Theorem pserdv 24183

Description: The derivative of a power series on its region of convergence. (Contributed by Mario Carneiro, 31-Mar-2015.)

Hypotheses

Ref	Expression
pserf.g	|- G = ( x e. CC |-> ( n e. NN0 |-> ( ( A ` n ) x. ( x ^ n ) ) ) )
pserf.f	|- F = ( y e. S |-> sum_ j e. NN0 ( ( G ` y ) ` j ) )
pserf.a	|- ( ph -> A : NN0 --> CC )
pserf.r	|- R = sup ( { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } , RR* , < )
psercn.s	|- S = ( `' abs " ( 0 [,) R ) )
psercn.m	|- M = if ( R e. RR , ( ( ( abs ` a ) + R ) / 2 ) , ( ( abs ` a ) + 1 ) )
pserdv.b	|- B = ( 0 ( ball ` ( abs o. - ) ) ( ( ( abs ` a ) + M ) / 2 ) )

Assertion

Ref	Expression
pserdv	|- ( ph -> ( CC _D F ) = ( y e. S |-> sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( y ^ k ) ) ) )

Distinct variable groups:   j, a, k, n, r, x, y, A   j, M, k, y   B, j, k, x, y   j, G, k, r, y   S, a, j, k, y   F, a   ph, a, j, k, y

Allowed substitution hints:   ph( x, n, r)   B( n, r, a)   R( x, y, j, k, n, r, a)   S( x, n, r)   F( x, y, j, k, n, r)   G( x, n, a)   M( x, n, r, a)

