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Theorem taylpval 26281

Description: Value of the Taylor polynomial. (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)
taylpval.x	⊢ (𝜑 → 𝑋 ∈ ℂ)

**Assertion**
Ref	Expression
taylpval	⊢ (𝜑 → (𝑇‘𝑋) = Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘)))

Distinct variable groups: 𝐵,𝑘 𝑘,𝐹 𝑘,𝑁 𝜑,𝑘 𝑆,𝑘 𝑘,𝑋 Allowed substitution hints: 𝐴(𝑘) 𝑇(𝑘)

Proof of Theorem taylpval Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		taylpfval.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
2		taylpfval.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
3		taylpfval.a	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑆)
4		taylpfval.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
5		taylpfval.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘𝑁))
6		taylpfval.t	. . 3 ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)
7	1, 2, 3, 4, 5, 6	taylpfval 26279	. 2 ⊢ (𝜑 → 𝑇 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))
8		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑘 ∈ (0...𝑁)) → 𝑥 = 𝑋)
9	8	oveq1d 7409	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑘 ∈ (0...𝑁)) → (𝑥 − 𝐵) = (𝑋 − 𝐵))
10	9	oveq1d 7409	. . . 4 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑥 − 𝐵)↑𝑘) = ((𝑋 − 𝐵)↑𝑘))
11	10	oveq2d 7410	. . 3 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑘 ∈ (0...𝑁)) → (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)) = (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘)))
12	11	sumeq2dv 15675	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)) = Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘)))
13		taylpval.x	. 2 ⊢ (𝜑 → 𝑋 ∈ ℂ)
14		sumex 15661	. . 3 ⊢ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘)) ∈ V
15	14	a1i 11	. 2 ⊢ (𝜑 → Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘)) ∈ V)
16	7, 12, 13, 15	fvmptd 6982	1 ⊢ (𝜑 → (𝑇‘𝑋) = Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 Vcvv 3455 ⊆ wss 3922 {cpr 4599 dom cdm 5646 ⟶wf 6515 ‘cfv 6519 (class class class)co 7394 ℂcc 11084 ℝcr 11085 0cc0 11086 · cmul 11091 − cmin 11423 / cdiv 11851 ℕ₀cn0 12458 ...cfz 13481 ↑cexp 14036 !cfa 14248 Σcsu 15659 D^𝑛 cdvn 25772 Tayl ctayl 26267

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 2702 ax-rep 5242 ax-sep 5259 ax-nul 5269 ax-pow 5328 ax-pr 5395 ax-un 7718 ax-inf2 9612 ax-cnex 11142 ax-resscn 11143 ax-1cn 11144 ax-icn 11145 ax-addcl 11146 ax-addrcl 11147 ax-mulcl 11148 ax-mulrcl 11149 ax-mulcom 11150 ax-addass 11151 ax-mulass 11152 ax-distr 11153 ax-i2m1 11154 ax-1ne0 11155 ax-1rid 11156 ax-rnegex 11157 ax-rrecex 11158 ax-cnre 11159 ax-pre-lttri 11160 ax-pre-lttrn 11161 ax-pre-ltadd 11162 ax-pre-mulgt0 11163 ax-pre-sup 11164 ax-addf 11165

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2880 df-ne 2928 df-nel 3032 df-ral 3047 df-rex 3056 df-rmo 3357 df-reu 3358 df-rab 3412 df-v 3457 df-sbc 3762 df-csb 3871 df-dif 3925 df-un 3927 df-in 3929 df-ss 3939 df-pss 3942 df-nul 4305 df-if 4497 df-pw 4573 df-sn 4598 df-pr 4600 df-tp 4602 df-op 4604 df-uni 4880 df-int 4919 df-iun 4965 df-iin 4966 df-br 5116 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5541 df-eprel 5546 df-po 5554 df-so 5555 df-fr 5599 df-se 5600 df-we 5601 df-xp 5652 df-rel 5653 df-cnv 5654 df-co 5655 df-dm 5656 df-rn 5657 df-res 5658 df-ima 5659 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6472 df-fun 6521 df-fn 6522 df-f 6523 df-f1 6524 df-fo 6525 df-f1o 6526 df-fv 6527 df-isom 6528 df-riota 7351 df-ov 7397 df-oprab 7398 df-mpo 7399 df-om 7851 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8269 df-wrecs 8300 df-recs 8349 df-rdg 8387 df-1o 8443 df-er 8682 df-map 8805 df-pm 8806 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-fsupp 9331 df-fi 9380 df-sup 9411 df-inf 9412 df-oi 9481 df-card 9910 df-pnf 11228 df-mnf 11229 df-xr 11230 df-ltxr 11231 df-le 11232 df-sub 11425 df-neg 11426 df-div 11852 df-nn 12198 df-2 12260 df-3 12261 df-4 12262 df-5 12263 df-6 12264 df-7 12265 df-8 12266 df-9 12267 df-n0 12459 df-z 12546 df-dec 12666 df-uz 12810 df-q 12922 df-rp 12966 df-xneg 13085 df-xadd 13086 df-xmul 13087 df-icc 13326 df-fz 13482 df-fzo 13629 df-seq 13977 df-exp 14037 df-fac 14249 df-hash 14306 df-cj 15075 df-re 15076 df-im 15077 df-sqrt 15211 df-abs 15212 df-clim 15461 df-sum 15660 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17186 df-plusg 17239 df-mulr 17240 df-starv 17241 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-rest 17391 df-topn 17392 df-0g 17410 df-gsum 17411 df-topgen 17412 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18874 df-minusg 18875 df-cntz 19255 df-cmn 19718 df-abl 19719 df-mgp 20056 df-ur 20097 df-ring 20150 df-cring 20151 df-psmet 21262 df-xmet 21263 df-met 21264 df-bl 21265 df-mopn 21266 df-fbas 21267 df-fg 21268 df-cnfld 21271 df-top 22787 df-topon 22804 df-topsp 22826 df-bases 22839 df-cld 22912 df-ntr 22913 df-cls 22914 df-nei 22991 df-lp 23029 df-perf 23030 df-cnp 23121 df-haus 23208 df-fil 23739 df-fm 23831 df-flim 23832 df-flf 23833 df-tsms 24020 df-xms 24214 df-ms 24215 df-limc 25774 df-dv 25775 df-dvn 25776 df-tayl 26269

This theorem is referenced by: (None)

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