taylf - Metamath Proof Explorer

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Theorem taylf 26575

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 26573	. . . . . 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 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
9	8	snssd 4754	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → {𝑥} ⊆ ℂ)
10	1, 2, 3, 4, 5	taylfvallem 26572	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))) ⊆ ℂ)
11		xpss12 5678	. . . . . . . . 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 596	. . . . . . . 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 3159	. . . . . . 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 5011	. . . . . . 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 3972	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ (ℂ × ℂ))
17		relxp 5681	. . . . 5 ⊢ Rel (ℂ × ℂ)
18		relss 5770	. . . . 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 26574	. . . . . . . 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 1960	. . . . . 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 21576	. . . . . . . . 9 ⊢ ℂ = (Base‘ℂ_fld)
24		cnring 21594	. . . . . . . . . 10 ⊢ ℂ_fld ∈ Ring
25		ringcmn 20410	. . . . . . . . . 10 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ CMnd)
26	24, 25	mp1i 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ℂ_fld ∈ CMnd)
27		cnfldtps 24985	. . . . . . . . . 10 ⊢ ℂ_fld ∈ TopSp
28	27	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ℂ_fld ∈ TopSp)
29		ovex 7452	. . . . . . . . . . 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 26571	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)) ∈ ℂ)
33	32	fmpttd 7114	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))):((0[,]𝑁) ∩ ℤ)⟶ℂ)
34		eqid 2765	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
35	34	cnfldhaus 24992	. . . . . . . . . 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 24344	. . . . . . . 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 418	. . . . . . 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 2649	. . . . . . 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 2574	. . . . . 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 1960	. . . 4 ⊢ (𝜑 → ∀𝑥∃*𝑦 𝑥𝑇𝑦)
44		dffun6 6551	. . . 4 ⊢ (Fun 𝑇 ↔ (Rel 𝑇 ∧ ∀𝑥∃*𝑦 𝑥𝑇𝑦))
45	19, 43, 44	sylanbrc 595	. . 3 ⊢ (𝜑 → Fun 𝑇)
46	45	funfnd 6571	. 2 ⊢ (𝜑 → 𝑇 Fn dom 𝑇)
47		rnss 5931	. . . 4 ⊢ (𝑇 ⊆ (ℂ × ℂ) → ran 𝑇 ⊆ ran (ℂ × ℂ))
48	16, 47	syl 18	. . 3 ⊢ (𝜑 → ran 𝑇 ⊆ ran (ℂ × ℂ))
49		rnxpss 6172	. . 3 ⊢ ran (ℂ × ℂ) ⊆ ℂ
50	48, 49	sstrdi 3950	. 2 ⊢ (𝜑 → ran 𝑇 ⊆ ℂ)
51		df-f 6544	. 2 ⊢ (𝑇:dom 𝑇⟶ℂ ↔ (𝑇 Fn dom 𝑇 ∧ ran 𝑇 ⊆ ℂ))
52	46, 50, 51	sylanbrc 595	1 ⊢ (𝜑 → 𝑇:dom 𝑇⟶ℂ)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∨ wo 861 ∀wal 1568 = wceq 1570 ∈ wcel 2146 ∃*wmo 2567 ∀wral 3081 Vcvv 3457 ∩ cin 3905 ⊆ wss 3906 {csn 4591 {cpr 4593 ∪ ciun 4958 class class class wbr 5111 ↦ cmpt 5194 × cxp 5661 dom cdm 5663 ran crn 5664 Rel wrel 5668 Fun wfun 6534 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 ℂcc 11113 ℝcr 11114 0cc0 11115 · cmul 11120 +∞cpnf 11255 − cmin 11456 / cdiv 11886 ℕ₀cn0 12519 ℤcz 12606 [,]cicc 13391 ↑cexp 14115 !cfa 14327 TopOpenctopn 17496 CMndccmn 19894 Ringcrg 20359 ℂ_fldccnfld 21572 TopSpctps 23139 Hauscha 23515 tsums ctsu 24334 D^𝑛 cdvn 26074 Tayl ctayl 26567

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193 ax-addf 11194

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-pm 8833 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-fi 9378 df-sup 9409 df-inf 9410 df-oi 9479 df-card 9941 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12520 df-z 12607 df-dec 12728 df-uz 12879 df-q 12989 df-rp 13033 df-xneg 13153 df-xadd 13154 df-xmul 13155 df-icc 13395 df-fz 13552 df-fzo 13700 df-seq 14056 df-exp 14116 df-fac 14328 df-hash 14385 df-cj 15174 df-re 15175 df-im 15176 df-sqrt 15310 df-abs 15311 df-struct 17229 df-sets 17246 df-slot 17264 df-ndx 17276 df-base 17292 df-plusg 17345 df-mulr 17346 df-starv 17347 df-tset 17351 df-ple 17352 df-ds 17354 df-unif 17355 df-rest 17497 df-topn 17498 df-0g 17516 df-gsum 17517 df-topgen 17518 df-mgm 18720 df-sgrp 18809 df-mnd 18825 df-grp 19047 df-minusg 19048 df-cntz 19431 df-cmn 19896 df-abl 19897 df-mgp 20261 df-ur 20308 df-ring 20361 df-cring 20362 df-psmet 21564 df-xmet 21565 df-met 21566 df-bl 21567 df-mopn 21568 df-fbas 21569 df-fg 21570 df-cnfld 21573 df-top 23101 df-topon 23118 df-topsp 23140 df-bases 23153 df-cld 23226 df-ntr 23227 df-cls 23228 df-nei 23305 df-lp 23343 df-perf 23344 df-cnp 23435 df-haus 23522 df-fil 24054 df-fm 24146 df-flim 24147 df-flf 24148 df-tsms 24335 df-xms 24528 df-ms 24529 df-limc 26076 df-dv 26077 df-dvn 26078 df-tayl 26569

This theorem is used by: tayl0 26576

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