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Theorem taylpval 26687

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 26685	. 2 ⊢ (𝜑 → 𝑇 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))
8		simplr 781	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑘 ∈ (0...𝑁)) → 𝑥 = 𝑋)
9	8	oveq1d 7433	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑘 ∈ (0...𝑁)) → (𝑥 − 𝐵) = (𝑋 − 𝐵))
10	9	oveq1d 7433	. . . 4 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑥 − 𝐵)↑𝑘) = ((𝑋 − 𝐵)↑𝑘))
11	10	oveq2d 7434	. . 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 15862	. 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 15848	. . 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 6999	1 ⊢ (𝜑 → (𝑇‘𝑋) = Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘)))

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 dom cdm 5651 ⟶wf 6533 ‘cfv 6537 (class class class)co 7418 ℂcc 11191 ℝcr 11192 0cc0 11193 · cmul 11198 − cmin 11534 / cdiv 11966 ℕ₀cn0 12599 ...cfz 13632 ↑cexp 14197 !cfa 14410 Σcsu 15846 D^𝑛 cdvn 26177 Tayl ctayl 26673

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 7749 ax-inf2 9635 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271 ax-addf 11272

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 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-1st 7999 df-2nd 8000 df-supp 8171 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-er 8710 df-map 8842 df-pm 8843 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9347 df-fi 9396 df-sup 9427 df-inf 9428 df-oi 9497 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-5 12401 df-6 12402 df-7 12403 df-8 12404 df-9 12405 df-n0 12600 df-z 12687 df-dec 12808 df-uz 12959 df-q 13069 df-rp 13114 df-xneg 13234 df-xadd 13235 df-xmul 13236 df-icc 13476 df-fz 13633 df-fzo 13782 df-seq 14138 df-exp 14198 df-fac 14411 df-hash 14468 df-cj 15259 df-re 15260 df-im 15261 df-sqrt 15395 df-abs 15396 df-clim 15648 df-sum 15847 df-struct 17318 df-sets 17335 df-slot 17353 df-ndx 17365 df-base 17381 df-plusg 17434 df-mulr 17435 df-starv 17436 df-tset 17440 df-ple 17441 df-ds 17443 df-unif 17444 df-rest 17586 df-topn 17587 df-0g 17605 df-gsum 17606 df-topgen 17607 df-mgm 18809 df-sgrp 18901 df-mnd 18917 df-grp 19140 df-minusg 19141 df-cntz 19524 df-cmn 19989 df-abl 19990 df-mgp 20354 df-ur 20401 df-ring 20454 df-cring 20455 df-psmet 21663 df-xmet 21664 df-met 21665 df-bl 21666 df-mopn 21667 df-fbas 21668 df-fg 21669 df-cnfld 21672 df-top 23205 df-topon 23222 df-topsp 23244 df-bases 23257 df-cld 23330 df-ntr 23331 df-cls 23332 df-nei 23409 df-lp 23447 df-perf 23448 df-cnp 23539 df-haus 23626 df-fil 24158 df-fm 24250 df-flim 24251 df-flf 24252 df-tsms 24439 df-xms 24632 df-ms 24633 df-limc 26179 df-dv 26180 df-dvn 26181 df-tayl 26675

This theorem is used by: (None)

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