Theorem coe1termlem 26312

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 4018	. . . 4 ⊢ ℂ ⊆ ℂ
2		coe1term.1	. . . . 5 ⊢ 𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑𝑁)))
3	2	ply1term 26258	. . . 4 ⊢ ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝐹 ∈ (Poly‘ℂ))
4	1, 3	mp3an1 1447	. . 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 11251	. . . . . 6 ⊢ 0 ∈ ℂ
8		ifcl 4576	. . . . . 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 7135	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)):ℕ0⟶ℂ)
12		eqid 2735	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))
13		eqeq1 2739	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝑛 = 𝑁 ↔ 𝑘 = 𝑁))
14	13	ifbid 4554	. . . . . . . 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 4576	. . . . . . . . . 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 7039	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) = if(𝑘 = 𝑁, 𝐴, 0))
20	19	neeq1d 2998	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) ≠ 0 ↔ if(𝑘 = 𝑁, 𝐴, 0) ≠ 0))
21		nn0re 12533	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
22	21	leidd 11827	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ≤ 𝑁)
23	22	ad2antlr 727	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝑁 ≤ 𝑁)
24		iffalse 4540	. . . . . . . . 9 ⊢ (¬ 𝑘 = 𝑁 → if(𝑘 = 𝑁, 𝐴, 0) = 0)
25	24	necon1ai 2966	. . . . . . . 8 ⊢ (if(𝑘 = 𝑁, 𝐴, 0) ≠ 0 → 𝑘 = 𝑁)
26	25	breq1d 5158	. . . . . . 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 3144	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ∀𝑘 ∈ ℕ0 (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
30		plyco0 26246	. . . . 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 26257	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘))))
34		elfznn0 13657	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0)
35	19	oveq1d 7446	. . . . . . 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 15735	. . . . 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 5250	. . . 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 2778	. . 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 26281	. 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 4537	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → if(𝑛 = 𝑁, 𝐴, 0) = 𝐴)
47	46, 12	fvmptg 7014	. . . . . . 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 2998	. . . . 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 26298	. . 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 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059   ⊆ wss 3963  ifcif 4531  {csn 4631   class class class wbr 5148   ↦ cmpt 5231   “ cima 5692  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  ℂcc 11151  0cc0 11153  1c1 11154   + caddc 11156   · cmul 11158   ≤ cle 11294  ℕ₀cn0 12524  ℤ_≥cuz 12876  ...cfz 13544  ↑cexp 14099  Σcsu 15719  Polycply 26238  coeffccoe 26240  degcdgr 26241

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-fl 13829  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-rlim 15522  df-sum 15720  df-0p 25719  df-ply 26242  df-coe 26244  df-dgr 26245

This theorem is referenced by:  coe1term  26313  dgr1term  26314