Theorem elplyd 15458

Description: Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.)

Hypotheses

Ref	Expression
elplyd.1	⊢ (𝜑 → 𝑆 ⊆ ℂ)
elplyd.2	⊢ (𝜑 → 𝑁 ∈ ℕ0)
elplyd.3	⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ∈ 𝑆)

Assertion

Ref	Expression
elplyd	⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(𝐴 · (𝑧↑𝑘))) ∈ (Poly‘𝑆))

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

Allowed substitution hint:   𝐴(𝑘)

Proof of Theorem elplyd

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nffvmpt1 5646	. . . . . . 7 ⊢ Ⅎ𝑘((𝑘 ∈ ℕ0 ↦ if(𝑘 ∈ (0...𝑁), 𝐴, 0))‘𝑗)
2		nfcv 2372	. . . . . . 7 ⊢ Ⅎ𝑘 ·
3		nfcv 2372	. . . . . . 7 ⊢ Ⅎ𝑘(𝑧↑𝑗)
4	1, 2, 3	nfov 6043	. . . . . 6 ⊢ Ⅎ𝑘(((𝑘 ∈ ℕ0 ↦ if(𝑘 ∈ (0...𝑁), 𝐴, 0))‘𝑗) · (𝑧↑𝑗))
5		nfcv 2372	. . . . . 6 ⊢ Ⅎ𝑗(((𝑘 ∈ ℕ0 ↦ if(𝑘 ∈ (0...𝑁), 𝐴, 0))‘𝑘) · (𝑧↑𝑘))
6		fveq2 5635	. . . . . . 7 ⊢ (𝑗 = 𝑘 → ((𝑘 ∈ ℕ0 ↦ if(𝑘 ∈ (0...𝑁), 𝐴, 0))‘𝑗) = ((𝑘 ∈ ℕ0 ↦ if(𝑘 ∈ (0...𝑁), 𝐴, 0))‘𝑘))
7		oveq2 6021	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑧↑𝑗) = (𝑧↑𝑘))
8	6, 7	oveq12d 6031	. . . . . 6 ⊢ (𝑗 = 𝑘 → (((𝑘 ∈ ℕ0 ↦ if(𝑘 ∈ (0...𝑁), 𝐴, 0))‘𝑗) · (𝑧↑𝑗)) = (((𝑘 ∈ ℕ0 ↦ if(𝑘 ∈ (0...𝑁), 𝐴, 0))‘𝑘) · (𝑧↑𝑘)))
9	4, 5, 8	cbvsumi 11916	. . . . 5 ⊢ Σ𝑗 ∈ (0...𝑁)(((𝑘 ∈ ℕ0 ↦ if(𝑘 ∈ (0...𝑁), 𝐴, 0))‘𝑗) · (𝑧↑𝑗)) = Σ𝑘 ∈ (0...𝑁)(((𝑘 ∈ ℕ0 ↦ if(𝑘 ∈ (0...𝑁), 𝐴, 0))‘𝑘) · (𝑧↑𝑘))
10		elfznn0 10342	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0)
11		iftrue 3608	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...𝑁) → if(𝑘 ∈ (0...𝑁), 𝐴, 0) = 𝐴)
12	11	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → if(𝑘 ∈ (0...𝑁), 𝐴, 0) = 𝐴)
13		elplyd.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ∈ 𝑆)
14	12, 13	eqeltrd 2306	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → if(𝑘 ∈ (0...𝑁), 𝐴, 0) ∈ 𝑆)
15		eqid 2229	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 ↦ if(𝑘 ∈ (0...𝑁), 𝐴, 0)) = (𝑘 ∈ ℕ0 ↦ if(𝑘 ∈ (0...𝑁), 𝐴, 0))
16	15	fvmpt2 5726	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0 ∧ if(𝑘 ∈ (0...𝑁), 𝐴, 0) ∈ 𝑆) → ((𝑘 ∈ ℕ0 ↦ if(𝑘 ∈ (0...𝑁), 𝐴, 0))‘𝑘) = if(𝑘 ∈ (0...𝑁), 𝐴, 0))
17	10, 14, 16	syl2an2 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑘 ∈ ℕ0 ↦ if(𝑘 ∈ (0...𝑁), 𝐴, 0))‘𝑘) = if(𝑘 ∈ (0...𝑁), 𝐴, 0))
18	17, 12	eqtrd 2262	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑘 ∈ ℕ0 ↦ if(𝑘 ∈ (0...𝑁), 𝐴, 0))‘𝑘) = 𝐴)
19	18	oveq1d 6028	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (((𝑘 ∈ ℕ0 ↦ if(𝑘 ∈ (0...𝑁), 𝐴, 0))‘𝑘) · (𝑧↑𝑘)) = (𝐴 · (𝑧↑𝑘)))
20	19	sumeq2dv 11922	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (0...𝑁)(((𝑘 ∈ ℕ0 ↦ if(𝑘 ∈ (0...𝑁), 𝐴, 0))‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑁)(𝐴 · (𝑧↑𝑘)))
21	9, 20	eqtrid 2274	. . . 4 ⊢ (𝜑 → Σ𝑗 ∈ (0...𝑁)(((𝑘 ∈ ℕ0 ↦ if(𝑘 ∈ (0...𝑁), 𝐴, 0))‘𝑗) · (𝑧↑𝑗)) = Σ𝑘 ∈ (0...𝑁)(𝐴 · (𝑧↑𝑘)))
22	21	mpteq2dv 4178	. . 3 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑁)(((𝑘 ∈ ℕ0 ↦ if(𝑘 ∈ (0...𝑁), 𝐴, 0))‘𝑗) · (𝑧↑𝑗))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(𝐴 · (𝑧↑𝑘))))
23		elplyd.1	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
24		0cnd 8165	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℂ)
