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Theorem taylf 26489

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 26487	. . . . . 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 489	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
9	8	snssd 4757	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → {𝑥} ⊆ ℂ)
10	1, 2, 3, 4, 5	taylfvallem 26486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))) ⊆ ℂ)
11		xpss12 5677	. . . . . . . . 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 595	. . . . . . . 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 3163	. . . . . . 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 5013	. . . . . . 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 237	. . . . . 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 3979	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ (ℂ × ℂ))
17		relxp 5680	. . . . 5 ⊢ Rel (ℂ × ℂ)
18		relss 5769	. . . . 5 ⊢ (𝑇 ⊆ (ℂ × ℂ) → (Rel (ℂ × ℂ) → Rel 𝑇))
19	16, 17, 18	mpisyl 22	. . . 4 ⊢ (𝜑 → Rel 𝑇)
20	1, 2, 3, 4, 5, 6	eltayl 26488	. . . . . . . 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 232	. . . . . . 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 1954	. . . . . 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 21494	. . . . . . . . 9 ⊢ ℂ = (Base‘ℂ_fld)
24		cnring 21512	. . . . . . . . . 10 ⊢ ℂ_fld ∈ Ring
25		ringcmn 20364	. . . . . . . . . 10 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ CMnd)
26	24, 25	mp1i 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ℂ_fld ∈ CMnd)
27		cnfldtps 24902	. . . . . . . . . 10 ⊢ ℂ_fld ∈ TopSp
28	27	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ℂ_fld ∈ TopSp)
29		ovex 7444	. . . . . . . . . . 11 ⊢ (0[,]𝑁) ∈ V
30	29	inex1 5288	. . . . . . . . . 10 ⊢ ((0[,]𝑁) ∩ ℤ) ∈ V
31	30	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((0[,]𝑁) ∩ ℤ) ∈ V)
32	1, 2, 3, 4, 5	taylfvallem1 26485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)) ∈ ℂ)
33	32	fmpttd 7111	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))):((0[,]𝑁) ∩ ℤ)⟶ℂ)
34		eqid 2769	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
35	34	cnfldhaus 24909	. . . . . . . . . 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 24261	. . . . . . . 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 417	. . . . . . 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 2653	. . . . . . 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 237	. . . . . 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 2578	. . . . . 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 66	. . . . 5 ⊢ (𝜑 → ∃*𝑦 𝑥𝑇𝑦)
43	42	alrimiv 1954	. . . 4 ⊢ (𝜑 → ∀𝑥∃*𝑦 𝑥𝑇𝑦)
44		dffun6 6548	. . . 4 ⊢ (Fun 𝑇 ↔ (Rel 𝑇 ∧ ∀𝑥∃*𝑦 𝑥𝑇𝑦))
45	19, 43, 44	sylanbrc 594	. . 3 ⊢ (𝜑 → Fun 𝑇)
46	45	funfnd 6568	. 2 ⊢ (𝜑 → 𝑇 Fn dom 𝑇)
47		rnss 5930	. . . 4 ⊢ (𝑇 ⊆ (ℂ × ℂ) → ran 𝑇 ⊆ ran (ℂ × ℂ))
48	16, 47	syl 18	. . 3 ⊢ (𝜑 → ran 𝑇 ⊆ ran (ℂ × ℂ))
49		rnxpss 6171	. . 3 ⊢ ran (ℂ × ℂ) ⊆ ℂ
50	48, 49	sstrdi 3957	. 2 ⊢ (𝜑 → ran 𝑇 ⊆ ℂ)
51		df-f 6541	. 2 ⊢ (𝑇:dom 𝑇⟶ℂ ↔ (𝑇 Fn dom 𝑇 ∧ ran 𝑇 ⊆ ℂ))
52	46, 50, 51	sylanbrc 594	1 ⊢ (𝜑 → 𝑇:dom 𝑇⟶ℂ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∨ wo 860 ∀wal 1565 = wceq 1567 ∈ wcel 2149 ∃*wmo 2571 ∀wral 3085 Vcvv 3463 ∩ cin 3912 ⊆ wss 3913 {csn 4594 {cpr 4596 ∪ ciun 4960 class class class wbr 5113 ↦ cmpt 5196 × cxp 5660 dom cdm 5662 ran crn 5663 Rel wrel 5667 Fun wfun 6531 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 ℂcc 11097 ℝcr 11098 0cc0 11099 · cmul 11104 +∞cpnf 11239 − cmin 11440 / cdiv 11870 ℕ₀cn0 12503 ℤcz 12590 [,]cicc 13374 ↑cexp 14096 !cfa 14308 TopOpenctopn 17473 CMndccmn 19849 Ringcrg 20314 ℂ_fldccnfld 21490 TopSpctps 23057 Hauscha 23433 tsums ctsu 24251 D^𝑛 cdvn 25991 Tayl ctayl 26481

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9609 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177 ax-addf 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 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 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8156 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-er 8693 df-map 8825 df-pm 8826 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9321 df-fi 9370 df-sup 9401 df-inf 9402 df-oi 9471 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-z 12591 df-dec 12711 df-uz 12862 df-q 12972 df-rp 13016 df-xneg 13136 df-xadd 13137 df-xmul 13138 df-icc 13378 df-fz 13535 df-fzo 13682 df-seq 14037 df-exp 14097 df-fac 14309 df-hash 14366 df-cj 15149 df-re 15150 df-im 15151 df-sqrt 15285 df-abs 15286 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-plusg 17322 df-mulr 17323 df-starv 17324 df-tset 17328 df-ple 17329 df-ds 17331 df-unif 17332 df-rest 17474 df-topn 17475 df-0g 17493 df-gsum 17494 df-topgen 17495 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-grp 19002 df-minusg 19003 df-cntz 19386 df-cmn 19851 df-abl 19852 df-mgp 20216 df-ur 20263 df-ring 20316 df-cring 20317 df-psmet 21482 df-xmet 21483 df-met 21484 df-bl 21485 df-mopn 21486 df-fbas 21487 df-fg 21488 df-cnfld 21491 df-top 23019 df-topon 23036 df-topsp 23058 df-bases 23071 df-cld 23144 df-ntr 23145 df-cls 23146 df-nei 23223 df-lp 23261 df-perf 23262 df-cnp 23353 df-haus 23440 df-fil 23971 df-fm 24063 df-flim 24064 df-flf 24065 df-tsms 24252 df-xms 24445 df-ms 24446 df-limc 25993 df-dv 25994 df-dvn 25995 df-tayl 26483

This theorem is referenced by: tayl0 26490

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