HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  occllem Structured version   Unicode version

Theorem occllem 24657
Description: Lemma for occl 24658. (Contributed by NM, 7-Aug-2000.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
occl.1  |-  ( ph  ->  A  C_  ~H )
occl.2  |-  ( ph  ->  F  e.  Cauchy )
occl.3  |-  ( ph  ->  F : NN --> ( _|_ `  A ) )
occl.4  |-  ( ph  ->  B  e.  A )
Assertion
Ref Expression
occllem  |-  ( ph  ->  ( (  ~~>v  `  F
)  .ih  B )  =  0 )

Proof of Theorem occllem
Dummy variables  x  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2438 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
21cnfldhaus 20339 . . 3  |-  ( TopOpen ` fld )  e.  Haus
32a1i 11 . 2  |-  ( ph  ->  ( TopOpen ` fld )  e.  Haus )
4 occl.2 . . . . . . 7  |-  ( ph  ->  F  e.  Cauchy )
5 ax-hcompl 24555 . . . . . . 7  |-  ( F  e.  Cauchy  ->  E. x  e.  ~H  F  ~~>v  x )
6 hlimf 24591 . . . . . . . . . 10  |-  ~~>v  : dom  ~~>v  --> ~H
7 ffn 5554 . . . . . . . . . 10  |-  (  ~~>v  : dom  ~~>v  --> ~H  ->  ~~>v  Fn  dom  ~~>v  )
86, 7ax-mp 5 . . . . . . . . 9  |-  ~~>v  Fn  dom  ~~>v
9 fnbr 5508 . . . . . . . . 9  |-  ( ( 
~~>v  Fn  dom  ~~>v  /\  F  ~~>v  x )  ->  F  e.  dom  ~~>v  )
108, 9mpan 670 . . . . . . . 8  |-  ( F 
~~>v  x  ->  F  e.  dom 
~~>v  )
1110rexlimivw 2832 . . . . . . 7  |-  ( E. x  e.  ~H  F  ~~>v  x  ->  F  e.  dom  ~~>v  )
124, 5, 113syl 20 . . . . . 6  |-  ( ph  ->  F  e.  dom  ~~>v  )
13 ffun 5556 . . . . . . 7  |-  (  ~~>v  : dom  ~~>v  --> ~H  ->  Fun  ~~>v  )
14 funfvbrb 5811 . . . . . . 7  |-  ( Fun  ~~>v 
->  ( F  e.  dom  ~~>v  <->  F  ~~>v  (  ~~>v  `  F )
) )
156, 13, 14mp2b 10 . . . . . 6  |-  ( F  e.  dom  ~~>v  <->  F  ~~>v  (  ~~>v  `  F ) )
1612, 15sylib 196 . . . . 5  |-  ( ph  ->  F  ~~>v  (  ~~>v  `  F
) )
17 eqid 2438 . . . . . . . 8  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
18 eqid 2438 . . . . . . . . 9  |-  ( normh  o. 
-h  )  =  (
normh  o.  -h  )
1917, 18hhims 24525 . . . . . . . 8  |-  ( normh  o. 
-h  )  =  (
IndMet `  <. <.  +h  ,  .h  >. ,  normh >. )
20 eqid 2438 . . . . . . . 8  |-  ( MetOpen `  ( normh  o.  -h  )
)  =  ( MetOpen `  ( normh  o.  -h  )
)
2117, 19, 20hhlm 24552 . . . . . . 7  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( normh  o.  -h  ) ) )  |`  ( ~H  ^m  NN ) )
22 resss 5129 . . . . . . 7  |-  ( ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) )  |`  ( ~H  ^m  NN ) ) 
C_  ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) )
2321, 22eqsstri 3381 . . . . . 6  |-  ~~>v  C_  ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) )
2423ssbri 4329 . . . . 5  |-  ( F 
~~>v  (  ~~>v  `  F )  ->  F ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) ) (  ~~>v  `  F )
)
2516, 24syl 16 . . . 4  |-  ( ph  ->  F ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) ) (  ~~>v  `  F )
)
2618hilxmet 24548 . . . . . 6  |-  ( normh  o. 
-h  )  e.  ( *Met `  ~H )
2720mopntopon 19989 . . . . . 6  |-  ( (
normh  o.  -h  )  e.  ( *Met `  ~H )  ->  ( MetOpen `  ( normh  o.  -h  )
)  e.  (TopOn `  ~H ) )
2826, 27mp1i 12 . . . . 5  |-  ( ph  ->  ( MetOpen `  ( normh  o. 