Proof of Theorem pserdv

Step	Hyp	Ref	Expression
1		dvfcn 23672	. . . . 5 |- ( CC _D F ) : dom ( CC _D F ) --> CC
2		ssid 3624	. . . . . . . . 9 |- CC C_ CC
3	2	a1i 11	. . . . . . . 8 |- ( ph -> CC C_ CC )
4		pserf.g	. . . . . . . . . 10 |- G = ( x e. CC |-> ( n e. NN0 |-> ( ( A ` n ) x. ( x ^ n ) ) ) )
5		pserf.f	. . . . . . . . . 10 |- F = ( y e. S |-> sum_ j e. NN0 ( ( G ` y ) ` j ) )
6		pserf.a	. . . . . . . . . 10 |- ( ph -> A : NN0 --> CC )
7		pserf.r	. . . . . . . . . 10 |- R = sup ( { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } , RR* , < )
8		psercn.s	. . . . . . . . . 10 |- S = ( `' abs " ( 0 [,) R ) )
9		psercn.m	. . . . . . . . . 10 |- M = if ( R e. RR , ( ( ( abs ` a ) + R ) / 2 ) , ( ( abs ` a ) + 1 ) )
10	4, 5, 6, 7, 8, 9	psercn 24180	. . . . . . . . 9 |- ( ph -> F e. ( S -cn-> CC ) )
11		cncff 22696	. . . . . . . . 9 |- ( F e. ( S -cn-> CC ) -> F : S --> CC )
12	10, 11	syl 17	. . . . . . . 8 |- ( ph -> F : S --> CC )
13		cnvimass 5485	. . . . . . . . . . 11 |- ( `' abs " ( 0 [,) R ) ) C_ dom abs
14		absf 14077	. . . . . . . . . . . 12 |- abs : CC --> RR
15	14	fdmi 6052	. . . . . . . . . . 11 |- dom abs = CC
16	13, 15	sseqtri 3637	. . . . . . . . . 10 |- ( `' abs " ( 0 [,) R ) ) C_ CC
17	8, 16	eqsstri 3635	. . . . . . . . 9 |- S C_ CC
18	17	a1i 11	. . . . . . . 8 |- ( ph -> S C_ CC )
19	3, 12, 18	dvbss 23665	. . . . . . 7 |- ( ph -> dom ( CC _D F ) C_ S )
20	2	a1i 11	. . . . . . . . . . . . 13 |- ( ( ph /\ a e. S ) -> CC C_ CC )
21	12	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ a e. S ) -> F : S --> CC )
22	17	a1i 11	. . . . . . . . . . . . 13 |- ( ( ph /\ a e. S ) -> S C_ CC )
23		pserdv.b	. . . . . . . . . . . . . 14 |- B = ( 0 ( ball ` ( abs o. - ) ) ( ( ( abs ` a ) + M ) / 2 ) )
24		cnxmet 22576	. . . . . . . . . . . . . . . 16 |- ( abs o. - ) e. ( *Met ` CC )
25	24	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( ph /\ a e. S ) -> ( abs o. - ) e. ( *Met ` CC ) )
26		0cnd 10033	. . . . . . . . . . . . . . 15 |- ( ( ph /\ a e. S ) -> 0 e. CC )
27	18	sselda 3603	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ a e. S ) -> a e. CC )
28	27	abscld 14175	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ a e. S ) -> ( abs ` a ) e. RR )
29	4, 5, 6, 7, 8, 9	psercnlem1 24179	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ a e. S ) -> ( M e. RR+ /\ ( abs ` a ) < M /\ M < R ) )
30	29	simp1d 1073	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ a e. S ) -> M e. RR+ )
31	30	rpred 11872	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ a e. S ) -> M e. RR )
32	28, 31	readdcld 10069	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ a e. S ) -> ( ( abs ` a ) + M ) e. RR )
33		0red 10041	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ a e. S ) -> 0 e. RR )
34	27	absge0d 14183	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ a e. S ) -> 0 <_ ( abs ` a ) )
35	28, 30	ltaddrpd 11905	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ a e. S ) -> ( abs ` a ) < ( ( abs ` a ) + M ) )
36	33, 28, 32, 34, 35	lelttrd 10195	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ a e. S ) -> 0 < ( ( abs ` a ) + M ) )
37	32, 36	elrpd 11869	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ a e. S ) -> ( ( abs ` a ) + M ) e. RR+ )
38	37	rphalfcld 11884	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ a e. S ) -> ( ( ( abs ` a ) + M ) / 2 ) e. RR+ )
39	38	rpxrd 11873	. . . . . . . . . . . . . . 15 |- ( ( ph /\ a e. S ) -> ( ( ( abs ` a ) + M ) / 2 ) e. RR* )
40		blssm 22223	. . . . . . . . . . . . . . 15 |- ( ( ( abs o. - ) e. ( *Met ` CC ) /\ 0 e. CC /\ ( ( ( abs ` a ) + M ) / 2 ) e. RR* ) -> ( 0 ( ball ` ( abs o. - ) ) ( ( ( abs ` a ) + M ) / 2 ) ) C_ CC )
41	25, 26, 39, 40	syl3anc 1326	. . . . . . . . . . . . . 14 |- ( ( ph /\ a e. S ) -> ( 0 ( ball ` ( abs o. - ) ) ( ( ( abs ` a ) + M ) / 2 ) ) C_ CC )
42	23, 41	syl5eqss 3649	. . . . . . . . . . . . 13 |- ( ( ph /\ a e. S ) -> B C_ CC )
43		eqid 2622	. . . . . . . . . . . . . 14 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
