Theorem dgrlb 26209

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 26206	. . . . 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 12475	. 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 12475	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑀 ∈ ℝ)
8		dgrub.1	. . . . . . . . . . . . 13 ⊢ 𝐴 = (coeff‘𝐹)
9	8	dgrlem 26202	. . . . . . . . . . . 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 6670	. . . . . . . . . 10 ⊢ (𝐴:ℕ0⟶(𝑆 ∪ {0}) → 𝐴 Fn ℕ0)
13		elpreima 7012	. . . . . . . . . 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 12475	. . . . . 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 4748	. . . . . . . 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 26205	. . . . . . . . . . . 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 26165	. . . . . . . . . . 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 11296	. . . . 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 12417	. . . . . . 7 ⊢ ℕ0 ⊆ ℝ
33		ltso 11225	. . . . . . 7 ⊢ < Or ℝ
34		soss 5560	. . . . . . 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 12512	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 0 ∈ ℤ)
38		cnvimass 6049	. . . . . . . 8 ⊢ (◡𝐴 “ (ℂ ∖ {0})) ⊆ dom 𝐴
39	38, 10	fssdm 6689	. . . . . . 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 12801	. . . . . . . 8 ⊢ ℕ0 = (ℤ≥‘0)
42	41	uzsupss 12865	. . . . . . 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 9377	. . . 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 26201	. . . . . 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 5112	. . 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 11295	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 3900   ∪ cun 3901   ⊆ wss 3903  {csn 4582   class class class wbr 5100   Or wor 5539  ^◡ccnv 5631   “ cima 5635   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  supcsup 9355  ℂcc 11036  ℝcr 11037  0cc0 11038  1c1 11039   + caddc 11041   < clt 11178   ≤ cle 11179  ℕ₀cn0 12413  ℤcz 12500  ℤ_≥cuz 12763  Polycply 26157  coeffccoe 26159  degcdgr 26160

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 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

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 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-of 7632  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-er 8645  df-map 8777  df-pm 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9357  df-inf 9358  df-oi 9427  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-3 12221  df-n0 12414  df-z 12501  df-uz 12764  df-rp 12918  df-fz 13436  df-fzo 13583  df-fl 13724  df-seq 13937  df-exp 13997  df-hash 14266  df-cj 15034  df-re 15035  df-im 15036  df-sqrt 15170  df-abs 15171  df-clim 15423  df-rlim 15424  df-sum 15622  df-0p 25639  df-ply 26161  df-coe 26163  df-dgr 26164

This theorem is referenced by:  coeidlem  26210  dgrle  26216  dgreq0  26239