Theorem taylply2 24122

Description: The Taylor polynomial is a polynomial of degree (at most) N. This version of taylply 24123 shows that the coefficients of T are in a subring of the complex numbers. (Contributed by Mario Carneiro, 1-Jan-2017.)

Hypotheses

Ref	Expression
taylpfval.s	|- ( ph -> S e. { RR , CC } )
taylpfval.f	|- ( ph -> F : A --> CC )
taylpfval.a	|- ( ph -> A C_ S )
taylpfval.n	|- ( ph -> N e. NN0 )
taylpfval.b	|- ( ph -> B e. dom ( ( S Dn F ) ` N ) )
taylpfval.t	|- T = ( N ( S Tayl F ) B )
taylply2.1	|- ( ph -> D e. (SubRing ` ℂfld ) )
taylply2.2	|- ( ph -> B e. D )
taylply2.3	|- ( ( ph /\ k e. ( 0 ... N ) ) -> ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) e. D )

Assertion

Ref	Expression
taylply2	|- ( ph -> ( T e. (Poly ` D ) /\ (deg ` T ) <_ N ) )

Distinct variable groups:   B, k   k, F   k, N   ph, k   D, k   S, k

Allowed substitution hints:   A( k)   T( k)

Proof of Theorem taylply2

Dummy variables u v x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		taylpfval.s	. . . . 5 |- ( ph -> S e. { RR , CC } )
2		taylpfval.f	. . . . 5 |- ( ph -> F : A --> CC )
3		taylpfval.a	. . . . 5 |- ( ph -> A C_ S )
4		taylpfval.n	. . . . 5 |- ( ph -> N e. NN0 )
5		taylpfval.b	. . . . 5 |- ( ph -> B e. dom ( ( S Dn F ) ` N ) )
6		taylpfval.t	. . . . 5 |- T = ( N ( S Tayl F ) B )
7	1, 2, 3, 4, 5, 6	taylpfval 24119	. . . 4 |- ( ph -> T = ( x e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) )
8		simpr 477	. . . . . 6 |- ( ( ph /\ x e. CC ) -> x e. CC )
9		cnex 10017	. . . . . . . . . . . . 13 |- CC e. _V
10	9	a1i 11	. . . . . . . . . . . 12 |- ( ph -> CC e. _V )
11		elpm2r 7875	. . . . . . . . . . . 12 |- ( ( ( CC e. _V /\ S e. { RR , CC } ) /\ ( F : A --> CC /\ A C_ S ) ) -> F e. ( CC ^pm S ) )
12	10, 1, 2, 3, 11	syl22anc 1327	. . . . . . . . . . 11 |- ( ph -> F e. ( CC ^pm S ) )
13		dvnbss 23691	. . . . . . . . . . 11 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ N e. NN0 ) -> dom ( ( S Dn F ) ` N ) C_ dom F )
14	1, 12, 4, 13	syl3anc 1326	. . . . . . . . . 10 |- ( ph -> dom ( ( S Dn F ) ` N ) C_ dom F )
15		fdm 6051	. . . . . . . . . . 11 |- ( F : A --> CC -> dom F = A )
16	2, 15	syl 17	. . . . . . . . . 10 |- ( ph -> dom F = A )
17	14, 16	sseqtrd 3641	. . . . . . . . 9 |- ( ph -> dom ( ( S Dn F ) ` N ) C_ A )
18		recnprss 23668	. . . . . . . . . . 11 |- ( S e. { RR , CC } -> S C_ CC )
19	1, 18	syl 17	. . . . . . . . . 10 |- ( ph -> S C_ CC )
20	3, 19	sstrd 3613	. . . . . . . . 9 |- ( ph -> A C_ CC )
21	17, 20	sstrd 3613	. . . . . . . 8 |- ( ph -> dom ( ( S Dn F ) ` N ) C_ CC )
22	21, 5	sseldd 3604	. . . . . . 7 |- ( ph -> B e. CC )
23	22	adantr 481	. . . . . 6 |- ( ( ph /\ x e. CC ) -> B e. CC )
24	8, 23	subcld 10392	. . . . 5 |- ( ( ph /\ x e. CC ) -> ( x - B ) e. CC )
25		df-idp 23945	. . . . . . . 8 |- Xp = ( _I |` CC )
26		mptresid 5456	. . . . . . . 8 |- ( x e. CC |-> x ) = ( _I |` CC )
27	25, 26	eqtr4i 2647	. . . . . . 7 |- Xp = ( x e. CC |-> x )
28	27	a1i 11	. . . . . 6 |- ( ph -> Xp = ( x e. CC |-> x ) )
29		fconstmpt 5163	. . . . . . 7 |- ( CC X. { B } ) = ( x e. CC |-> B )
30	29	a1i 11	. . . . . 6 |- ( ph -> ( CC X. { B } ) = ( x e. CC |-> B ) )