25	24	snssd 3816	. . . . 5 ⊢ (𝜑 → {0} ⊆ ℂ)
26	23, 25	unssd 3381	. . . 4 ⊢ (𝜑 → (𝑆 ∪ {0}) ⊆ ℂ)
27		elplyd.2	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
28		elun1 3372	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ (𝑆 ∪ {0}))
29	13, 28	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ∈ (𝑆 ∪ {0}))
30	29	adantlr 477	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ∈ (𝑆 ∪ {0}))
31		ssun2 3369	. . . . . . . 8 ⊢ {0} ⊆ (𝑆 ∪ {0})
32		c0ex 8166	. . . . . . . . 9 ⊢ 0 ∈ V
33	32	snss 3806	. . . . . . . 8 ⊢ (0 ∈ (𝑆 ∪ {0}) ↔ {0} ⊆ (𝑆 ∪ {0}))
34	31, 33	mpbir 146	. . . . . . 7 ⊢ 0 ∈ (𝑆 ∪ {0})
35	34	a1i 9	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ (0...𝑁)) → 0 ∈ (𝑆 ∪ {0}))
36		nn0z 9492	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℤ)
37	36	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℤ)
38		0zd 9484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 0 ∈ ℤ)
39	27	nn0zd 9593	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
40	39	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℤ)
41		fzdcel 10268	. . . . . . 7 ⊢ ((𝑘 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑘 ∈ (0...𝑁))
42	37, 38, 40, 41	syl3anc 1271	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → DECID 𝑘 ∈ (0...𝑁))
43	30, 35, 42	ifcldadc 3633	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → if(𝑘 ∈ (0...𝑁), 𝐴, 0) ∈ (𝑆 ∪ {0}))
44	43	fmpttd 5798	. . . 4 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ if(𝑘 ∈ (0...𝑁), 𝐴, 0)):ℕ0⟶(𝑆 ∪ {0}))
45		elplyr 15457	. . . 4 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ 𝑁 ∈ ℕ0 ∧ (𝑘 ∈ ℕ0 ↦ if(𝑘 ∈ (0...𝑁), 𝐴, 0)):ℕ0⟶(𝑆 ∪ {0})) → (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑁)(((𝑘 ∈ ℕ0 ↦ if(𝑘 ∈ (0...𝑁), 𝐴, 0))‘𝑗) · (𝑧↑𝑗))) ∈ (Poly‘(𝑆 ∪ {0})))
46	26, 27, 44, 45	syl3anc 1271	. . 3 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑁)(((𝑘 ∈ ℕ0 ↦ if(𝑘 ∈ (0...𝑁), 𝐴, 0))‘𝑗) · (𝑧↑𝑗))) ∈ (Poly‘(𝑆 ∪ {0})))
47	22, 46	eqeltrrd 2307	. 2 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(𝐴 · (𝑧↑𝑘))) ∈ (Poly‘(𝑆 ∪ {0})))
48		plyun0 15453	. 2 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
49	47, 48	eleqtrdi 2322	1 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(𝐴 · (𝑧↑𝑘))) ∈ (Poly‘𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  DECID wdc 839   = wceq 1395   ∈ wcel 2200   ∪ cun 3196   ⊆ wss 3198  ifcif 3603  {csn 3667   ↦ cmpt 4148  ⟶wf 5320  ‘cfv 5324  (class class class)co 6013  ℂcc 8023  0cc0 8025   · cmul 8030  ℕ₀cn0 9395  ℤcz 9472  ...cfz 10236  ↑cexp 10793  Σcsu 11907  Polycply 15445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-addass 8127  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-0id 8133  ax-rnegex 8134  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-ltadd 8141

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-map 6814  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-inn 9137  df-n0 9396  df-z 9473  df-uz 9749  df-fz 10237  df-seqfrec 10703  df-sumdc 11908  df-ply 15447

This theorem is referenced by:  ply1term  15460  plyaddlem  15466  plymullem  15467  plycj  15478  dvply2g  15483