Theorem dgrlb 26372

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 26369	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
3	1, 2	eqeltrid 2865	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
4	3	3ad2ant1 1149	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑁 ∈ ℕ0)
5	4	nn0red 12565	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑁 ∈ ℝ)
6		simp2 1153	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑀 ∈ ℕ0)
7	6	nn0red 12565	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑀 ∈ ℝ)
8		dgrub.1	. . . . . . . . . . . . 13 ⊢ 𝐴 = (coeff‘𝐹)
9	8	dgrlem 26365	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴:ℕ0⟶(𝑆 ∪ {0}) ∧ ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛))
10	9	simpld 499	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
11	10	3ad2ant1 1149	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
12		ffn 6705	. . . . . . . . . 10 ⊢ (𝐴:ℕ0⟶(𝑆 ∪ {0}) → 𝐴 Fn ℕ0)
13		elpreima 7053	. . . . . . . . . 10 ⊢ (𝐴 Fn ℕ0 → (𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0})) ↔ (𝑦 ∈ ℕ0 ∧ (𝐴‘𝑦) ∈ (ℂ ∖ {0}))))
14	11, 12, 13	3syl 19	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → (𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0})) ↔ (𝑦 ∈ ℕ0 ∧ (𝐴‘𝑦) ∈ (ℂ ∖ {0}))))
15	14	biimpa 481	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → (𝑦 ∈ ℕ0 ∧ (𝐴‘𝑦) ∈ (ℂ ∖ {0})))
16	15	simpld 499	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → 𝑦 ∈ ℕ0)
17	16	nn0red 12565	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → 𝑦 ∈ ℝ)
18	7	adantr 485	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → 𝑀 ∈ ℝ)
19		eldifsni 4757	. . . . . . . 8 ⊢ ((𝐴‘𝑦) ∈ (ℂ ∖ {0}) → (𝐴‘𝑦) ≠ 0)
20	15, 19	simpl2im 512	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → (𝐴‘𝑦) ≠ 0)
21		simp3 1154	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})
22	8	coef3 26368	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
23	22	3ad2ant1 1149	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝐴:ℕ0⟶ℂ)
24		plyco0 26328	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0} ↔ ∀𝑦 ∈ ℕ0 ((𝐴‘𝑦) ≠ 0 → 𝑦 ≤ 𝑀)))
25	6, 23, 24	syl2anc 595	. . . . . . . . . 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 235	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → ∀𝑦 ∈ ℕ0 ((𝐴‘𝑦) ≠ 0 → 𝑦 ≤ 𝑀))
27	26	r19.21bi 3255	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ ℕ0) → ((𝐴‘𝑦) ≠ 0 → 𝑦 ≤ 𝑀))
28	16, 27	syldan 602	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → ((𝐴‘𝑦) ≠ 0 → 𝑦 ≤ 𝑀))
29	20, 28	mpd 16	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → 𝑦 ≤ 𝑀)
30	17, 18, 29	lensymd 11360	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) ∧ 𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0}))) → ¬ 𝑀 < 𝑦)
31	30	ralrimiva 3155	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → ∀𝑦 ∈ (◡𝐴 “ (ℂ ∖ {0})) ¬ 𝑀 < 𝑦)
32		nn0ssre 12507	. . . . . . 7 ⊢ ℕ0 ⊆ ℝ
33		ltso 11289	. . . . . . 7 ⊢ < Or ℝ
34		soss 5589	. . . . . . 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 12602	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 0 ∈ ℤ)
38		cnvimass 6084	. . . . . . . 8 ⊢ (◡𝐴 “ (ℂ ∖ {0})) ⊆ dom 𝐴
39	38, 10	fssdm 6725	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → (◡𝐴 “ (ℂ ∖ {0})) ⊆ ℕ0)
40	9	simprd 500	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛)
41		nn0uz 12899	. . . . . . . 8 ⊢ ℕ0 = (ℤ≥‘0)
42	41	uzsupss 12963	. . . . . . 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 1396	. . . . . 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 1149	. . . . 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 9421	. . . 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 711	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → ¬ 𝑀 < sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
47	8	dgrval 26364	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
48	1, 47	eqtrid 2808	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑁 = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
49	48	3ad2ant1 1149	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑁 = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
50	49	breq2d 5120	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → (𝑀 < 𝑁 ↔ 𝑀 < sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < )))
51	46, 50	mtbird 328	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → ¬ 𝑀 < 𝑁)
52	5, 7, 51	nltled 11359	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑁 ≤ 𝑀)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141   ≠ wne 2956  ∀wral 3077  ∃wrex 3087   ∖ cdif 3901   ∪ cun 3902   ⊆ wss 3904  {csn 4588   class class class wbr 5108   Or wor 5568  ^◡ccnv 5660   “ cima 5664   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  supcsup 9399  ℂcc 11097  ℝcr 11098  0cc0 11099  1c1 11100   + caddc 11102   < clt 11242   ≤ cle 11243  ℕ₀cn0 12503  ℤcz 12590  ℤ_≥cuz 12861  Polycply 26320  coeffccoe 26322  degcdgr 26323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9609  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-pre-sup 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-map 8825  df-pm 8826  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-sup 9401  df-inf 9402  df-oi 9471  df-card 9924  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-nn 12233  df-2 12302  df-3 12303  df-n0 12504  df-z 12591  df-uz 12862  df-rp 13016  df-fz 13535  df-fzo 13683  df-fl 13825  df-seq 14038  df-exp 14098  df-hash 14367  df-cj 15150  df-re 15151  df-im 15152  df-sqrt 15286  df-abs 15287  df-clim 15539  df-rlim 15540  df-sum 15738  df-0p 25808  df-ply 26324  df-coe 26326  df-dgr 26327

This theorem is referenced by:  coeidlem  26373  dgrle  26379  dgreq0  26401