31	10, 8, 23, 28, 30	offval2 6914	. . . . 5 |- ( ph -> ( Xp oF - ( CC X. { B } ) ) = ( x e. CC |-> ( x - B ) ) )
32		eqidd 2623	. . . . 5 |- ( ph -> ( y e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) ) = ( y e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) ) )
33		oveq1 6657	. . . . . . 7 |- ( y = ( x - B ) -> ( y ^ k ) = ( ( x - B ) ^ k ) )
34	33	oveq2d 6666	. . . . . 6 |- ( y = ( x - B ) -> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) = ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) )
35	34	sumeq2sdv 14435	. . . . 5 |- ( y = ( x - B ) -> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) = sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) )
36	24, 31, 32, 35	fmptco 6396	. . . 4 |- ( ph -> ( ( y e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) ) o. ( Xp oF - ( CC X. { B } ) ) ) = ( x e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) )
37	7, 36	eqtr4d 2659	. . 3 |- ( ph -> T = ( ( y e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) ) o. ( Xp oF - ( CC X. { B } ) ) ) )
38		taylply2.1	. . . . . 6 |- ( ph -> D e. (SubRing ` ℂfld ) )
39		cnfldbas 19750	. . . . . . 7 |- CC = ( Base ` ℂfld )
40	39	subrgss 18781	. . . . . 6 |- ( D e. (SubRing ` ℂfld ) -> D C_ CC )
41	38, 40	syl 17	. . . . 5 |- ( ph -> D C_ CC )
42		taylply2.3	. . . . 5 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) e. D )
43	41, 4, 42	elplyd 23958	. . . 4 |- ( ph -> ( y e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) ) e. (Poly ` D ) )
44		cnfld1 19771	. . . . . . . 8 |- 1 = ( 1r ` ℂfld )
45	44	subrg1cl 18788	. . . . . . 7 |- ( D e. (SubRing ` ℂfld ) -> 1 e. D )
46	38, 45	syl 17	. . . . . 6 |- ( ph -> 1 e. D )
47		plyid 23965	. . . . . 6 |- ( ( D C_ CC /\ 1 e. D ) -> Xp e. (Poly ` D ) )
48	41, 46, 47	syl2anc 693	. . . . 5 |- ( ph -> Xp e. (Poly ` D ) )
49		taylply2.2	. . . . . 6 |- ( ph -> B e. D )
50		plyconst 23962	. . . . . 6 |- ( ( D C_ CC /\ B e. D ) -> ( CC X. { B } ) e. (Poly ` D ) )
51	41, 49, 50	syl2anc 693	. . . . 5 |- ( ph -> ( CC X. { B } ) e. (Poly ` D ) )
52		subrgsubg 18786	. . . . . . 7 |- ( D e. (SubRing ` ℂfld ) -> D e. (SubGrp ` ℂfld ) )
53	38, 52	syl 17	. . . . . 6 |- ( ph -> D e. (SubGrp ` ℂfld ) )
54		cnfldadd 19751	. . . . . . . 8 |- + = ( +g ` ℂfld )
55	54	subgcl 17604	. . . . . . 7 |- ( ( D e. (SubGrp ` ℂfld ) /\ u e. D /\ v e. D ) -> ( u + v ) e. D )
56	55	3expb 1266	. . . . . 6 |- ( ( D e. (SubGrp ` ℂfld ) /\ ( u e. D /\ v e. D ) ) -> ( u + v ) e. D )
57	53, 56	sylan 488	. . . . 5 |- ( ( ph /\ ( u e. D /\ v e. D ) ) -> ( u + v ) e. D )
58		cnfldmul 19752	. . . . . . . 8 |- x. = ( .r ` ℂfld )
59	58	subrgmcl 18792	. . . . . . 7 |- ( ( D e. (SubRing ` ℂfld ) /\ u e. D /\ v e. D ) -> ( u x. v ) e. D )
60	59	3expb 1266	. . . . . 6 |- ( ( D e. (SubRing ` ℂfld ) /\ ( u e. D /\ v e. D ) ) -> ( u x. v ) e. D )
61	38, 60	sylan 488	. . . . 5 |- ( ( ph /\ ( u e. D /\ v e. D ) ) -> ( u x. v ) e. D )
62		ax-1cn 9994	. . . . . . 7 |- 1 e. CC
63		cnfldneg 19772	. . . . . . 7 |- ( 1 e. CC -> ( ( invg ` ℂfld ) ` 1 ) = -u 1 )
64	62, 63	ax-mp 5	. . . . . 6 |- ( ( invg ` ℂfld ) ` 1 ) = -u 1
65		eqid 2622	. . . . . . . 8 |- ( invg ` ℂfld ) = ( invg ` ℂfld )
66	65	subginvcl 17603	. . . . . . 7 |- ( ( D e. (SubGrp ` ℂfld ) /\ 1 e. D ) -> ( ( invg ` ℂfld ) ` 1 ) e. D )
