Theorem dgrlb 25397

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}) → 𝑁 ≤ 𝑀)

Proof of Theorem dgrlb

Dummy variables 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dgrub.2	. . . . 5 ⊢ 𝑁 = (deg‘𝐹)
2		dgrcl 25394	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
3	1, 2	eqeltrid 2843	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
4	3	3ad2ant1 1132	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑁 ∈ ℕ0)
5	4	nn0red 12294	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑁 ∈ ℝ)
6		simp2 1136	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑀 ∈ ℕ0)
7	6	nn0red 12294	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑀 ∈ ℝ)
8		dgrub.1	. . . . . . . . . . . . 13 ⊢ 𝐴 = (coeff‘𝐹)
9	8	dgrlem 25390	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴:ℕ0⟶(𝑆 ∪ {0}) ∧ ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛))
10	9	simpld 495	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
11	10	3ad2ant1 1132	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
12		ffn 6600	. . . . . . . . . 10 ⊢ (𝐴:ℕ0⟶(𝑆 ∪ {0}) → 𝐴 Fn ℕ0)
13		elpreima 6935	. . . . . . . . . 10 ⊢ (𝐴 Fn ℕ0 → (𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0})) ↔ (𝑦 ∈ ℕ0 ∧ (𝐴‘𝑦) ∈ (ℂ ∖ {0}))))
14	11, 12, 13	3syl 18	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → (𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0})) ↔ (𝑦 ∈ ℕ0 ∧ (𝐴‘𝑦) ∈ (ℂ ∖ {0}))))
15	14	biimpa 477	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → (𝑦 ∈ ℕ0 ∧ (𝐴‘𝑦) ∈ (ℂ ∖ {0})))
16	15	simpld 495	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → 𝑦 ∈ ℕ0)
17	16	nn0red 12294	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → 𝑦 ∈ ℝ)
18	7	adantr 481	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → 𝑀 ∈ ℝ)
19		eldifsni 4723	. . . . . . . 8 ⊢ ((𝐴‘𝑦) ∈ (ℂ ∖ {0}) → (𝐴‘𝑦) ≠ 0)
20	15, 19	simpl2im 504	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → (𝐴‘𝑦) ≠ 0)
21		simp3 1137	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})
22	8	coef3 25393	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
23	22	3ad2ant1 1132	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝐴:ℕ0⟶ℂ)
24		plyco0 25353	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0} ↔ ∀𝑦 ∈ ℕ0 ((𝐴‘𝑦) ≠ 0 → 𝑦 ≤ 𝑀)))
25	6, 23, 24	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → ((𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0} ↔ ∀𝑦 ∈ ℕ0 ((𝐴‘𝑦) ≠ 0 → 𝑦 ≤ 𝑀)))
26	21, 25	mpbid 231	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → ∀𝑦 ∈ ℕ0 ((𝐴‘𝑦) ≠ 0 → 𝑦 ≤ 𝑀))
27	26	r19.21bi 3134	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ ℕ0) → ((𝐴‘𝑦) ≠ 0 → 𝑦 ≤ 𝑀))
28	16, 27	syldan 591	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → ((𝐴‘𝑦) ≠ 0 → 𝑦 ≤ 𝑀))
29	20, 28	mpd 15	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → 𝑦 ≤ 𝑀)
30	17, 18, 29	lensymd 11126	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → ¬ 𝑀 < 𝑦)
31	30	ralrimiva 3103	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → ∀𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0})) ¬ 𝑀 < 𝑦)
32		nn0ssre 12237	. . . . . . 7 ⊢ ℕ0 ⊆ ℝ
33		ltso 11055	. . . . . . 7 ⊢ < Or ℝ
34		soss 5523	. . . . . . 7 ⊢ (ℕ0 ⊆ ℝ → ( < Or ℝ → < Or ℕ0))
35	32, 33, 34	mp2 9	. . . . . 6 ⊢ < Or ℕ0
36	35	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → < Or ℕ0)
37		0zd 12331	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 0 ∈ ℤ)
38		cnvimass 5989	. . . . . . . 8 ⊢ (◡𝐴 “ (ℂ ∖ {0})) ⊆ dom 𝐴
39	38, 10	fssdm 6620	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → (◡𝐴 “ (ℂ ∖ {0})) ⊆ ℕ0)
40	9	simprd 496	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛)
41		nn0uz 12620	. . . . . . . 8 ⊢ ℕ0 = (ℤ≥‘0)
42	41	uzsupss 12680	. . . . . . 7 ⊢ ((0 ∈ ℤ ∧ (◡𝐴 “ (ℂ ∖ {0})) ⊆ ℕ0 ∧ ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛) → ∃𝑛 ∈ ℕ0 (∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0})) ¬ 𝑛 < 𝑥 ∧ ∀𝑥 ∈ ℕ0 (𝑥 < 𝑛 → ∃𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 < 𝑦)))
43	37, 39, 40, 42	syl3anc 1370	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℕ0 (∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0})) ¬ 𝑛 < 𝑥 ∧ ∀𝑥 ∈ ℕ0 (𝑥 < 𝑛 → ∃𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 < 𝑦)))
44	43	3ad2ant1 1132	. . . . 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)))
45	36, 44	supnub 9221	. . . 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, < )))
46	6, 31, 45	mp2and 696	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → ¬ 𝑀 < sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
47	8	dgrval 25389	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
48	1, 47	eqtrid 2790	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑁 = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
49	48	3ad2ant1 1132	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑁 = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
50	49	breq2d 5086	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → (𝑀 < 𝑁 ↔ 𝑀 < sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < )))
51	46, 50	mtbird 325	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → ¬ 𝑀 < 𝑁)
52	5, 7, 51	nltled 11125	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑁 ≤ 𝑀)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065   ∖ cdif 3884   ∪ cun 3885   ⊆ wss 3887  {csn 4561   class class class wbr 5074   Or wor 5502  ^◡ccnv 5588   “ cima 5592   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  supcsup 9199  ℂcc 10869  ℝcr 10870  0cc0 10871  1c1 10872   + caddc 10874   < clt 11009   ≤ cle 11010  ℕ₀cn0 12233  ℤcz 12319  ℤ_≥cuz 12582  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:  coeidlem  25398  dgrle  25404  dgreq0  25426