Theorem coe1termlem 26557

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 3953	. . . 4 ⊢ ℂ ⊆ ℂ
2		coe1term.1	. . . . 5 ⊢ 𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑𝑁)))
3	2	ply1term 26502	. . . 4 ⊢ ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝐹 ∈ (Poly‘ℂ))
4	1, 3	mp3an1 1477	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝐹 ∈ (Poly‘ℂ))
5		simpr 490	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
6		simpl 488	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝐴 ∈ ℂ)
7		0cn 11279	. . . . . 6 ⊢ 0 ∈ ℂ
8		ifcl 4528	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → if(𝑛 = 𝑁, 𝐴, 0) ∈ ℂ)
9	6, 7, 8	sylancl 598	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → if(𝑛 = 𝑁, 𝐴, 0) ∈ ℂ)
10	9	adantr 486	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → if(𝑛 = 𝑁, 𝐴, 0) ∈ ℂ)
11	10	fmpttd 7107	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)):ℕ0⟶ℂ)
12		eqid 2761	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))
13		eqeq1 2765	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝑛 = 𝑁 ↔ 𝑘 = 𝑁))
14	13	ifbid 4506	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → if(𝑛 = 𝑁, 𝐴, 0) = if(𝑘 = 𝑁, 𝐴, 0))
15		simpr 490	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
16		ifcl 4528	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → if(𝑘 = 𝑁, 𝐴, 0) ∈ ℂ)
17	6, 7, 16	sylancl 598	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → if(𝑘 = 𝑁, 𝐴, 0) ∈ ℂ)
18	17	adantr 486	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → if(𝑘 = 𝑁, 𝐴, 0) ∈ ℂ)
19	12, 14, 15, 18	fvmptd3 7009	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) = if(𝑘 = 𝑁, 𝐴, 0))
20	19	neeq1d 3015	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) ≠ 0 ↔ if(𝑘 = 𝑁, 𝐴, 0) ≠ 0))
21		nn0re 12596	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
22	21	leidd 11863	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ≤ 𝑁)
23	22	ad2antlr 740	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → 𝑁 ≤ 𝑁)
24		iffalse 4491	. . . . . . . . 9 ⊢ (¬ 𝑘 = 𝑁 → if(𝑘 = 𝑁, 𝐴, 0) = 0)
25	24	necon1ai 2983	. . . . . . . 8 ⊢ (if(𝑘 = 𝑁, 𝐴, 0) ≠ 0 → 𝑘 = 𝑁)
26	25	breq1d 5113	. . . . . . 7 ⊢ (if(𝑘 = 𝑁, 𝐴, 0) ≠ 0 → (𝑘 ≤ 𝑁 ↔ 𝑁 ≤ 𝑁))
27	23, 26	syl5ibrcom 250	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (if(𝑘 = 𝑁, 𝐴, 0) ≠ 0 → 𝑘 ≤ 𝑁))
28	20, 27	sylbid 243	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0) → (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
29	28	ralrimiva 3155	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ∀𝑘 ∈ ℕ0 (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
30		plyco0 26490	. . . . 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 596	. . . 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 260	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)) “ (ℤ≥‘(𝑁 + 1))) = {0})
33	2	ply1termlem 26501	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘))))
34		elfznn0 13734	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0)
35	19	oveq1d 7427	. . . . . . 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 605	. . . . . 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 15849	. . . . 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 5199	. . . 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 2799	. . 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 26526	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (coeff‘𝐹) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)))
41	4	adantr 486	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ≠ 0) → 𝐹 ∈ (Poly‘ℂ))
42	5	adantr 486	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ≠ 0) → 𝑁 ∈ ℕ0)
43	11	adantr 486	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ≠ 0) → (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)):ℕ0⟶ℂ)
44	32	adantr 486	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ≠ 0) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)) “ (ℤ≥‘(𝑁 + 1))) = {0})
45	39	adantr 486	. . . 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 4488	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → if(𝑛 = 𝑁, 𝐴, 0) = 𝐴)
47	46, 12	fvmptg 6983	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℂ) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑁) = 𝐴)
48	47	ancoms 464	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑁) = 𝐴)
49	48	neeq1d 3015	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑁) ≠ 0 ↔ 𝐴 ≠ 0))
50	49	biimpar 483	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ≠ 0) → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0))‘𝑁) ≠ 0)
51	41, 42, 43, 44, 45, 50	dgreq 26543	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝐴 ≠ 0) → (deg‘𝐹) = 𝑁)
52	51	ex 418	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 ≠ 0 → (deg‘𝐹) = 𝑁))
53	40, 52	jca 521	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((coeff‘𝐹) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)) ∧ (𝐴 ≠ 0 → (deg‘𝐹) = 𝑁)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077   ⊆ wss 3899  ifcif 4482  {csn 4584   class class class wbr 5103   ↦ cmpt 5186   “ cima 5654  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412  ℂcc 11179  0cc0 11181  1c1 11182   + caddc 11184   · cmul 11186   ≤ cle 11325  ℕ₀cn0 12587  ℤ_≥cuz 12946  ...cfz 13620  ↑cexp 14184  Σcsu 15833  Polycply 26482  coeffccoe 26484  degcdgr 26485

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-inf2 9626  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258  ax-pre-sup 11259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7682  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-er 8701  df-map 8833  df-pm 8834  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9418  df-inf 9419  df-oi 9488  df-card 10001  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-div 11955  df-nn 12317  df-2 12386  df-3 12387  df-n0 12588  df-z 12675  df-uz 12947  df-rp 13102  df-fz 13621  df-fzo 13769  df-fl 13912  df-seq 14125  df-exp 14185  df-hash 14455  df-cj 15246  df-re 15247  df-im 15248  df-sqrt 15382  df-abs 15383  df-clim 15635  df-rlim 15636  df-sum 15834  df-0p 25971  df-ply 26486  df-coe 26488  df-dgr 26489

This theorem is used by:  coe1term  26558  dgr1term  26559