Theorem taylf 24115

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	|- ( ph -> S e. { RR , CC } )
taylfval.f	|- ( ph -> F : A --> CC )
taylfval.a	|- ( ph -> A C_ S )
taylfval.n	|- ( ph -> ( N e. NN0 \/ N = +oo ) )
taylfval.b	|- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> B e. dom ( ( S Dn F ) ` k ) )
taylfval.t	|- T = ( N ( S Tayl F ) B )

Assertion

Ref	Expression
taylf	|- ( ph -> T : dom T --> CC )

Distinct variable groups:   B, k   k, F   ph, 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 |- ( ph -> S e. { RR , CC } )
2		taylfval.f	. . . . . . 7 |- ( ph -> F : A --> CC )
3		taylfval.a	. . . . . . 7 |- ( ph -> A C_ S )
4		taylfval.n	. . . . . . 7 |- ( ph -> ( N e. NN0 \/ N = +oo ) )
5		taylfval.b	. . . . . . 7 |- ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> B e. dom ( ( S Dn F ) ` k ) )
6		taylfval.t	. . . . . . 7 |- T = ( N ( S Tayl F ) B )
7	1, 2, 3, 4, 5, 6	taylfval 24113	. . . . . 6 |- ( ph -> T = U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) )
8		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ x e. CC ) -> x e. CC )
9	8	snssd 4340	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> { x } C_ CC )
10	1, 2, 3, 4, 5	taylfvallem 24112	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) C_ CC )
11		xpss12 5225	. . . . . . . . 9 |- ( ( { x } C_ CC /\ (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) C_ CC ) -> ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) C_ ( CC X. CC ) )
12	9, 10, 11	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ x e. CC ) -> ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) C_ ( CC X. CC ) )
13	12	ralrimiva 2966	. . . . . . 7 |- ( ph -> A. x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) C_ ( CC X. CC ) )
14		iunss 4561	. . . . . . 7 |- ( U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) C_ ( CC X. CC ) <-> A. x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) C_ ( CC X. CC ) )
15	13, 14	sylibr 224	. . . . . 6 |- ( ph -> U_ x e. CC ( { x } X. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) C_ ( CC X. CC ) )
16	7, 15	eqsstrd 3639	. . . . 5 |- ( ph -> T C_ ( CC X. CC ) )
17		relxp 5227	. . . . 5 |- Rel ( CC X. CC )
18		relss 5206	. . . . 5 |- ( T C_ ( CC X. CC ) -> ( Rel ( CC X. CC ) -> Rel T ) )
19	16, 17, 18	mpisyl 21	. . . 4 |- ( ph -> Rel T )
20	1, 2, 3, 4, 5, 6	eltayl 24114	. . . . . . . 8 |- ( ph -> ( x T y <-> ( x e. CC /\ y e. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) ) )
21	20	biimpd 219	. . . . . . 7 |- ( ph -> ( x T y -> ( x e. CC /\ y e. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) ) )
22	21	alrimiv 1855	. . . . . 6 |- ( ph -> A. y ( x T y -> ( x e. CC /\ y e. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) ) )
23		cnfldbas 19750	. . . . . . . . 9 |- CC = ( Base ` ℂfld )
24		cnring 19768	. . . . . . . . . 10 |- ℂfld e. Ring
25		ringcmn 18581	. . . . . . . . . 10 |- (ℂfld e. Ring -> ℂfld e. CMnd )
26	24, 25	mp1i 13	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> ℂfld e. CMnd )
27		cnfldtps 22581	. . . . . . . . . 10 |- ℂfld e. TopSp
28	27	a1i 11	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> ℂfld e. TopSp )
29		ovex 6678	. . . . . . . . . . 11 |- ( 0 [,] N ) e. _V
30	29	inex1 4799	. . . . . . . . . 10 |- ( ( 0 [,] N ) i^i ZZ ) e. _V
31	30	a1i 11	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> ( ( 0 [,] N ) i^i ZZ ) e. _V )
32	1, 2, 3, 4, 5	taylfvallem1 24111	. . . . . . . . . 10 |- ( ( ( ph /\ x e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) e. CC )
33		eqid 2622	. . . . . . . . . 10 |- ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) = ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) )
34	32, 33	fmptd 6385	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) : ( ( 0 [,] N ) i^i ZZ ) --> CC )
35		eqid 2622	. . . . . . . . 9 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
36	35	cnfldhaus 22588	. . . . . . . . . 10 |- ( TopOpen ` ℂfld ) e. Haus
37	36	a1i 11	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> ( TopOpen ` ℂfld ) e. Haus )
38	23, 26, 28, 31, 34, 35, 37	haustsms 21939	. . . . . . . 8 |- ( ( ph /\ x e. CC ) -> E* y y e. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) )
39	38	ex 450	. . . . . . 7 |- ( ph -> ( x e. CC -> E* y y e. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) )
40		moanimv 2531	. . . . . . 7 |- ( E* y ( x e. CC /\ y e. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) <-> ( x e. CC -> E* y y e. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) )
41	39, 40	sylibr 224	. . . . . 6 |- ( ph -> E* y ( x e. CC /\ y e. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) )
42		moim 2519	. . . . . 6 |- ( A. y ( x T y -> ( x e. CC /\ y e. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) ) -> ( E* y ( x e. CC /\ y e. (ℂfld tsums ( k e. ( ( 0 [,] N ) i^i ZZ ) |-> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) ) ) -> E* y x T y ) )
43	22, 41, 42	sylc 65	. . . . 5 |- ( ph -> E* y x T y )
44	43	alrimiv 1855	. . . 4 |- ( ph -> A. x E* y x T y )
45		dffun6 5903	. . . 4 |- ( Fun T <-> ( Rel T /\ A. x E* y x T y ) )
46	19, 44, 45	sylanbrc 698	. . 3 |- ( ph -> Fun T )
47		funfn 5918	. . 3 |- ( Fun T <-> T Fn dom T )
48	46, 47	sylib 208	. 2 |- ( ph -> T Fn dom T )
49		rnss 5354	. . . 4 |- ( T C_ ( CC X. CC ) -> ran T C_ ran ( CC X. CC ) )
50	16, 49	syl 17	. . 3 |- ( ph -> ran T C_ ran ( CC X. CC ) )
51		rnxpss 5566	. . 3 |- ran ( CC X. CC ) C_ CC
52	50, 51	syl6ss 3615	. 2 |- ( ph -> ran T C_ CC )
53		df-f 5892	. 2 |- ( T : dom T --> CC <-> ( T Fn dom T /\ ran T C_ CC ) )
54	48, 52, 53	sylanbrc 698	1 |- ( ph -> T : dom T --> CC )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   E*wmo 2471   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   {csn 4177   {cpr 4179   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ran crn 5115   Rel wrel 5119   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   x. cmul 9941   +oocpnf 10071   - cmin 10266   / cdiv 10684   NN0cn0 11292   ZZcz 11377   [,]cicc 12178   ^cexp 12860   !cfa 13060   TopOpenctopn 16082  CMndccmn 18193   Ringcrg 18547  ℂ_fldccnfld 19746   TopSpctps 20736   Hauscha 21112   tsums ctsu 21929   Dncdvn 23628   Tayl ctayl 24107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-fac 13061  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cnp 21032  df-haus 21119  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-tsms 21930  df-xms 22125  df-ms 22126  df-limc 23630  df-dv 23631  df-dvn 23632  df-tayl 24109

This theorem is referenced by:  tayl0  24116