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Theorem eltayl 24513

Description: Value of the Taylor series as a relation (elementhood in the domain here expresses that the series is convergent). (Contributed by Mario Carneiro, 30-Dec-2016.)

**Hypotheses**
Ref	Expression
taylfval.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
taylfval.f	⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
taylfval.a	⊢ (𝜑 → 𝐴 ⊆ 𝑆)
taylfval.n	⊢ (𝜑 → (𝑁 ∈ ℕ₀ ∨ 𝑁 = +∞))
taylfval.b	⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘𝑘))
taylfval.t	⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)

**Assertion**
Ref	Expression
eltayl	⊢ (𝜑 → (𝑋𝑇𝑌 ↔ (𝑋 ∈ ℂ ∧ 𝑌 ∈ (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘)))))))

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

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

Step	Hyp	Ref	Expression
1		taylfval.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
2		taylfval.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
3		taylfval.a	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝑆)
4		taylfval.n	. . . 4 ⊢ (𝜑 → (𝑁 ∈ ℕ₀ ∨ 𝑁 = +∞))
5		taylfval.b	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘𝑘))
6		taylfval.t	. . . 4 ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)
7	1, 2, 3, 4, 5, 6	taylfval 24512	. . 3 ⊢ (𝜑 → 𝑇 = ∪ 𝑥 ∈ ℂ ({𝑥} × (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))))
8	7	eleq2d 2892	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ 𝑇 ↔ ⟨𝑋, 𝑌⟩ ∈ ∪ 𝑥 ∈ ℂ ({𝑥} × (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))))))
9		df-br 4874	. . 3 ⊢ (𝑋𝑇𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ 𝑇)
10	9	bicomi 216	. 2 ⊢ (⟨𝑋, 𝑌⟩ ∈ 𝑇 ↔ 𝑋𝑇𝑌)
11		oveq1 6912	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 − 𝐵) = (𝑋 − 𝐵))
12	11	oveq1d 6920	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥 − 𝐵)↑𝑘) = ((𝑋 − 𝐵)↑𝑘))
13	12	oveq2d 6921	. . . . 5 ⊢ (𝑥 = 𝑋 → (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)) = (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘)))
14	13	mpteq2dv 4968	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))) = (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘))))
15	14	oveq2d 6921	. . 3 ⊢ (𝑥 = 𝑋 → (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))) = (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘)))))
16	15	opeliunxp2 5493	. 2 ⊢ (⟨𝑋, 𝑌⟩ ∈ ∪ 𝑥 ∈ ℂ ({𝑥} × (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))) ↔ (𝑋 ∈ ℂ ∧ 𝑌 ∈ (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘))))))
17	8, 10, 16	3bitr3g 305	1 ⊢ (𝜑 → (𝑋𝑇𝑌 ↔ (𝑋 ∈ ℂ ∧ 𝑌 ∈ (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋 − 𝐵)↑𝑘)))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∨ wo 880 = wceq 1658 ∈ wcel 2166 ∩ cin 3797 ⊆ wss 3798 {csn 4397 {cpr 4399 ⟨cop 4403 ∪ ciun 4740 class class class wbr 4873 ↦ cmpt 4952 × cxp 5340 dom cdm 5342 ⟶wf 6119 ‘cfv 6123 (class class class)co 6905 ℂcc 10250 ℝcr 10251 0cc0 10252 · cmul 10257 +∞cpnf 10388 − cmin 10585 / cdiv 11009 ℕ₀cn0 11618 ℤcz 11704 [,]cicc 12466 ↑cexp 13154 !cfa 13353 ℂ_fldccnfld 20106 tsums ctsu 22299 D^𝑛 cdvn 24027 Tayl ctayl 24506

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 ax-addf 10331 ax-mulf 10332

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-iin 4743 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-supp 7560 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-pm 8125 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-fsupp 8545 df-fi 8586 df-sup 8617 df-inf 8618 df-oi 8684 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-q 12072 df-rp 12113 df-xneg 12232 df-xadd 12233 df-xmul 12234 df-icc 12470 df-fz 12620 df-fzo 12761 df-seq 13096 df-exp 13155 df-fac 13354 df-hash 13411 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-plusg 16318 df-mulr 16319 df-starv 16320 df-tset 16324 df-ple 16325 df-ds 16327 df-unif 16328 df-rest 16436 df-topn 16437 df-0g 16455 df-gsum 16456 df-topgen 16457 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-grp 17779 df-minusg 17780 df-cntz 18100 df-cmn 18548 df-abl 18549 df-mgp 18844 df-ur 18856 df-ring 18903 df-cring 18904 df-psmet 20098 df-xmet 20099 df-met 20100 df-bl 20101 df-mopn 20102 df-fbas 20103 df-fg 20104 df-cnfld 20107 df-top 21069 df-topon 21086 df-topsp 21108 df-bases 21121 df-cld 21194 df-ntr 21195 df-cls 21196 df-nei 21273 df-lp 21311 df-perf 21312 df-cnp 21403 df-haus 21490 df-fil 22020 df-fm 22112 df-flim 22113 df-flf 22114 df-tsms 22300 df-xms 22495 df-ms 22496 df-limc 24029 df-dv 24030 df-dvn 24031 df-tayl 24508

This theorem is referenced by: taylf 24514 tayl0 24515

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