-h  ) )  e.  (TopOn `  ~H )
)
2928cnmptid 19209 . . . . 5  |-  ( ph  ->  ( x  e.  ~H  |->  x )  e.  ( ( MetOpen `  ( normh  o. 
-h  ) )  Cn  ( MetOpen `  ( normh  o. 
-h  ) ) ) )
30 occl.1 . . . . . . 7  |-  ( ph  ->  A  C_  ~H )
31 occl.4 . . . . . . 7  |-  ( ph  ->  B  e.  A )
3230, 31sseldd 3352 . . . . . 6  |-  ( ph  ->  B  e.  ~H )
3328, 28, 32cnmptc 19210 . . . . 5  |-  ( ph  ->  ( x  e.  ~H  |->  B )  e.  ( ( MetOpen `  ( normh  o. 
-h  ) )  Cn  ( MetOpen `  ( normh  o. 
-h  ) ) ) )
3417hhnv 24518 . . . . . 6  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec
3517hhip 24530 . . . . . . 7  |-  .ih  =  ( .iOLD `  <. <.  +h  ,  .h  >. ,  normh >.
)
3635, 19, 20, 1dipcn 24069 . . . . . 6  |-  ( <. <.  +h  ,  .h  >. , 
normh >.  e.  NrmCVec  ->  .ih  e.  ( ( ( MetOpen `  ( normh  o.  -h  )
)  tX  ( MetOpen `  ( normh  o.  -h  )
) )  Cn  ( TopOpen
` fld
) ) )
3734, 36mp1i 12 . . . . 5  |-  ( ph  ->  .ih  e.  ( ( ( MetOpen `  ( normh  o. 
-h  ) )  tX  ( MetOpen `  ( normh  o. 
-h  ) ) )  Cn  ( TopOpen ` fld ) ) )
3828, 29, 33, 37cnmpt12f 19214 . . . 4  |-  ( ph  ->  ( x  e.  ~H  |->  ( x  .ih  B ) )  e.  ( (
MetOpen `  ( normh  o.  -h  ) )  Cn  ( TopOpen
` fld
) ) )
3925, 38lmcn 18884 . . 3  |-  ( ph  ->  ( ( x  e. 
~H  |->  ( x  .ih  B ) )  o.  F
) ( ~~> t `  ( TopOpen ` fld ) ) ( ( x  e.  ~H  |->  ( x  .ih  B ) ) `  (  ~~>v  `  F ) ) )
40 occl.3 . . . . . . . . . . 11  |-  ( ph  ->  F : NN --> ( _|_ `  A ) )
4140ffvelrnda 5838 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e.  ( _|_ `  A
) )
42 ocel 24635 . . . . . . . . . . . 12  |-  ( A 
C_  ~H  ->  ( ( F `  k )  e.  ( _|_ `  A
)  <->  ( ( F `
 k )  e. 
~H  /\  A. x  e.  A  ( ( F `  k )  .ih  x )  =  0 ) ) )
4330, 42syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( ( F `  k )  e.  ( _|_ `  A )  <-> 
( ( F `  k )  e.  ~H  /\ 
A. x  e.  A  ( ( F `  k )  .ih  x
)  =  0 ) ) )
4443adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( F `  k )  e.  ( _|_ `  A
)  <->  ( ( F `
 k )  e. 
~H  /\  A. x  e.  A  ( ( F `  k )  .ih  x )  =  0 ) ) )
4541, 44mpbid 210 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( F `  k )  e.  ~H  /\  A. x  e.  A  (
( F `  k
)  .ih  x )  =  0 ) )
4645simpld 459 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e. 
~H )
47 oveq1 6093 . . . . . . . . 9  |-  ( x  =  ( F `  k )  ->  (
x  .ih  B )  =  ( ( F `
 k )  .ih  B ) )
48 eqid 2438 . . . . . . . . 9  |-  ( x  e.  ~H  |->  ( x 
.ih  B ) )  =  ( x  e. 
~H  |->  ( x  .ih  B ) )
49 ovex 6111 . . . . . . . . 9  |-  ( ( F `  k ) 
.ih  B )  e. 
