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Theorem taylply 26258

Description: The Taylor polynomial is a polynomial of degree (at most) 𝑁. (Contributed by Mario Carneiro, 31-Dec-2016.)

**Hypotheses**
Ref	Expression
taylpfval.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
taylpfval.f	⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
taylpfval.a	⊢ (𝜑 → 𝐴 ⊆ 𝑆)
taylpfval.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
taylpfval.b	⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘𝑁))
taylpfval.t	⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)

**Assertion**
Ref	Expression
taylply	⊢ (𝜑 → (𝑇 ∈ (Poly‘ℂ) ∧ (deg‘𝑇) ≤ 𝑁))

Proof of Theorem taylply Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		taylpfval.s	. 2 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
2		taylpfval.f	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
3		taylpfval.a	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝑆)
4		taylpfval.n	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
5		taylpfval.b	. 2 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘𝑁))
6		taylpfval.t	. 2 ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)
7		cnring 21281	. . 3 ⊢ ℂ_fld ∈ Ring
8		cnfldbas 21249	. . . 4 ⊢ ℂ = (Base‘ℂ_fld)
9	8	subrgid 20442	. . 3 ⊢ (ℂ_fld ∈ Ring → ℂ ∈ (SubRing‘ℂ_fld))
10	7, 9	mp1i 13	. 2 ⊢ (𝜑 → ℂ ∈ (SubRing‘ℂ_fld))
11		cnex 11078	. . . . . . . 8 ⊢ ℂ ∈ V
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℂ ∈ V)
13		elpm2r 8763	. . . . . . 7 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
14	12, 1, 2, 3, 13	syl22anc 838	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm 𝑆))
15		dvnbss 25811	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom 𝐹)
16	1, 14, 4, 15	syl3anc 1373	. . . . 5 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom 𝐹)
17	2, 16	fssdmd 6664	. . . 4 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ 𝐴)
18		recnprss 25786	. . . . . 6 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
19	1, 18	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
20	3, 19	sstrd 3942	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
21	17, 20	sstrd 3942	. . 3 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ ℂ)
22	21, 5	sseldd 3932	. 2 ⊢ (𝜑 → 𝐵 ∈ ℂ)
23	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑆 ∈ {ℝ, ℂ})
24	14	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
25		elfznn0 13511	. . . . . 6 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
26	25	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ₀)
27		dvnf 25810	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑘 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘𝑘):dom ((𝑆 D^𝑛 𝐹)‘𝑘)⟶ℂ)
28	23, 24, 26, 27	syl3anc 1373	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D^𝑛 𝐹)‘𝑘):dom ((𝑆 D^𝑛 𝐹)‘𝑘)⟶ℂ)
29		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ (0...𝑁))
30		dvn2bss 25813	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑘 ∈ (0...𝑁)) → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom ((𝑆 D^𝑛 𝐹)‘𝑘))
31	23, 24, 29, 30	syl3anc 1373	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom ((𝑆 D^𝑛 𝐹)‘𝑘))
32	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘𝑁))
33	31, 32	sseldd 3932	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘𝑘))
34	28, 33	ffvelcdmd 7012	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) ∈ ℂ)
35	26	faccld 14179	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℕ)
36	35	nncnd 12132	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℂ)
37	35	nnne0d 12166	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ≠ 0)
38	34, 36, 37	divcld 11888	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) ∈ ℂ)
39	1, 2, 3, 4, 5, 6, 10, 22, 38	taylply2 26256	1 ⊢ (𝜑 → (𝑇 ∈ (Poly‘ℂ) ∧ (deg‘𝑇) ≤ 𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 Vcvv 3433 ⊆ wss 3899 {cpr 4575 class class class wbr 5088 dom cdm 5613 ⟶wf 6472 ‘cfv 6476 (class class class)co 7340 ↑_pm cpm 8745 ℂcc 10995 ℝcr 10996 0cc0 10997 ≤ cle 11138 ℕ₀cn0 12372 ...cfz 13398 !cfa 14168 Ringcrg 20105 SubRingcsubrg 20438 ℂ_fldccnfld 21245 D^𝑛 cdvn 25746 Polycply 26070 degcdgr 26073 Tayl ctayl 26241

