Theorem dgrlb 26213

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 26210	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
3	1, 2	eqeltrid 2841	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
4	3	3ad2ant1 1134	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑁 ∈ ℕ0)
5	4	nn0red 12488	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑁 ∈ ℝ)
6		simp2 1138	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑀 ∈ ℕ0)
7	6	nn0red 12488	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑀 ∈ ℝ)
8		dgrub.1	. . . . . . . . . . . . 13 ⊢ 𝐴 = (coeff‘𝐹)
9	8	dgrlem 26206	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴:ℕ0⟶(𝑆 ∪ {0}) ∧ ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛))
10	9	simpld 494	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
11	10	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
12		ffn 6660	. . . . . . . . . 10 ⊢ (𝐴:ℕ0⟶(𝑆 ∪ {0}) → 𝐴 Fn ℕ0)
13		elpreima 7002	. . . . . . . . . 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 476	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → (𝑦 ∈ ℕ0 ∧ (𝐴‘𝑦) ∈ (ℂ ∖ {0})))
16	15	simpld 494	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → 𝑦 ∈ ℕ0)
17	16	nn0red 12488	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → 𝑦 ∈ ℝ)
18	7	adantr 480	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → 𝑀 ∈ ℝ)
19		eldifsni 4734	. . . . . . . 8 ⊢ ((𝐴‘𝑦) ∈ (ℂ ∖ {0}) → (𝐴‘𝑦) ≠ 0)
20	15, 19	simpl2im 503	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → (𝐴‘𝑦) ≠ 0)
21		simp3 1139	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})
22	8	coef3 26209	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
23	22	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝐴:ℕ0⟶ℂ)
24		plyco0 26169	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0} ↔ ∀𝑦 ∈ ℕ0 ((𝐴‘𝑦) ≠ 0 → 𝑦 ≤ 𝑀)))
25	6, 23, 24	syl2anc 585	. . . . . . . . . 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 232	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → ∀𝑦 ∈ ℕ0 ((𝐴‘𝑦) ≠ 0 → 𝑦 ≤ 𝑀))
27	26	r19.21bi 3230	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ ℕ0) → ((𝐴‘𝑦) ≠ 0 → 𝑦 ≤ 𝑀))
28	16, 27	syldan 592	. . . . . . 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 11286	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → ¬ 𝑀 < 𝑦)
31	30	ralrimiva 3130	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → ∀𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0})) ¬ 𝑀 < 𝑦)
32		nn0ssre 12430	. . . . . . 7 ⊢ ℕ0 ⊆ ℝ
33		ltso 11215	. . . . . . 7 ⊢ < Or ℝ
34		soss 5550	. . . . . . 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 12525	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 0 ∈ ℤ)
38		cnvimass 6039	. . . . . . . 8 ⊢ (◡𝐴 “ (ℂ ∖ {0})) ⊆ dom 𝐴
39	38, 10	fssdm 6679	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → (◡𝐴 “ (ℂ ∖ {0})) ⊆ ℕ0)
40	9	simprd 495	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛)
41		nn0uz 12815	. . . . . . . 8 ⊢ ℕ0 = (ℤ≥‘0)
42	41	uzsupss 12879	. . . . . . 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 1374	. . . . . 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 1134	. . . . 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 9366	. . . 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 700	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → ¬ 𝑀 < sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
47	8	dgrval 26205	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
48	1, 47	eqtrid 2784	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑁 = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
49	48	3ad2ant1 1134	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑁 = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
50	49	breq2d 5098	. . 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 11285	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑁 ≤ 𝑀)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062   ∖ cdif 3887   ∪ cun 3888   ⊆ wss 3890  {csn 4568   class class class wbr 5086   Or wor 5529  ^◡ccnv 5621   “ cima 5625   Fn wfn 6485  ⟶wf 6486  ‘cfv 6490  (class class class)co 7358  supcsup 9344  ℂcc 11025  ℝcr 11026  0cc0 11027  1c1 11028   + caddc 11030   < clt 11168   ≤ cle 11169  ℕ₀cn0 12426  ℤcz 12513  ℤ_≥cuz 12777  Polycply 26161  coeffccoe 26163  degcdgr 26164

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680  ax-inf2 9551  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104  ax-pre-sup 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-er 8634  df-map 8766  df-pm 8767  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-sup 9346  df-inf 9347  df-oi 9416  df-card 9852  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-div 11797  df-nn 12164  df-2 12233  df-3 12234  df-n0 12427  df-z 12514  df-uz 12778  df-rp 12932  df-fz 13451  df-fzo 13598  df-fl 13740  df-seq 13953  df-exp 14013  df-hash 14282  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15439  df-rlim 15440  df-sum 15638  df-0p 25646  df-ply 26165  df-coe 26167  df-dgr 26168

This theorem is referenced by:  coeidlem  26214  dgrle  26220  dgreq0  26242