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Theorem taylf 25425

Description: The Taylor series defines a function on a subset of the complex numbers. (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
taylf	⊢ (𝜑 → 𝑇:dom 𝑇⟶ℂ)

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

Proof of Theorem taylf Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		taylfval.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
2		taylfval.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
3		taylfval.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ 𝑆)
4		taylfval.n	. . . . . . 7 ⊢ (𝜑 → (𝑁 ∈ ℕ₀ ∨ 𝑁 = +∞))
5		taylfval.b	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘𝑘))
6		taylfval.t	. . . . . . 7 ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)
7	1, 2, 3, 4, 5, 6	taylfval 25423	. . . . . 6 ⊢ (𝜑 → 𝑇 = ∪ 𝑥 ∈ ℂ ({𝑥} × (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))))
8		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
9	8	snssd 4739	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → {𝑥} ⊆ ℂ)
10	1, 2, 3, 4, 5	taylfvallem 25422	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))) ⊆ ℂ)
11		xpss12 5595	. . . . . . . . 9 ⊢ (({𝑥} ⊆ ℂ ∧ (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))) ⊆ ℂ) → ({𝑥} × (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))) ⊆ (ℂ × ℂ))
12	9, 10, 11	syl2anc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ({𝑥} × (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))) ⊆ (ℂ × ℂ))
13	12	ralrimiva 3107	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ ℂ ({𝑥} × (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))) ⊆ (ℂ × ℂ))
14		iunss 4971	. . . . . . 7 ⊢ (∪ 𝑥 ∈ ℂ ({𝑥} × (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))) ⊆ (ℂ × ℂ) ↔ ∀𝑥 ∈ ℂ ({𝑥} × (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))) ⊆ (ℂ × ℂ))
15	13, 14	sylibr 233	. . . . . 6 ⊢ (𝜑 → ∪ 𝑥 ∈ ℂ ({𝑥} × (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))) ⊆ (ℂ × ℂ))
16	7, 15	eqsstrd 3955	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ (ℂ × ℂ))
17		relxp 5598	. . . . 5 ⊢ Rel (ℂ × ℂ)
18		relss 5682	. . . . 5 ⊢ (𝑇 ⊆ (ℂ × ℂ) → (Rel (ℂ × ℂ) → Rel 𝑇))
19	16, 17, 18	mpisyl 21	. . . 4 ⊢ (𝜑 → Rel 𝑇)
20	1, 2, 3, 4, 5, 6	eltayl 25424	. . . . . . . 8 ⊢ (𝜑 → (𝑥𝑇𝑦 ↔ (𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))))))
21	20	biimpd 228	. . . . . . 7 ⊢ (𝜑 → (𝑥𝑇𝑦 → (𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))))))
22	21	alrimiv 1931	. . . . . 6 ⊢ (𝜑 → ∀𝑦(𝑥𝑇𝑦 → (𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))))))
23		cnfldbas 20514	. . . . . . . . 9 ⊢ ℂ = (Base‘ℂ_fld)
24		cnring 20532	. . . . . . . . . 10 ⊢ ℂ_fld ∈ Ring
25		ringcmn 19735	. . . . . . . . . 10 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ CMnd)
26	24, 25	mp1i 13	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ℂ_fld ∈ CMnd)
27		cnfldtps 23847	. . . . . . . . . 10 ⊢ ℂ_fld ∈ TopSp
28	27	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ℂ_fld ∈ TopSp)
29		ovex 7288	. . . . . . . . . . 11 ⊢ (0[,]𝑁) ∈ V
30	29	inex1 5236	. . . . . . . . . 10 ⊢ ((0[,]𝑁) ∩ ℤ) ∈ V
