Theorem dgrle 23803

Description: Given an explicit expression for a polynomial, the degree is at most the highest term in the sum. (Contributed by Mario Carneiro, 24-Jul-2014.)

Hypotheses

Ref	Expression
dgrle.1	⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
dgrle.2	⊢ (𝜑 → 𝑁 ∈ ℕ0)
dgrle.3	⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ∈ ℂ)
dgrle.4	⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(𝐴 · (𝑧↑𝑘))))

Assertion

Ref	Expression
dgrle	⊢ (𝜑 → (deg‘𝐹) ≤ 𝑁)

Distinct variable groups:   𝑧,𝐴   𝑧,𝑘,𝑁   𝜑,𝑘,𝑧

Allowed substitution hints:   𝐴(𝑘)   𝑆(𝑧,𝑘)   𝐹(𝑧,𝑘)

Proof of Theorem dgrle

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dgrle.1	. 2 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
2		dgrle.2	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
3		dgrle.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ∈ ℂ)
4		dgrle.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(𝐴 · (𝑧↑𝑘))))
5	1, 2, 3, 4	coeeq2 23802	. . . . . . . . 9 ⊢ (𝜑 → (coeff‘𝐹) = (𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ 𝑁, 𝐴, 0)))
6	5	ad2antrr 758	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ ¬ 𝑚 ≤ 𝑁) → (coeff‘𝐹) = (𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ 𝑁, 𝐴, 0)))
7	6	fveq1d 6105	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ ¬ 𝑚 ≤ 𝑁) → ((coeff‘𝐹)‘𝑚) = ((𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ 𝑁, 𝐴, 0))‘𝑚))
8		nfcv 2751	. . . . . . . . . 10 ⊢ Ⅎ𝑘𝑚
9		nfv 1830	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 ¬ 𝑚 ≤ 𝑁
10		nffvmpt1 6111	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘((𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ 𝑁, 𝐴, 0))‘𝑚)
11	10	nfeq1 2764	. . . . . . . . . . 11 ⊢ Ⅎ𝑘((𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ 𝑁, 𝐴, 0))‘𝑚) = 0
12	9, 11	nfim 1813	. . . . . . . . . 10 ⊢ Ⅎ𝑘(¬ 𝑚 ≤ 𝑁 → ((𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ 𝑁, 𝐴, 0))‘𝑚) = 0)
13		breq1 4586	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → (𝑘 ≤ 𝑁 ↔ 𝑚 ≤ 𝑁))
14	13	notbid 307	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (¬ 𝑘 ≤ 𝑁 ↔ ¬ 𝑚 ≤ 𝑁))
15		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → ((𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ 𝑁, 𝐴, 0))‘𝑘) = ((𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ 𝑁, 𝐴, 0))‘𝑚))
16	15	eqeq1d 2612	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (((𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ 𝑁, 𝐴, 0))‘𝑘) = 0 ↔ ((𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ 𝑁, 𝐴, 0))‘𝑚) = 0))
17	14, 16	imbi12d 333	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → ((¬ 𝑘 ≤ 𝑁 → ((𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ 𝑁, 𝐴, 0))‘𝑘) = 0) ↔ (¬ 𝑚 ≤ 𝑁 → ((𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ 𝑁, 𝐴, 0))‘𝑚) = 0)))
18		iffalse 4045	. . . . . . . . . . . . 13 ⊢ (¬ 𝑘 ≤ 𝑁 → if(𝑘 ≤ 𝑁, 𝐴, 0) = 0)
19	18	fveq2d 6107	. . . . . . . . . . . 12 ⊢ (¬ 𝑘 ≤ 𝑁 → ( I ‘if(𝑘 ≤ 𝑁, 𝐴, 0)) = ( I ‘0))
20		0cn 9911	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℂ
