taylply - Metamath Proof Explorer

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Theorem taylply 26708

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 21712	. . 3 ⊢ ℂ_fld ∈ Ring
8		cnfldbas 21694	. . . 4 ⊢ ℂ = (Base‘ℂ_fld)
9	8	subrgid 20837	. . 3 ⊢ (ℂ_fld ∈ Ring → ℂ ∈ (SubRing‘ℂ_fld))
10	7, 9	mp1i 14	. 2 ⊢ (𝜑 → ℂ ∈ (SubRing‘ℂ_fld))
11		cnex 11291	. . . . . . . 8 ⊢ ℂ ∈ V
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℂ ∈ V)
13		elpm2r 8876	. . . . . . 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 26260	. . . . . 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 6736	. . . 4 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ 𝐴)
18		recnprss 26236	. . . . . 6 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
19	1, 18	syl 18	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
20	3, 19	sstrd 3922	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
21	17, 20	sstrd 3922	. . 3 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ ℂ)
22	21, 5	sseldd 3913	. 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 13764	. . . . . 6 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
26	25	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ₀)
27		dvnf 26259	. . . . 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 26262	. . . . . 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 3913	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘𝑘))
34	28, 33	ffvelcdmd 7093	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) ∈ ℂ)
35	26	faccld 14438	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℕ)
36	35	nncnd 12361	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℂ)
37	35	nnne0d 12398	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ≠ 0)
38	34, 36, 37	divcld 12103	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) ∈ ℂ)
39	1, 2, 3, 4, 5, 6, 10, 22, 38	taylply2 26707	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 1569 ∈ wcel 2146 Vcvv 3453 ⊆ wss 3899 {cpr 4589 class class class wbr 5105 dom cdm 5654 ⟶wf 6542 ‘cfv 6546 (class class class)co 7427 ↑_pm cpm 8859 ℂcc 11208 ℝcr 11209 0cc0 11210 ≤ cle 11354 ℕ₀cn0 12616 ...cfz 13649 !cfa 14427 Ringcrg 20471 SubRingcsubrg 20833 ℂ_fldccnfld 21690 D^𝑛 cdvn 26196 Polycply 26514 degcdgr 26517 Tayl ctayl 26692

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5234 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758 ax-inf2 9652 ax-cnex 11266 ax-resscn 11267 ax-1cn 11268 ax-icn 11269 ax-addcl 11270 ax-addrcl 11271 ax-mulcl 11272 ax-mulrcl 11273 ax-mulcom 11274 ax-addass 11275 ax-mulass 11276 ax-distr 11277 ax-i2m1 11278 ax-1ne0 11279 ax-1rid 11280 ax-rnegex 11281 ax-rrecex 11282 ax-cnre 11283 ax-pre-lttri 11284 ax-pre-lttrn 11285 ax-pre-ltadd 11286 ax-pre-mulgt0 11287 ax-pre-sup 11288 ax-addf 11289

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-pss 4016 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4838 df-uni 4870 df-int 4910 df-iun 4955 df-iin 4956 df-br 5106 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5549 df-eprel 5554 df-po 5562 df-so 5563 df-fr 5607 df-se 5608 df-we 5609 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6310 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7384 df-ov 7430 df-oprab 7431 df-mpo 7432 df-of 7700 df-om 7885 df-1st 8008 df-2nd 8009 df-supp 8188 df-frecs 8310 df-wrecs 8341 df-recs 8390 df-rdg 8429 df-1o 8487 df-er 8728 df-map 8860 df-pm 8861 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9365 df-fi 9413 df-sup 9444 df-inf 9445 df-oi 9514 df-card 10030 df-pnf 11355 df-mnf 11356 df-xr 11357 df-ltxr 11358 df-le 11359 df-sub 11553 df-neg 11554 df-div 11984 df-nn 12346 df-2 12415 df-3 12416 df-4 12417 df-5 12418 df-6 12419 df-7 12420 df-8 12421 df-9 12422 df-n0 12617 df-z 12704 df-dec 12825 df-uz 12976 df-q 13086 df-rp 13131 df-xneg 13251 df-xadd 13252 df-xmul 13253 df-icc 13493 df-fz 13650 df-fzo 13799 df-fl 13942 df-seq 14155 df-exp 14215 df-fac 14428 df-hash 14485 df-cj 15278 df-re 15279 df-im 15280 df-sqrt 15414 df-abs 15415 df-clim 15667 df-rlim 15668 df-sum 15866 df-struct 17337 df-sets 17354 df-slot 17372 df-ndx 17384 df-base 17400 df-ress 17421 df-plusg 17453 df-mulr 17454 df-starv 17455 df-tset 17459 df-ple 17460 df-ds 17462 df-unif 17463 df-rest 17605 df-topn 17606 df-0g 17624 df-gsum 17625 df-topgen 17626 df-mgm 18828 df-sgrp 18920 df-mnd 18936 df-grp 19159 df-minusg 19160 df-subg 19345 df-cntz 19543 df-cmn 20008 df-abl 20009 df-mgp 20373 df-rng 20387 df-ur 20420 df-ring 20473 df-cring 20474 df-subrng 20810 df-subrg 20834 df-psmet 21682 df-xmet 21683 df-met 21684 df-bl 21685 df-mopn 21686 df-fbas 21687 df-fg 21688 df-cnfld 21691 df-top 23224 df-topon 23241 df-topsp 23263 df-bases 23276 df-cld 23349 df-ntr 23350 df-cls 23351 df-nei 23428 df-lp 23466 df-perf 23467 df-cnp 23558 df-haus 23645 df-fil 24177 df-fm 24269 df-flim 24270 df-flf 24271 df-tsms 24458 df-xms 24651 df-ms 24652 df-0p 26003 df-limc 26198 df-dv 26199 df-dvn 26200 df-ply 26518 df-idp 26519 df-coe 26520 df-dgr 26521 df-tayl 26694

This theorem is used by: (None)

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