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Theorem taylply 26349

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 21335	. . 3 ⊢ ℂ_fld ∈ Ring
8		cnfldbas 21300	. . . 4 ⊢ ℂ = (Base‘ℂ_fld)
9	8	subrgid 20524	. . 3 ⊢ (ℂ_fld ∈ Ring → ℂ ∈ (SubRing‘ℂ_fld))
10	7, 9	mp1i 13	. 2 ⊢ (𝜑 → ℂ ∈ (SubRing‘ℂ_fld))
11		cnex 11221	. . . . . . . 8 ⊢ ℂ ∈ V
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℂ ∈ V)
13		elpm2r 8864	. . . . . . 7 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
14	12, 1, 2, 3, 13	syl22anc 837	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm 𝑆))
15		dvnbss 25902	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom 𝐹)
16	1, 14, 4, 15	syl3anc 1368	. . . . 5 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom 𝐹)
17	2, 16	fssdmd 6741	. . . 4 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ 𝐴)
18		recnprss 25877	. . . . . 6 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
19	1, 18	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
20	3, 19	sstrd 3987	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
21	17, 20	sstrd 3987	. . 3 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ ℂ)
22	21, 5	sseldd 3977	. 2 ⊢ (𝜑 → 𝐵 ∈ ℂ)
23	1	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑆 ∈ {ℝ, ℂ})
24	14	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
25		elfznn0 13629	. . . . . 6 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
26	25	adantl 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ₀)
27		dvnf 25901	. . . . 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 1368	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D^𝑛 𝐹)‘𝑘):dom ((𝑆 D^𝑛 𝐹)‘𝑘)⟶ℂ)
29		simpr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ (0...𝑁))
30		dvn2bss 25904	. . . . . 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 1368	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom ((𝑆 D^𝑛 𝐹)‘𝑘))
32	5	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘𝑁))
33	31, 32	sseldd 3977	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘𝑘))
34	28, 33	ffvelcdmd 7094	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) ∈ ℂ)
35	26	faccld 14279	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℕ)
36	35	nncnd 12261	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℂ)
37	35	nnne0d 12295	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ≠ 0)
38	34, 36, 37	divcld 12023	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) ∈ ℂ)
39	1, 2, 3, 4, 5, 6, 10, 22, 38	taylply2 26347	1 ⊢ (𝜑 → (𝑇 ∈ (Poly‘ℂ) ∧ (deg‘𝑇) ≤ 𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3461 ⊆ wss 3944 {cpr 4632 class class class wbr 5149 dom cdm 5678 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ↑_pm cpm 8846 ℂcc 11138 ℝcr 11139 0cc0 11140 ≤ cle 11281 ℕ₀cn0 12505 ...cfz 13519 !cfa 14268 Ringcrg 20185 SubRingcsubrg 20518 ℂ_fldccnfld 21296 D^𝑛 cdvn 25837 Polycply 26163 degcdgr 26166 Tayl ctayl 26332

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 ax-addf 11219

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-fi 9436 df-sup 9467 df-inf 9468 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-icc 13366 df-fz 13520 df-fzo 13663 df-fl 13793 df-seq 14003 df-exp 14063 df-fac 14269 df-hash 14326 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-clim 15468 df-rlim 15469 df-sum 15669 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-rest 17407 df-topn 17408 df-0g 17426 df-gsum 17427 df-topgen 17428 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18901 df-minusg 18902 df-subg 19086 df-cntz 19280 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-cring 20188 df-subrng 20495 df-subrg 20520 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22893 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-lp 23084 df-perf 23085 df-cnp 23176 df-haus 23263 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-tsms 24075 df-xms 24270 df-ms 24271 df-0p 25643 df-limc 25839 df-dv 25840 df-dvn 25841 df-ply 26167 df-idp 26168 df-coe 26169 df-dgr 26170 df-tayl 26334

This theorem is referenced by: (None)

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