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Theorem dgrlb 23796
 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 1055 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝑀 ∈ ℕ0)
2 dgrub.1 . . . . . . . . . . . . . 14 𝐴 = (coeff‘𝐹)
32dgrlem 23789 . . . . . . . . . . . . 13 (𝐹 ∈ (Poly‘𝑆) → (𝐴:ℕ0⟶(𝑆 ∪ {0}) ∧ ∃𝑛 ∈ ℤ ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛))
43simpld 474 . . . . . . . . . . . 12 (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
543ad2ant1 1075 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
6 ffn 5958 . . . . . . . . . . 11 (𝐴:ℕ0⟶(𝑆 ∪ {0}) → 𝐴 Fn ℕ0)
7 elpreima 6245 . . . . . . . . . . 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 500 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → (𝑦 ∈ ℕ0 ∧ (𝐴𝑦) ∈ (ℂ ∖ {0})))
109simprd 478 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → (𝐴𝑦) ∈ (ℂ ∖ {0}))
11 eldifsni 4261 . . . . . . . 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 474 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → 𝑦 ∈ ℕ0)
14 simp3 1056 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})
152coef3 23792 . . . . . . . . . . . 12 (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
16153ad2ant1 1075 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝐴:ℕ0⟶ℂ)
17 plyco0 23752 . . . . . . . . . . 11 ((𝑀 ∈ ℕ0𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ‘(𝑀 + 1))) = {0} ↔ ∀𝑦 ∈ ℕ0 ((𝐴𝑦) ≠ 0 → 𝑦𝑀)))
181, 16, 17syl2anc 691 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → ((𝐴 “ (ℤ‘(𝑀 + 1))) = {0} ↔ ∀𝑦 ∈ ℕ0 ((𝐴𝑦) ≠ 0 → 𝑦𝑀)))
1914, 18mpbid 221 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → ∀𝑦 ∈ ℕ0 ((𝐴𝑦) ≠ 0 → 𝑦𝑀))
2019r19.21bi 2916 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ ℕ0) → ((𝐴𝑦) ≠ 0 → 𝑦𝑀))
2113, 20syldan 486 . . . . . . 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 11229 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → 𝑦 ∈ ℝ)
241nn0red 11229 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝑀 ∈ ℝ)
2524adantr 480 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → 𝑀 ∈ ℝ)
2623, 25lenltd 10062 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → (𝑦𝑀 ↔ ¬ 𝑀 < 𝑦))
2722, 26mpbid 221 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))) → ¬ 𝑀 < 𝑦)
2827ralrimiva 2949 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → ∀𝑦 ∈ (𝐴 “ (ℂ ∖ {0})) ¬ 𝑀 < 𝑦)
29 nn0ssre 11173 . . . . . . 7 0 ⊆ ℝ
30 ltso 9997 . . . . . . 7 < Or ℝ
31 soss 4977 . . . . . . 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 11266 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → 0 ∈ ℤ)
35 cnvimass 5404 . . . . . . . 8 (𝐴 “ (ℂ ∖ {0})) ⊆ dom 𝐴
36 fdm 5964 . . . . . . . . 9 (𝐴:ℕ0⟶(𝑆 ∪ {0}) → dom 𝐴 = ℕ0)
374, 36syl 17 . . . . . . . 8 (𝐹 ∈ (Poly‘𝑆) → dom 𝐴 = ℕ0)
3835, 37syl5sseq 3616 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℂ ∖ {0})) ⊆ ℕ0)
393simprd 478 . . . . . . 7 (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛)
40 nn0uz 11598 . . . . . . . 8 0 = (ℤ‘0)
4140uzsupss 11656 . . . . . . 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 1318 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℕ0 (∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0})) ¬ 𝑛 < 𝑥 ∧ ∀𝑥 ∈ ℕ0 (𝑥 < 𝑛 → ∃𝑦 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥 < 𝑦)))
43423ad2ant1 1075 . . . . 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 8251 . . . 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 711 . . 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 23788 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) = sup((𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
4846, 47syl5eq 2656 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → 𝑁 = sup((𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
49483ad2ant1 1075 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝑁 = sup((𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
5049breq2d 4595 . . 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 23793 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
5346, 52syl5eqel 2692 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
54533ad2ant1 1075 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝑁 ∈ ℕ0)
5554nn0red 11229 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝑁 ∈ ℝ)
5655, 24lenltd 10062 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → (𝑁𝑀 ↔ ¬ 𝑀 < 𝑁))
5751, 56mpbird 246 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  {csn 4125   class class class wbr 4583   Or wor 4958  ◡ccnv 5037  dom cdm 5038   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  supcsup 8229  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  Polycply 23744  coeffccoe 23746  degcdgr 23747 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-0p 23243  df-ply 23748  df-coe 23750  df-dgr 23751 This theorem is referenced by:  coeidlem  23797  dgrle  23803  dgreq0  23825
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