21		fvi 6165	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℂ → ( I ‘0) = 0)
22	20, 21	ax-mp 5	. . . . . . . . . . . 12 ⊢ ( I ‘0) = 0
23	19, 22	syl6eq 2660	. . . . . . . . . . 11 ⊢ (¬ 𝑘 ≤ 𝑁 → ( I ‘if(𝑘 ≤ 𝑁, 𝐴, 0)) = 0)
24		eqid 2610	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ 𝑁, 𝐴, 0)) = (𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ 𝑁, 𝐴, 0))
25	24	fvmpt2i 6199	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ 𝑁, 𝐴, 0))‘𝑘) = ( I ‘if(𝑘 ≤ 𝑁, 𝐴, 0)))
26	25	eqeq1d 2612	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → (((𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ 𝑁, 𝐴, 0))‘𝑘) = 0 ↔ ( I ‘if(𝑘 ≤ 𝑁, 𝐴, 0)) = 0))
27	23, 26	syl5ibr 235	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → (¬ 𝑘 ≤ 𝑁 → ((𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ 𝑁, 𝐴, 0))‘𝑘) = 0))
28	8, 12, 17, 27	vtoclgaf 3244	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ0 → (¬ 𝑚 ≤ 𝑁 → ((𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ 𝑁, 𝐴, 0))‘𝑚) = 0))
29	28	imp 444	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ0 ∧ ¬ 𝑚 ≤ 𝑁) → ((𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ 𝑁, 𝐴, 0))‘𝑚) = 0)
30	29	adantll 746	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ ¬ 𝑚 ≤ 𝑁) → ((𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ 𝑁, 𝐴, 0))‘𝑚) = 0)
31	7, 30	eqtrd 2644	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ ¬ 𝑚 ≤ 𝑁) → ((coeff‘𝐹)‘𝑚) = 0)
32	31	ex 449	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (¬ 𝑚 ≤ 𝑁 → ((coeff‘𝐹)‘𝑚) = 0))
33	32	necon1ad 2799	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (((coeff‘𝐹)‘𝑚) ≠ 0 → 𝑚 ≤ 𝑁))
34	33	ralrimiva 2949	. . 3 ⊢ (𝜑 → ∀𝑚 ∈ ℕ0 (((coeff‘𝐹)‘𝑚) ≠ 0 → 𝑚 ≤ 𝑁))
35		eqid 2610	. . . . . 6 ⊢ (coeff‘𝐹) = (coeff‘𝐹)
36	35	coef3 23792	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
37	1, 36	syl 17	. . . 4 ⊢ (𝜑 → (coeff‘𝐹):ℕ0⟶ℂ)
38		plyco0 23752	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (coeff‘𝐹):ℕ0⟶ℂ) → (((coeff‘𝐹) “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑚 ∈ ℕ0 (((coeff‘𝐹)‘𝑚) ≠ 0 → 𝑚 ≤ 𝑁)))
39	2, 37, 38	syl2anc 691	. . 3 ⊢ (𝜑 → (((coeff‘𝐹) “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑚 ∈ ℕ0 (((coeff‘𝐹)‘𝑚) ≠ 0 → 𝑚 ≤ 𝑁)))
40	34, 39	mpbird 246	. 2 ⊢ (𝜑 → ((coeff‘𝐹) “ (ℤ≥‘(𝑁 + 1))) = {0})
41		eqid 2610	. . 3 ⊢ (deg‘𝐹) = (deg‘𝐹)
42	35, 41	dgrlb 23796	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑁 ∈ ℕ0 ∧ ((coeff‘𝐹) “ (ℤ≥‘(𝑁 + 1))) = {0}) → (deg‘𝐹) ≤ 𝑁)
43	1, 2, 40, 42	syl3anc 1318	1 ⊢ (𝜑 → (deg‘𝐹) ≤ 𝑁)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ifcif 4036  {csn 4125   class class class wbr 4583   ↦ cmpt 4643   I cid 4948   “ cima 5041  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   ≤ cle 9954  ℕ₀cn0 11169  ℤ_≥cuz 11563  ...cfz 12197  ↑cexp 12722  Σcsu 14264  Polycply 23744  coeffccoe 23746  degcdgr 23747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-0p 23243  df-ply 23748  df-coe 23750  df-dgr 23751

This theorem is referenced by:  dgreq  23804  0dgr  23805  coeaddlem  23809  coemullem  23810  taylply2  23926