 (𝐹 ∈ (Poly‘𝑆) → (𝐴:ℕ0⟶(𝑆 ∪ {0}) ∧ ∃𝑛 ∈ ℤ ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛))
43simpld 477 . . . . . . . . . . . 12 (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
543ad2ant1 1127 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
6 ffn 6198 . . . . . . . . . . 11 (𝐴:ℕ0⟶(𝑆 ∪ {0}) → 𝐴 Fn ℕ0)
7 elpreima 6492 . . . . . . . . . . 11 (𝐴 Fn ℕ0 → (𝑦 ∈ (𝐴 “ (ℂ ∖ {0})) ↔ (𝑦 ∈ ℕ0 ∧ (𝐴𝑦) ∈ (ℂ ∖ {0}))))
85, 6, 73syl 18 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → (𝑦 ∈ (𝐴 “ (ℂ ∖ {0})) ↔ (𝑦 ∈ ℕ0 ∧ (𝐴𝑦) ∈ (ℂ ∖ {0}))))
98biimpa 502 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → (𝑦 ∈ ℕ0 ∧ (𝐴𝑦) ∈ (ℂ ∖ {0})))
109simprd 482 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → (𝐴𝑦) ∈ (ℂ ∖ {0}))
11 eldifsni 4458 . . . . . . . 8 ((𝐴𝑦) ∈ (ℂ ∖ {0}) → (𝐴𝑦) ≠ 0)
1210, 11syl 17 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → (𝐴𝑦) ≠ 0)
139simpld 477 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → 𝑦 ∈ ℕ0)
14 simp3 1132 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})
152coef3 24179 . . . . . . . . . . . 12 (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
16153ad2ant1 1127 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝐴:ℕ0⟶ℂ)
17 plyco0 24139 . . . . . . . . . . 11 ((𝑀 ∈ ℕ0𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ‘(𝑀 + 1))) = {0} ↔ ∀𝑦 ∈ ℕ0 ((𝐴𝑦) ≠ 0 → 𝑦𝑀)))
181, 16, 17syl2anc 696 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → ((𝐴 “ (ℤ‘(𝑀 + 1))) = {0} ↔ ∀𝑦 ∈ ℕ0 ((𝐴𝑦) ≠ 0 → 𝑦𝑀)))
1914, 18mpbid 222 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → ∀𝑦 ∈ ℕ0 ((𝐴𝑦) ≠ 0 → 𝑦𝑀))
2019r19.21bi 3062 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ ℕ0) → ((𝐴𝑦) ≠ 0 → 𝑦𝑀))
2113, 20syldan 488 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → ((𝐴𝑦) ≠ 0 → 𝑦𝑀))
2212, 21mpd 15 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → 𝑦𝑀)
2313nn0red 11536 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → 𝑦 ∈ ℝ)
241nn0red 11536 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝑀 ∈ ℝ)
2524adantr 472 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → 𝑀 ∈ ℝ)
2623, 25lenltd 10367 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → (𝑦𝑀 ↔ ¬ 𝑀 < 𝑦))
2722, 26mpbid 222 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → ¬ 𝑀 < 𝑦)
2827ralrimiva 3096 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → ∀𝑦 ∈ (𝐴 “ (ℂ ∖ {0})) ¬ 𝑀 < 𝑦)
29 nn0ssre 11480 . . . . . . 7 0 ⊆ ℝ
30 ltso 10302 . . . . . . 7 < Or ℝ
31 soss 5197 . . . . . . 7 (ℕ0 ⊆ ℝ → ( < Or ℝ → < Or ℕ0))
3229, 30, 31mp2 9 . . . . . 6 < Or ℕ0
3332a1i 11 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → < Or ℕ0)
34 0zd 11573 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → 0 ∈ ℤ)
35 cnvimass 5635 . . . . . . . 8 (𝐴 “ (ℂ ∖ {0})) ⊆ dom 𝐴
36 fdm 6204 . . . . . . . . 9 (𝐴:ℕ0⟶(𝑆 ∪ {0}) → dom 𝐴 = ℕ0)
374, 36syl 17 . . . . . . . 8 (𝐹 ∈ (Poly‘𝑆) → dom 𝐴 = ℕ0)
3835, 37syl5sseq 3786 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℂ ∖ {0})) ⊆ ℕ0)
393simprd 482 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛)
40 nn0uz 11907 . . . . . . . 8 0 = (ℤ‘0)
4140uzsupss 11965 . . . . . . 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)))
4234, 38, 39, 41syl3anc 1473 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℕ0 (∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0})) ¬ 𝑛 < 𝑥 ∧ ∀𝑥 ∈ ℕ0 (𝑥 < 𝑛 → ∃𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥 < 𝑦)))
43423ad2ant1 1127 . . . . 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)))
4433, 43supnub 8525 . . . 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, < )))
451, 28, 44mp2and 717 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → ¬ 𝑀 < sup((𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
46 dgrub.2 . . . . . 6 𝑁 = (deg‘𝐹)
472dgrval 24175 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) = sup((𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
4846, 47syl5eq 2798 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → 𝑁 = sup((𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
49483ad2ant1 1127 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝑁 = sup((𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
5049breq2d 4808 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → (𝑀 < 𝑁𝑀 < sup((𝐴 “ (ℂ ∖ {0})), ℕ0, < )))
5145, 50mtbird 314 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → ¬ 𝑀 < 𝑁)
52 dgrcl 24180 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
5346, 52syl5eqel 2835 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
54533ad2ant1 1127 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝑁 ∈ ℕ0)
5554nn0red 11536 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝑁 ∈ ℝ)
5655, 24lenltd 10367 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → (𝑁𝑀 ↔ ¬ 𝑀 < 𝑁))
5751, 56mpbird 247 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 196  wa 383  w3a 1072   = wceq 1624  wcel 2131  wne 2924  wral 3042  wrex 3043  cdif 3704  cun 3705  wss 3707  {csn 4313   class class class wbr 4796   Or wor 5178  ccnv 5257  dom cdm 5258  cima 5261   Fn wfn 6036  wf 6037  cfv 6041  (class class class)co 6805  supcsup 8503  cc 10118  cr 10119  0cc0 10120  1c1 10121   + caddc 10123   < clt 10258  cle 10259  0cn0 11476  cz 11561  cuz 11871  Polycply 24131  coeffccoe 24133  degcdgr 24134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-inf2 8703  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197  ax-pre-sup 10198  ax-addf 10199
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-fal 1630  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-se 5218  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-isom 6050  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-of 7054  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-oadd 7725  df-er 7903  df-map 8017  df-pm 8018  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-sup 8505  df-inf 8506  df-oi 8572  df-card 8947  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-div 10869  df-nn 11205  df-2 11263  df-3 11264  df-n0 11477  df-z 11562  df-uz 11872  df-rp 12018  df-fz 12512  df-fzo 12652  df-fl 12779  df-seq 12988  df-exp 13047  df-hash 13304  df-cj 14030  df-re 14031  df-im 14032  df-sqrt 14166  df-abs 14167  df-clim 14410  df-rlim 14411  df-sum 14608  df-0p 23628  df-ply 24135  df-coe 24137  df-dgr 24138
This theorem is referenced by:  coeidlem  24184  dgrle  24190  dgreq0  24212
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