taylply - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > taylply		Structured version Visualization version GIF version

Theorem taylply 26696

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 21700	. . 3 ⊢ ℂ_fld ∈ Ring
8		cnfldbas 21682	. . . 4 ⊢ ℂ = (Base‘ℂ_fld)
9	8	subrgid 20825	. . 3 ⊢ (ℂ_fld ∈ Ring → ℂ ∈ (SubRing‘ℂ_fld))
10	7, 9	mp1i 14	. 2 ⊢ (𝜑 → ℂ ∈ (SubRing‘ℂ_fld))
11		cnex 11281	. . . . . . . 8 ⊢ ℂ ∈ V
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℂ ∈ V)
13		elpm2r 8865	. . . . . . 7 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
14	12, 1, 2, 3, 13	syl22anc 852	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm 𝑆))
15		dvnbss 26248	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom 𝐹)
16	1, 14, 4, 15	syl3anc 1398	. . . . 5 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom 𝐹)
17	2, 16	fssdmd 6728	. . . 4 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ 𝐴)
18		recnprss 26224	. . . . . 6 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
19	1, 18	syl 18	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
20	3, 19	sstrd 3941	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
21	17, 20	sstrd 3941	. . 3 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ ℂ)
22	21, 5	sseldd 3932	. 2 ⊢ (𝜑 → 𝐵 ∈ ℂ)
23	1	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑆 ∈ {ℝ, ℂ})
24	14	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
25		elfznn0 13754	. . . . . 6 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
26	25	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ₀)
27		dvnf 26247	. . . . 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 1398	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D^𝑛 𝐹)‘𝑘):dom ((𝑆 D^𝑛 𝐹)‘𝑘)⟶ℂ)
29		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ (0...𝑁))
30		dvn2bss 26250	. . . . . 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 1398	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom ((𝑆 D^𝑛 𝐹)‘𝑘))
32	5	adantr 486	. . . . 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 7085	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) ∈ ℂ)
35	26	faccld 14428	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℕ)
36	35	nncnd 12351	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℂ)
37	35	nnne0d 12388	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ≠ 0)
38	34, 36, 37	divcld 12093	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) ∈ ℂ)
39	1, 2, 3, 4, 5, 6, 10, 22, 38	taylply2 26695	1 ⊢ (𝜑 → (𝑇 ∈ (Poly‘ℂ) ∧ (deg‘𝑇) ≤ 𝑁))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 Vcvv 3451 ⊆ wss 3899 {cpr 4586 class class class wbr 5103 dom cdm 5651 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ↑_pm cpm 8848 ℂcc 11198 ℝcr 11199 0cc0 11200 ≤ cle 11344 ℕ₀cn0 12606 ...cfz 13639 !cfa 14417 Ringcrg 20459 SubRingcsubrg 20821 ℂ_fldccnfld 21678 D^𝑛 cdvn 26184 Polycply 26502 degcdgr 26505 Tayl ctayl 26680

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 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279

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-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 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 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-er 8717 df-map 8849 df-pm 8850 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-icc 13483 df-fz 13640 df-fzo 13789 df-fl 13932 df-seq 14145 df-exp 14205 df-fac 14418 df-hash 14475 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-clim 15655 df-rlim 15656 df-sum 15854 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-rest 17593 df-topn 17594 df-0g 17612 df-gsum 17613 df-topgen 17614 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-grp 19147 df-minusg 19148 df-subg 19333 df-cntz 19531 df-cmn 19996 df-abl 19997 df-mgp 20361 df-rng 20375 df-ur 20408 df-ring 20461 df-cring 20462 df-subrng 20798 df-subrg 20822 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-fbas 21675 df-fg 21676 df-cnfld 21679 df-top 23212 df-topon 23229 df-topsp 23251 df-bases 23264 df-cld 23337 df-ntr 23338 df-cls 23339 df-nei 23416 df-lp 23454 df-perf 23455 df-cnp 23546 df-haus 23633 df-fil 24165 df-fm 24257 df-flim 24258 df-flf 24259 df-tsms 24446 df-xms 24639 df-ms 24640 df-0p 25991 df-limc 26186 df-dv 26187 df-dvn 26188 df-ply 26506 df-idp 26507 df-coe 26508 df-dgr 26509 df-tayl 26682

This theorem is used by: (None)

W3C validator