Theorem coe1termlem 26170

Description: The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Hypothesis

Ref	Expression
coe1term.1	⊢ 𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑𝑁)))

Assertion

Ref	Expression
coe1termlem	⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((coeff‘𝐹) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)) ∧ (𝐴 ≠ 0 → (deg‘𝐹) = 𝑁)))

Distinct variable groups:   𝑧,𝑛,𝐴   𝑛,𝑁,𝑧

Allowed substitution hints:   𝐹(𝑧,𝑛)

Proof of Theorem coe1termlem

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3972	. . . 4 ⊢ ℂ ⊆ ℂ
2		coe1term.1	. . . . 5 ⊢ 𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑𝑁)))
3	2	ply1term 26116	. . . 4 ⊢ ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝐹 ∈ (Poly‘ℂ))
4	1, 3	mp3an1 1450	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝐹 ∈ (Poly‘ℂ))
5		simpr 484	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
6		simpl 482	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝐴 ∈ ℂ)
7		0cn 11173	. . . . . 6 ⊢ 0 ∈ ℂ
8		ifcl 4537	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → if(𝑛 = 𝑁, 𝐴, 0) ∈ ℂ)
9	6, 7, 8	sylancl 586	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → if(𝑛 = 𝑁, 𝐴, 0) ∈ ℂ)
10	9	adantr 480	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → if(𝑛 = 𝑁, 𝐴, 0) ∈ ℂ)
11	10	fmpttd 7090	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)):ℕ0⟶ℂ)
12		eqid 2730	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))
13		eqeq1 2734	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝑛 = 𝑁 ↔ 𝑘 = 𝑁))
14	13	ifbid 4515	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → if(𝑛 = 𝑁, 𝐴, 0) = if(𝑘 = 𝑁, 𝐴, 0))
15		simpr 484	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
16		ifcl 4537	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → if(𝑘 = 𝑁, 𝐴, 0) ∈ ℂ)
17	6, 7, 16	sylancl 586	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → if(𝑘 = 𝑁, 𝐴, 0) ∈ ℂ)
18	17	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → if(𝑘 = 𝑁, 𝐴, 0) ∈ ℂ)
19	12, 14, 15, 18	fvmptd3 6994	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) = if(𝑘 = 𝑁, 𝐴, 0))
20	19	neeq1d 2985	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) ≠ 0 ↔ if(𝑘 = 𝑁, 𝐴, 0) ≠ 0))
21		nn0re 12458	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
22	21	leidd 11751	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ≤ 𝑁)
23	22	ad2antlr 727	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝑁 ≤ 𝑁)
24		iffalse 4500	. . . . . . . . 9 ⊢ (¬ 𝑘 = 𝑁 → if(𝑘 = 𝑁, 𝐴, 0) = 0)
25	24	necon1ai 2953	. . . . . . . 8 ⊢ (if(𝑘 = 𝑁, 𝐴, 0) ≠ 0 → 𝑘 = 𝑁)
26	25	breq1d 5120	. . . . . . 7 ⊢ (if(𝑘 = 𝑁, 𝐴, 0) ≠ 0 → (𝑘 ≤ 𝑁 ↔ 𝑁 ≤ 𝑁))
27	23, 26	syl5ibrcom 247	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (if(𝑘 = 𝑁, 𝐴, 0) ≠ 0 → 𝑘 ≤ 𝑁))
28	20, 27	sylbid 240	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
29	28	ralrimiva 3126	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ∀𝑘 ∈ ℕ0 (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
30		plyco0 26104	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)):ℕ0⟶ℂ) → (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)) “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)))
31	5, 11, 30	syl2anc 584	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)) “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)))
32	29, 31	mpbird 257	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)) “ (ℤ≥‘(𝑁 + 1))) = {0})
33	2	ply1termlem 26115	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘))))
34		elfznn0 13588	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0)
35	19	oveq1d 7405	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) · (𝑧↑𝑘)) = (if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘)))
36	34, 35	sylan2 593	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑁)) → (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) · (𝑧↑𝑘)) = (if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘)))
37	36	sumeq2dv 15675	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑁)(((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘)))
38	37	mpteq2dv 5204	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘))))
39	33, 38	eqtr4d 2768	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) · (𝑧↑𝑘))))
40	4, 5, 11, 32, 39	coeeq 26139	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (coeff‘𝐹) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)))
41	4	adantr 480	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ≠ 0) → 𝐹 ∈ (Poly‘ℂ))
42	5	adantr 480	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ≠ 0) → 𝑁 ∈ ℕ0)
43	11	adantr 480	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ≠ 0) → (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)):ℕ0⟶ℂ)
44	32	adantr 480	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ≠ 0) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)) “ (ℤ≥‘(𝑁 + 1))) = {0})
45	39	adantr 480	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ≠ 0) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) · (𝑧↑𝑘))))
46		iftrue 4497	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → if(𝑛 = 𝑁, 𝐴, 0) = 𝐴)
47	46, 12	fvmptg 6969	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℂ) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑁) = 𝐴)
48	47	ancoms 458	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑁) = 𝐴)
49	48	neeq1d 2985	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑁) ≠ 0 ↔ 𝐴 ≠ 0))
50	49	biimpar 477	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ≠ 0) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑁) ≠ 0)
51	41, 42, 43, 44, 45, 50	dgreq 26156	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ≠ 0) → (deg‘𝐹) = 𝑁)
52	51	ex 412	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 ≠ 0 → (deg‘𝐹) = 𝑁))
53	40, 52	jca 511	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((coeff‘𝐹) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)) ∧ (𝐴 ≠ 0 → (deg‘𝐹) = 𝑁)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045   ⊆ wss 3917  ifcif 4491  {csn 4592   class class class wbr 5110   ↦ cmpt 5191   “ cima 5644  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  ℂcc 11073  0cc0 11075  1c1 11076   + caddc 11078   · cmul 11080   ≤ cle 11216  ℕ₀cn0 12449  ℤ_≥cuz 12800  ...cfz 13475  ↑cexp 14033  Σcsu 15659  Polycply 26096  coeffccoe 26098  degcdgr 26099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-map 8804  df-pm 8805  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-inf 9401  df-oi 9470  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-z 12537  df-uz 12801  df-rp 12959  df-fz 13476  df-fzo 13623  df-fl 13761  df-seq 13974  df-exp 14034  df-hash 14303  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-clim 15461  df-rlim 15462  df-sum 15660  df-0p 25578  df-ply 26100  df-coe 26102  df-dgr 26103

This theorem is referenced by:  coe1term  26171  dgr1term  26172