67	53, 46, 66	syl2anc 693	. . . . . 6 |- ( ph -> ( ( invg ` ℂfld ) ` 1 ) e. D )
68	64, 67	syl5eqelr 2706	. . . . 5 |- ( ph -> -u 1 e. D )
69	48, 51, 57, 61, 68	plysub 23975	. . . 4 |- ( ph -> ( Xp oF - ( CC X. { B } ) ) e. (Poly ` D ) )
70	43, 69, 57, 61	plyco 23997	. . 3 |- ( ph -> ( ( y e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) ) o. ( Xp oF - ( CC X. { B } ) ) ) e. (Poly ` D ) )
71	37, 70	eqeltrd 2701	. 2 |- ( ph -> T e. (Poly ` D ) )
72	37	fveq2d 6195	. . . 4 |- ( ph -> (deg ` T ) = (deg ` ( ( y e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) ) o. ( Xp oF - ( CC X. { B } ) ) ) ) )
73		eqid 2622	. . . . 5 |- (deg ` ( y e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) ) ) = (deg ` ( y e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) ) )
74		eqid 2622	. . . . 5 |- (deg ` ( Xp oF - ( CC X. { B } ) ) ) = (deg ` ( Xp oF - ( CC X. { B } ) ) )
75	73, 74, 43, 69	dgrco 24031	. . . 4 |- ( ph -> (deg ` ( ( y e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) ) o. ( Xp oF - ( CC X. { B } ) ) ) ) = ( (deg ` ( y e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) ) ) x. (deg ` ( Xp oF - ( CC X. { B } ) ) ) ) )
76		eqid 2622	. . . . . . . . 9 |- ( Xp oF - ( CC X. { B } ) ) = ( Xp oF - ( CC X. { B } ) )
77	76	plyremlem 24059	. . . . . . . 8 |- ( B e. CC -> ( ( Xp oF - ( CC X. { B } ) ) e. (Poly ` CC ) /\ (deg ` ( Xp oF - ( CC X. { B } ) ) ) = 1 /\ ( `' ( Xp oF - ( CC X. { B } ) ) " { 0 } ) = { B } ) )
78	22, 77	syl 17	. . . . . . 7 |- ( ph -> ( ( Xp oF - ( CC X. { B } ) ) e. (Poly ` CC ) /\ (deg ` ( Xp oF - ( CC X. { B } ) ) ) = 1 /\ ( `' ( Xp oF - ( CC X. { B } ) ) " { 0 } ) = { B } ) )
79	78	simp2d 1074	. . . . . 6 |- ( ph -> (deg ` ( Xp oF - ( CC X. { B } ) ) ) = 1 )
80	79	oveq2d 6666	. . . . 5 |- ( ph -> ( (deg ` ( y e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) ) ) x. (deg ` ( Xp oF - ( CC X. { B } ) ) ) ) = ( (deg ` ( y e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) ) ) x. 1 ) )
81		dgrcl 23989	. . . . . . . 8 |- ( ( y e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) ) e. (Poly ` D ) -> (deg ` ( y e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) ) ) e. NN0 )
82	43, 81	syl 17	. . . . . . 7 |- ( ph -> (deg ` ( y e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) ) ) e. NN0 )
83	82	nn0cnd 11353	. . . . . 6 |- ( ph -> (deg ` ( y e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) ) ) e. CC )
84	83	mulid1d 10057	. . . . 5 |- ( ph -> ( (deg ` ( y e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) ) ) x. 1 ) = (deg ` ( y e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) ) ) )
85	80, 84	eqtrd 2656	. . . 4 |- ( ph -> ( (deg ` ( y e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) ) ) x. (deg ` ( Xp oF - ( CC X. { B } ) ) ) ) = (deg ` ( y e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) ) ) )
86	72, 75, 85	3eqtrd 2660	. . 3 |- ( ph -> (deg ` T ) = (deg ` ( y e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) ) ) )
87	1	adantr 481	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... N ) ) -> S e. { RR , CC } )
88	12	adantr 481	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... N ) ) -> F e. ( CC ^pm S ) )
