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Theorem taylf 26322

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 26320	. . . . . 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 4763	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → {𝑥} ⊆ ℂ)
10	1, 2, 3, 4, 5	taylfvallem 26319	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))) ⊆ ℂ)
11		xpss12 5637	. . . . . . . . 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 584	. . . . . . . 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 3126	. . . . . . 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 4998	. . . . . . 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 234	. . . . . 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 3966	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ (ℂ × ℂ))
17		relxp 5640	. . . . 5 ⊢ Rel (ℂ × ℂ)
18		relss 5729	. . . . 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 26321	. . . . . . . 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 229	. . . . . . 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 1928	. . . . . 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 21311	. . . . . . . . 9 ⊢ ℂ = (Base‘ℂ_fld)
24		cnring 21343	. . . . . . . . . 10 ⊢ ℂ_fld ∈ Ring
25		ringcmn 20215	. . . . . . . . . 10 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ CMnd)
26	24, 25	mp1i 13	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ℂ_fld ∈ CMnd)
27		cnfldtps 24719	. . . . . . . . . 10 ⊢ ℂ_fld ∈ TopSp
28	27	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ℂ_fld ∈ TopSp)
29		ovex 7389	. . . . . . . . . . 11 ⊢ (0[,]𝑁) ∈ V
30	29	inex1 5260	. . . . . . . . . 10 ⊢ ((0[,]𝑁) ∩ ℤ) ∈ V
31	30	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((0[,]𝑁) ∩ ℤ) ∈ V)
32	1, 2, 3, 4, 5	taylfvallem1 26318	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)) ∈ ℂ)
33	32	fmpttd 7058	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))):((0[,]𝑁) ∩ ℤ)⟶ℂ)
34		eqid 2734	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
35	34	cnfldhaus 24726	. . . . . . . . . 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 24078	. . . . . . . 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 2617	. . . . . . 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 234	. . . . . 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 2542	. . . . . 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 1928	. . . 4 ⊢ (𝜑 → ∀𝑥∃*𝑦 𝑥𝑇𝑦)
44		dffun6 6501	. . . 4 ⊢ (Fun 𝑇 ↔ (Rel 𝑇 ∧ ∀𝑥∃*𝑦 𝑥𝑇𝑦))
45	19, 43, 44	sylanbrc 583	. . 3 ⊢ (𝜑 → Fun 𝑇)
46	45	funfnd 6521	. 2 ⊢ (𝜑 → 𝑇 Fn dom 𝑇)
47		rnss 5886	. . . 4 ⊢ (𝑇 ⊆ (ℂ × ℂ) → ran 𝑇 ⊆ ran (ℂ × ℂ))
48	16, 47	syl 17	. . 3 ⊢ (𝜑 → ran 𝑇 ⊆ ran (ℂ × ℂ))
49		rnxpss 6128	. . 3 ⊢ ran (ℂ × ℂ) ⊆ ℂ
50	48, 49	sstrdi 3944	. 2 ⊢ (𝜑 → ran 𝑇 ⊆ ℂ)
51		df-f 6494	. 2 ⊢ (𝑇:dom 𝑇⟶ℂ ↔ (𝑇 Fn dom 𝑇 ∧ ran 𝑇 ⊆ ℂ))
52	46, 50, 51	sylanbrc 583	1 ⊢ (𝜑 → 𝑇:dom 𝑇⟶ℂ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 ∀wal 1539 = wceq 1541 ∈ wcel 2113 ∃*wmo 2535 ∀wral 3049 Vcvv 3438 ∩ cin 3898 ⊆ wss 3899 {csn 4578 {cpr 4580 ∪ ciun 4944 class class class wbr 5096 ↦ cmpt 5177 × cxp 5620 dom cdm 5622 ran crn 5623 Rel wrel 5627 Fun wfun 6484 Fn wfn 6485 ⟶wf 6486 ‘cfv 6490 (class class class)co 7356 ℂcc 11022 ℝcr 11023 0cc0 11024 · cmul 11029 +∞cpnf 11161 − cmin 11362 / cdiv 11792 ℕ₀cn0 12399 ℤcz 12486 [,]cicc 13262 ↑cexp 13982 !cfa 14194 TopOpenctopn 17339 CMndccmn 19707 Ringcrg 20166 ℂ_fldccnfld 21307 TopSpctps 22874 Hauscha 23250 tsums ctsu 24068 D^𝑛 cdvn 25819 Tayl ctayl 26314

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102 ax-addf 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8763 df-pm 8764 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-fi 9312 df-sup 9343 df-inf 9344 df-oi 9413 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-q 12860 df-rp 12904 df-xneg 13024 df-xadd 13025 df-xmul 13026 df-icc 13266 df-fz 13422 df-fzo 13569 df-seq 13923 df-exp 13983 df-fac 14195 df-hash 14252 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-plusg 17188 df-mulr 17189 df-starv 17190 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-rest 17340 df-topn 17341 df-0g 17359 df-gsum 17360 df-topgen 17361 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18864 df-minusg 18865 df-cntz 19244 df-cmn 19709 df-abl 19710 df-mgp 20074 df-ur 20115 df-ring 20168 df-cring 20169 df-psmet 21299 df-xmet 21300 df-met 21301 df-bl 21302 df-mopn 21303 df-fbas 21304 df-fg 21305 df-cnfld 21308 df-top 22836 df-topon 22853 df-topsp 22875 df-bases 22888 df-cld 22961 df-ntr 22962 df-cls 22963 df-nei 23040 df-lp 23078 df-perf 23079 df-cnp 23170 df-haus 23257 df-fil 23788 df-fm 23880 df-flim 23881 df-flf 23882 df-tsms 24069 df-xms 24262 df-ms 24263 df-limc 25821 df-dv 25822 df-dvn 25823 df-tayl 26316

This theorem is referenced by: tayl0 26323

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