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 2701 ax-rep 5214 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5367 ax-un 7662 ax-inf2 9525 ax-cnex 11053 ax-resscn 11054 ax-1cn 11055 ax-icn 11056 ax-addcl 11057 ax-addrcl 11058 ax-mulcl 11059 ax-mulrcl 11060 ax-mulcom 11061 ax-addass 11062 ax-mulass 11063 ax-distr 11064 ax-i2m1 11065 ax-1ne0 11066 ax-1rid 11067 ax-rnegex 11068 ax-rrecex 11069 ax-cnre 11070 ax-pre-lttri 11071 ax-pre-lttrn 11072 ax-pre-ltadd 11073 ax-pre-mulgt0 11074 ax-pre-sup 11075 ax-addf 11076

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3393 df-v 3435 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4895 df-iun 4940 df-iin 4941 df-br 5089 df-opab 5151 df-mpt 5170 df-tr 5196 df-id 5508 df-eprel 5513 df-po 5521 df-so 5522 df-fr 5566 df-se 5567 df-we 5568 df-xp 5619 df-rel 5620 df-cnv 5621 df-co 5622 df-dm 5623 df-rn 5624 df-res 5625 df-ima 5626 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-isom 6485 df-riota 7297 df-ov 7343 df-oprab 7344 df-mpo 7345 df-of 7604 df-om 7791 df-1st 7915 df-2nd 7916 df-supp 8085 df-frecs 8205 df-wrecs 8236 df-recs 8285 df-rdg 8323 df-1o 8379 df-er 8616 df-map 8746 df-pm 8747 df-en 8864 df-dom 8865 df-sdom 8866 df-fin 8867 df-fsupp 9240 df-fi 9289 df-sup 9320 df-inf 9321 df-oi 9390 df-card 9823 df-pnf 11139 df-mnf 11140 df-xr 11141 df-ltxr 11142 df-le 11143 df-sub 11337 df-neg 11338 df-div 11766 df-nn 12117 df-2 12179 df-3 12180 df-4 12181 df-5 12182 df-6 12183 df-7 12184 df-8 12185 df-9 12186 df-n0 12373 df-z 12460 df-dec 12580 df-uz 12724 df-q 12838 df-rp 12882 df-xneg 13002 df-xadd 13003 df-xmul 13004 df-icc 13243 df-fz 13399 df-fzo 13546 df-fl 13684 df-seq 13897 df-exp 13957 df-fac 14169 df-hash 14226 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-clim 15382 df-rlim 15383 df-sum 15581 df-struct 17045 df-sets 17062 df-slot 17080 df-ndx 17092 df-base 17108 df-ress 17129 df-plusg 17161 df-mulr 17162 df-starv 17163 df-tset 17167 df-ple 17168 df-ds 17170 df-unif 17171 df-rest 17313 df-topn 17314 df-0g 17332 df-gsum 17333 df-topgen 17334 df-mgm 18501 df-sgrp 18580 df-mnd 18596 df-grp 18802 df-minusg 18803 df-subg 18989 df-cntz 19183 df-cmn 19648 df-abl 19649 df-mgp 20013 df-rng 20025 df-ur 20054 df-ring 20107 df-cring 20108 df-subrng 20415 df-subrg 20439 df-psmet 21237 df-xmet 21238 df-met 21239 df-bl 21240 df-mopn 21241 df-fbas 21242 df-fg 21243 df-cnfld 21246 df-top 22763 df-topon 22780 df-topsp 22802 df-bases 22815 df-cld 22888 df-ntr 22889 df-cls 22890 df-nei 22967 df-lp 23005 df-perf 23006 df-cnp 23097 df-haus 23184 df-fil 23715 df-fm 23807 df-flim 23808 df-flf 23809 df-tsms 23996 df-xms 24189 df-ms 24190 df-0p 25552 df-limc 25748 df-dv 25749 df-dvn 25750 df-ply 26074 df-idp 26075 df-coe 26076 df-dgr 26077 df-tayl 26243

This theorem is referenced by: (None)

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