31	30	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((0[,]𝑁) ∩ ℤ) ∈ V)
32	1, 2, 3, 4, 5	taylfvallem1 25421	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)) ∈ ℂ)
33	32	fmpttd 6971	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))):((0[,]𝑁) ∩ ℤ)⟶ℂ)
34		eqid 2738	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
35	34	cnfldhaus 23854	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) ∈ Haus
36	35	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (TopOpen‘ℂ_fld) ∈ Haus)
37	23, 26, 28, 31, 33, 34, 36	haustsms 23195	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ∃*𝑦 𝑦 ∈ (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))))
38	37	ex 412	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℂ → ∃*𝑦 𝑦 ∈ (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))))
39		moanimv 2621	. . . . . . 7 ⊢ (∃𝑦(𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))) ↔ (𝑥 ∈ ℂ → ∃𝑦 𝑦 ∈ (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))))
40	38, 39	sylibr 233	. . . . . 6 ⊢ (𝜑 → ∃*𝑦(𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))))
41		moim 2544	. . . . . 6 ⊢ (∀𝑦(𝑥𝑇𝑦 → (𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))))) → (∃𝑦(𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))) → ∃𝑦 𝑥𝑇𝑦))
42	22, 40, 41	sylc 65	. . . . 5 ⊢ (𝜑 → ∃*𝑦 𝑥𝑇𝑦)
43	42	alrimiv 1931	. . . 4 ⊢ (𝜑 → ∀𝑥∃*𝑦 𝑥𝑇𝑦)
44		dffun6 6433	. . . 4 ⊢ (Fun 𝑇 ↔ (Rel 𝑇 ∧ ∀𝑥∃*𝑦 𝑥𝑇𝑦))
45	19, 43, 44	sylanbrc 582	. . 3 ⊢ (𝜑 → Fun 𝑇)
46	45	funfnd 6449	. 2 ⊢ (𝜑 → 𝑇 Fn dom 𝑇)
47		rnss 5837	. . . 4 ⊢ (𝑇 ⊆ (ℂ × ℂ) → ran 𝑇 ⊆ ran (ℂ × ℂ))
48	16, 47	syl 17	. . 3 ⊢ (𝜑 → ran 𝑇 ⊆ ran (ℂ × ℂ))
49		rnxpss 6064	. . 3 ⊢ ran (ℂ × ℂ) ⊆ ℂ
50	48, 49	sstrdi 3929	. 2 ⊢ (𝜑 → ran 𝑇 ⊆ ℂ)
51		df-f 6422	. 2 ⊢ (𝑇:dom 𝑇⟶ℂ ↔ (𝑇 Fn dom 𝑇 ∧ ran 𝑇 ⊆ ℂ))
52	46, 50, 51	sylanbrc 582	1 ⊢ (𝜑 → 𝑇:dom 𝑇⟶ℂ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 843 ∀wal 1537 = wceq 1539 ∈ wcel 2108 ∃*wmo 2538 ∀wral 3063 Vcvv 3422 ∩ cin 3882 ⊆ wss 3883 {csn 4558 {cpr 4560 ∪ ciun 4921 class class class wbr 5070 ↦ cmpt 5153 × cxp 5578 dom cdm 5580 ran crn 5581 Rel wrel 5585 Fun wfun 6412 Fn wfn 6413 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 · cmul 10807 +∞cpnf 10937 − cmin 11135 / cdiv 11562 ℕ₀cn0 12163 ℤcz 12249 [,]cicc 13011 ↑cexp 13710 !cfa 13915 TopOpenctopn 17049 CMndccmn 19301 Ringcrg 19698 ℂ_fldccnfld 20510 TopSpctps 21989 Hauscha 22367 tsums ctsu 23185 D^𝑛 cdvn 24933 Tayl ctayl 25417

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-icc 13015 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-fac 13916 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-plusg 16901 df-mulr 16902 df-starv 16903 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-grp 18495 df-minusg 18496 df-cntz 18838 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cnp 22287 df-haus 22374 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-tsms 23186 df-xms 23381 df-ms 23382 df-limc 24935 df-dv 24936 df-dvn 24937 df-tayl 25419

This theorem is referenced by: tayl0 25426

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