44	43	cnfldtop 22587	. . . . . . . . . . . . . . . 16 |- ( TopOpen ` ℂfld ) e. Top
45	43	cnfldtopon 22586	. . . . . . . . . . . . . . . . . 18 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
46	45	toponunii 20721	. . . . . . . . . . . . . . . . 17 |- CC = U. ( TopOpen ` ℂfld )
47	46	restid 16094	. . . . . . . . . . . . . . . 16 |- ( ( TopOpen ` ℂfld ) e. Top -> ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld ) )
48	44, 47	ax-mp 5	. . . . . . . . . . . . . . 15 |- ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld )
49	48	eqcomi 2631	. . . . . . . . . . . . . 14 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
50	43, 49	dvres 23675	. . . . . . . . . . . . 13 |- ( ( ( CC C_ CC /\ F : S --> CC ) /\ ( S C_ CC /\ B C_ CC ) ) -> ( CC _D ( F |` B ) ) = ( ( CC _D F ) |` ( ( int ` ( TopOpen ` ℂfld ) ) ` B ) ) )
51	20, 21, 22, 42, 50	syl22anc 1327	. . . . . . . . . . . 12 |- ( ( ph /\ a e. S ) -> ( CC _D ( F |` B ) ) = ( ( CC _D F ) |` ( ( int ` ( TopOpen ` ℂfld ) ) ` B ) ) )
52		resss 5422	. . . . . . . . . . . 12 |- ( ( CC _D F ) |` ( ( int ` ( TopOpen ` ℂfld ) ) ` B ) ) C_ ( CC _D F )
53	51, 52	syl6eqss 3655	. . . . . . . . . . 11 |- ( ( ph /\ a e. S ) -> ( CC _D ( F |` B ) ) C_ ( CC _D F ) )
54		dmss 5323	. . . . . . . . . . 11 |- ( ( CC _D ( F |` B ) ) C_ ( CC _D F ) -> dom ( CC _D ( F |` B ) ) C_ dom ( CC _D F ) )
55	53, 54	syl 17	. . . . . . . . . 10 |- ( ( ph /\ a e. S ) -> dom ( CC _D ( F |` B ) ) C_ dom ( CC _D F ) )
56	4, 5, 6, 7, 8, 9	pserdvlem1 24181	. . . . . . . . . . . . . 14 |- ( ( ph /\ a e. S ) -> ( ( ( ( abs ` a ) + M ) / 2 ) e. RR+ /\ ( abs ` a ) < ( ( ( abs ` a ) + M ) / 2 ) /\ ( ( ( abs ` a ) + M ) / 2 ) < R ) )
57	4, 5, 6, 7, 8, 56	psercnlem2 24178	. . . . . . . . . . . . 13 |- ( ( ph /\ a e. S ) -> ( a e. ( 0 ( ball ` ( abs o. - ) ) ( ( ( abs ` a ) + M ) / 2 ) ) /\ ( 0 ( ball ` ( abs o. - ) ) ( ( ( abs ` a ) + M ) / 2 ) ) C_ ( `' abs " ( 0 [,] ( ( ( abs ` a ) + M ) / 2 ) ) ) /\ ( `' abs " ( 0 [,] ( ( ( abs ` a ) + M ) / 2 ) ) ) C_ S ) )
58	57	simp1d 1073	. . . . . . . . . . . 12 |- ( ( ph /\ a e. S ) -> a e. ( 0 ( ball ` ( abs o. - ) ) ( ( ( abs ` a ) + M ) / 2 ) ) )
59	58, 23	syl6eleqr 2712	. . . . . . . . . . 11 |- ( ( ph /\ a e. S ) -> a e. B )
60	4, 5, 6, 7, 8, 9, 23	pserdvlem2 24182	. . . . . . . . . . . . 13 |- ( ( ph /\ a e. S ) -> ( CC _D ( F |` B ) ) = ( y e. B |-> sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( y ^ k ) ) ) )
61	60	dmeqd 5326	. . . . . . . . . . . 12 |- ( ( ph /\ a e. S ) -> dom ( CC _D ( F |` B ) ) = dom ( y e. B |-> sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( y ^ k ) ) ) )
62		dmmptg 5632	. . . . . . . . . . . . 13 |- ( A. y e. B sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( y ^ k ) ) e. _V -> dom ( y e. B |-> sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( y ^ k ) ) ) = B )
63		sumex 14418	. . . . . . . . . . . . . 14 |- sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( y ^ k ) ) e. _V
64	63	a1i 11	. . . . . . . . . . . . 13 |- ( y e. B -> sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( y ^ k ) ) e. _V )
65	62, 64	mprg 2926	. . . . . . . . . . . 12 |- dom ( y e. B |-> sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( y ^ k ) ) ) = B
66	61, 65	syl6eq 2672	. . . . . . . . . . 11 |- ( ( ph /\ a e. S ) -> dom ( CC _D ( F |` B ) ) = B )
67	59, 66	eleqtrrd 2704	. . . . . . . . . 10 |- ( ( ph /\ a e. S ) -> a e. dom ( CC _D ( F |` B ) ) )
68	55, 67	sseldd 3604	. . . . . . . . 9 |- ( ( ph /\ a e. S ) -> a e. dom ( CC _D F ) )
69	68	ex 450	. . . . . . . 8 |- ( ph -> ( a e. S -> a e. dom ( CC _D F ) ) )
70	69	ssrdv 3609	. . . . . . 7 |- ( ph -> S C_ dom ( CC _D F ) )
71	19, 70	eqssd 3620	. . . . . 6 |- ( ph -> dom ( CC _D F ) = S )
72	71	feq2d 6031	. . . . 5 |- ( ph -> ( ( CC _D F ) : dom ( CC _D F ) --> CC <-> ( CC _D F ) : S --> CC ) )
73	1, 72	mpbii 223	. . . 4 |- ( ph -> ( CC _D F ) : S --> CC )
74	73	feqmptd 6249	. . 3 |- ( ph -> ( CC _D F ) = ( a e. S |-> ( ( CC _D F ) ` a ) ) )
75		ffun 6048	. . . . . . 7 |- ( ( CC _D F ) : dom ( CC _D F ) --> CC -> Fun ( CC _D F ) )
