Theorem dgrlb 24826

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 24823	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
3	1, 2	eqeltrid 2917	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
4	3	3ad2ant1 1129	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑁 ∈ ℕ0)
5	4	nn0red 11957	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑁 ∈ ℝ)
6		simp2 1133	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑀 ∈ ℕ0)
7	6	nn0red 11957	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑀 ∈ ℝ)
8		dgrub.1	. . . . . . . . . . . . 13 ⊢ 𝐴 = (coeff‘𝐹)
9	8	dgrlem 24819	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴:ℕ0⟶(𝑆 ∪ {0}) ∧ ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛))
10	9	simpld 497	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
11	10	3ad2ant1 1129	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
12		ffn 6514	. . . . . . . . . 10 ⊢ (𝐴:ℕ0⟶(𝑆 ∪ {0}) → 𝐴 Fn ℕ0)
13		elpreima 6828	. . . . . . . . . 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 479	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → (𝑦 ∈ ℕ0 ∧ (𝐴‘𝑦) ∈ (ℂ ∖ {0})))
16	15	simpld 497	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → 𝑦 ∈ ℕ0)
17	16	nn0red 11957	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → 𝑦 ∈ ℝ)
18	7	adantr 483	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → 𝑀 ∈ ℝ)
19		eldifsni 4722	. . . . . . . 8 ⊢ ((𝐴‘𝑦) ∈ (ℂ ∖ {0}) → (𝐴‘𝑦) ≠ 0)
20	15, 19	simpl2im 506	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → (𝐴‘𝑦) ≠ 0)
21		simp3 1134	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})
22	8	coef3 24822	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
23	22	3ad2ant1 1129	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝐴:ℕ0⟶ℂ)
24		plyco0 24782	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0} ↔ ∀𝑦 ∈ ℕ0 ((𝐴‘𝑦) ≠ 0 → 𝑦 ≤ 𝑀)))
25	6, 23, 24	syl2anc 586	. . . . . . . . . 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 234	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → ∀𝑦 ∈ ℕ0 ((𝐴‘𝑦) ≠ 0 → 𝑦 ≤ 𝑀))
27	26	r19.21bi 3208	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ ℕ0) → ((𝐴‘𝑦) ≠ 0 → 𝑦 ≤ 𝑀))
28	16, 27	syldan 593	. . . . . . 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 10791	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → ¬ 𝑀 < 𝑦)
31	30	ralrimiva 3182	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → ∀𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0})) ¬ 𝑀 < 𝑦)
32		nn0ssre 11902	. . . . . . 7 ⊢ ℕ0 ⊆ ℝ
33		ltso 10721	. . . . . . 7 ⊢ < Or ℝ
34		soss 5493	. . . . . . 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 11994	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 0 ∈ ℤ)
38		cnvimass 5949	. . . . . . . 8 ⊢ (◡𝐴 “ (ℂ ∖ {0})) ⊆ dom 𝐴
39	38, 10	fssdm 6530	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → (◡𝐴 “ (ℂ ∖ {0})) ⊆ ℕ0)
40	9	simprd 498	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛)
41		nn0uz 12281	. . . . . . . 8 ⊢ ℕ0 = (ℤ≥‘0)
42	41	uzsupss 12341	. . . . . . 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 1367	. . . . . 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 1129	. . . . 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 8926	. . . 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 697	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → ¬ 𝑀 < sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
47	8	dgrval 24818	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
48	1, 47	syl5eq 2868	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑁 = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
49	48	3ad2ant1 1129	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑁 = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
50	49	breq2d 5078	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → (𝑀 < 𝑁 ↔ 𝑀 < sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < )))
51	46, 50	mtbird 327	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → ¬ 𝑀 < 𝑁)
52	5, 7, 51	nltled 10790	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑁 ≤ 𝑀)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  ∃wrex 3139   ∖ cdif 3933   ∪ cun 3934   ⊆ wss 3936  {csn 4567   class class class wbr 5066   Or wor 5473  ^◡ccnv 5554   “ cima 5558   Fn wfn 6350  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156  supcsup 8904  ℂcc 10535  ℝcr 10536  0cc0 10537  1c1 10538   + caddc 10540   < clt 10675   ≤ cle 10676  ℕ₀cn0 11898  ℤcz 11982  ℤ_≥cuz 12244  Polycply 24774  coeffccoe 24776  degcdgr 24777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615  ax-addf 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7409  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-map 8408  df-pm 8409  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-sup 8906  df-inf 8907  df-oi 8974  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-nn 11639  df-2 11701  df-3 11702  df-n0 11899  df-z 11983  df-uz 12245  df-rp 12391  df-fz 12894  df-fzo 13035  df-fl 13163  df-seq 13371  df-exp 13431  df-hash 13692  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-clim 14845  df-rlim 14846  df-sum 15043  df-0p 24271  df-ply 24778  df-coe 24780  df-dgr 24781

This theorem is referenced by:  coeidlem  24827  dgrle  24833  dgreq0  24855