89		elfznn0 12433	. . . . . . . 8 |- ( k e. ( 0 ... N ) -> k e. NN0 )
90	89	adantl 482	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... N ) ) -> k e. NN0 )
91		dvnf 23690	. . . . . . 7 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ k e. NN0 ) -> ( ( S Dn F ) ` k ) : dom ( ( S Dn F ) ` k ) --> CC )
92	87, 88, 90, 91	syl3anc 1326	. . . . . 6 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( ( S Dn F ) ` k ) : dom ( ( S Dn F ) ` k ) --> CC )
93		simpr 477	. . . . . . . 8 |- ( ( ph /\ k e. ( 0 ... N ) ) -> k e. ( 0 ... N ) )
94		dvn2bss 23693	. . . . . . . 8 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) /\ k e. ( 0 ... N ) ) -> dom ( ( S Dn F ) ` N ) C_ dom ( ( S Dn F ) ` k ) )
95	87, 88, 93, 94	syl3anc 1326	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... N ) ) -> dom ( ( S Dn F ) ` N ) C_ dom ( ( S Dn F ) ` k ) )
96	5	adantr 481	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... N ) ) -> B e. dom ( ( S Dn F ) ` N ) )
97	95, 96	sseldd 3604	. . . . . 6 |- ( ( ph /\ k e. ( 0 ... N ) ) -> B e. dom ( ( S Dn F ) ` k ) )
98	92, 97	ffvelrnd 6360	. . . . 5 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( ( ( S Dn F ) ` k ) ` B ) e. CC )
99		faccl 13070	. . . . . . 7 |- ( k e. NN0 -> ( ! ` k ) e. NN )
100	90, 99	syl 17	. . . . . 6 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( ! ` k ) e. NN )
101	100	nncnd 11036	. . . . 5 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( ! ` k ) e. CC )
102	100	nnne0d 11065	. . . . 5 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( ! ` k ) =/= 0 )
103	98, 101, 102	divcld 10801	. . . 4 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) e. CC )
104	43, 4, 103, 32	dgrle 23999	. . 3 |- ( ph -> (deg ` ( y e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( y ^ k ) ) ) ) <_ N )
105	86, 104	eqbrtrd 4675	. 2 |- ( ph -> (deg ` T ) <_ N )
106	71, 105	jca 554	1 |- ( ph -> ( T e. (Poly ` D ) /\ (deg ` T ) <_ N ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {csn 4177   {cpr 4179   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   X. cxp 5112   `'ccnv 5113   dom cdm 5114   |` cres 5116   "cima 5117   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   ^pm cpm 7858   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   NN0cn0 11292   ...cfz 12326   ^cexp 12860   !cfa 13060   sum_csu 14416   invgcminusg 17423  SubGrpcsubg 17588  SubRingcsubrg 18776  ℂ_fldccnfld 19746   Dncdvn 23628  Polycply 23940   Xpcidp 23941  degcdgr 23943   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-fal 1489  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-of 6897  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-fl 12593  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-clim 14219  df-rlim 14220  df-sum 14417  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  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-subg 17591  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-subrg 18778  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-0p 23437  df-limc 23630  df-dv 23631  df-dvn 23632  df-ply 23944  df-idp 23945  df-coe 23946  df-dgr 23947  df-tayl 24109

This theorem is referenced by:  taylply  24123  taylthlem2  24128