76	1, 75	mp1i 13	. . . . . 6 |- ( ( ph /\ a e. S ) -> Fun ( CC _D F ) )
77		funssfv 6209	. . . . . 6 |- ( ( Fun ( CC _D F ) /\ ( CC _D ( F |` B ) ) C_ ( CC _D F ) /\ a e. dom ( CC _D ( F |` B ) ) ) -> ( ( CC _D F ) ` a ) = ( ( CC _D ( F |` B ) ) ` a ) )
78	76, 53, 67, 77	syl3anc 1326	. . . . 5 |- ( ( ph /\ a e. S ) -> ( ( CC _D F ) ` a ) = ( ( CC _D ( F |` B ) ) ` a ) )
79	60	fveq1d 6193	. . . . 5 |- ( ( ph /\ a e. S ) -> ( ( CC _D ( F |` B ) ) ` a ) = ( ( y e. B |-> sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( y ^ k ) ) ) ` a ) )
80		oveq1 6657	. . . . . . . . 9 |- ( y = a -> ( y ^ k ) = ( a ^ k ) )
81	80	oveq2d 6666	. . . . . . . 8 |- ( y = a -> ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( y ^ k ) ) = ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( a ^ k ) ) )
82	81	sumeq2sdv 14435	. . . . . . 7 |- ( y = a -> sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( y ^ k ) ) = sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( a ^ k ) ) )
83		eqid 2622	. . . . . . 7 |- ( y e. B |-> sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( y ^ k ) ) ) = ( y e. B |-> sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( y ^ k ) ) )
84		sumex 14418	. . . . . . 7 |- sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( a ^ k ) ) e. _V
85	82, 83, 84	fvmpt 6282	. . . . . 6 |- ( a e. B -> ( ( y e. B |-> sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( y ^ k ) ) ) ` a ) = sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( a ^ k ) ) )
86	59, 85	syl 17	. . . . 5 |- ( ( ph /\ a e. S ) -> ( ( y e. B |-> sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( y ^ k ) ) ) ` a ) = sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( a ^ k ) ) )
87	78, 79, 86	3eqtrd 2660	. . . 4 |- ( ( ph /\ a e. S ) -> ( ( CC _D F ) ` a ) = sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( a ^ k ) ) )
88	87	mpteq2dva 4744	. . 3 |- ( ph -> ( a e. S |-> ( ( CC _D F ) ` a ) ) = ( a e. S |-> sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( a ^ k ) ) ) )
89	74, 88	eqtrd 2656	. 2 |- ( ph -> ( CC _D F ) = ( a e. S |-> sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( a ^ k ) ) ) )
90		oveq1 6657	. . . . 5 |- ( a = y -> ( a ^ k ) = ( y ^ k ) )
91	90	oveq2d 6666	. . . 4 |- ( a = y -> ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( a ^ k ) ) = ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( y ^ k ) ) )
92	91	sumeq2sdv 14435	. . 3 |- ( a = y -> sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( a ^ k ) ) = sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( y ^ k ) ) )
93	92	cbvmptv 4750	. 2 |- ( a e. S |-> sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( a ^ k ) ) ) = ( y e. S |-> sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( y ^ k ) ) )
94	89, 93	syl6eq 2672	1 |- ( ph -> ( CC _D F ) = ( y e. S |-> sum_ k e. NN0 ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( y ^ k ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   |` cres 5116   "cima 5117   o. ccom 5118   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   supcsup 8346   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   RR*cxr 10073   < clt 10074   - cmin 10266   / cdiv 10684   2c2 11070   NN0cn0 11292   RR+crp 11832   [,)cico 12177   [,]cicc 12178   seqcseq 12801   ^cexp 12860   abscabs 13974   ~~> cli 14215   sum_csu 14416   ↾_t crest 16081   TopOpenctopn 16082   *Metcxmt 19731   ballcbl 19733  ℂ_fldccnfld 19746   Topctop 20698   intcnt 20821   -cn->ccncf 22679   _D cdv 23627

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-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  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-cda 8990  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-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  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-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  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-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-ulm 24131

This theorem is referenced by:  pserdv2  24184