_V
5047, 48, 49fvmpt 5769 . . . . . . . 8  |-  ( ( F `  k )  e.  ~H  ->  (
( x  e.  ~H  |->  ( x  .ih  B ) ) `  ( F `
 k ) )  =  ( ( F `
 k )  .ih  B ) )
5146, 50syl 16 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( x  e.  ~H  |->  ( x  .ih  B ) ) `  ( F `
 k ) )  =  ( ( F `
 k )  .ih  B ) )
5231adantr 465 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  B  e.  A )
5345simprd 463 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  A. x  e.  A  ( ( F `  k )  .ih  x )  =  0 )
54 oveq2 6094 . . . . . . . . . 10  |-  ( x  =  B  ->  (
( F `  k
)  .ih  x )  =  ( ( F `
 k )  .ih  B ) )
5554eqeq1d 2446 . . . . . . . . 9  |-  ( x  =  B  ->  (
( ( F `  k )  .ih  x
)  =  0  <->  (
( F `  k
)  .ih  B )  =  0 ) )
5655rspcv 3064 . . . . . . . 8  |-  ( B  e.  A  ->  ( A. x  e.  A  ( ( F `  k )  .ih  x
)  =  0  -> 
( ( F `  k )  .ih  B
)  =  0 ) )
5752, 53, 56sylc 60 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( F `  k ) 
.ih  B )  =  0 )
5851, 57eqtrd 2470 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( x  e.  ~H  |->  ( x  .ih  B ) ) `  ( F `
 k ) )  =  0 )
59 ocss 24639 . . . . . . . . 9  |-  ( A 
C_  ~H  ->  ( _|_ `  A )  C_  ~H )
6030, 59syl 16 . . . . . . . 8  |-  ( ph  ->  ( _|_ `  A
)  C_  ~H )
61 fss 5562 . . . . . . . 8  |-  ( ( F : NN --> ( _|_ `  A )  /\  ( _|_ `  A )  C_  ~H )  ->  F : NN
--> ~H )
6240, 60, 61syl2anc 661 . . . . . . 7  |-  ( ph  ->  F : NN --> ~H )
63 fvco3 5763 . . . . . . 7  |-  ( ( F : NN --> ~H  /\  k  e.  NN )  ->  ( ( ( x  e.  ~H  |->  ( x 
.ih  B ) )  o.  F ) `  k )  =  ( ( x  e.  ~H  |->  ( x  .ih  B ) ) `  ( F `
 k ) ) )
6462, 63sylan 471 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( x  e.  ~H  |->  ( x  .ih  B ) )  o.  F ) `
 k )  =  ( ( x  e. 
~H  |->  ( x  .ih  B ) ) `  ( F `  k )
) )
65 c0ex 9372 . . . . . . . 8  |-  0  e.  _V
6665fvconst2 5928 . . . . . . 7  |-  ( k  e.  NN  ->  (
( NN  X.  {
0 } ) `  k )  =  0 )
6766adantl 466 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( NN  X.  { 0 } ) `  k
)  =  0 )
6858, 64, 673eqtr4d 2480 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( x  e.  ~H  |->  ( x  .ih  B ) )  o.  F ) `
 k )  =  ( ( NN  X.  { 0 } ) `
 k ) )
6968ralrimiva 2794 . . . 4  |-  ( ph  ->  A. k  e.  NN  ( ( ( x  e.  ~H  |->  ( x 
.ih  B ) )  o.  F ) `  k )  =  ( ( NN  X.  {
0 } ) `  k ) )
70 ovex 6111 . . . . . . . 8  |-  ( x 
.ih  B )  e. 
_V
7170, 48fnmpti 5534 . . . . . . 7  |-  ( x  e.  ~H  |->  ( x 
.ih  B ) )  Fn  ~H
7271a1i 11 . . . . . 6  |-  ( ph  ->  ( x  e.  ~H  |->  ( x  .ih  B ) )  Fn  ~H )
73 fnfco 5572 . . . . . 6  |-  ( ( ( x  e.  ~H  |->  ( x  .ih  B ) )  Fn  ~H  /\  F : NN --> ~H )  ->  ( ( x  e. 
~H  |->  ( x  .ih  B ) )  o.  F
)  Fn  NN )
7472, 62, 73syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( x  e. 
~H  |->  ( x  .ih  B ) )  o.  F
)  Fn  NN )
7565fconst 5591 . . . . . 6  |-  ( NN 
X.  { 0 } ) : NN --> { 0 }
76 ffn 5554 . . . . . 6  |-  ( ( NN  X.  { 0 } ) : NN --> { 0 }  ->  ( NN  X.  { 0 } )  Fn  NN )
7775, 76ax-mp 5 . . . . 5  |-  ( NN 
X.  { 0 } )  Fn  NN
78 eqfnfv 5792 . . . . 5  |-  ( ( ( ( x  e. 
~H  |->  ( x  .ih  B ) )  o.  F
)  Fn  NN  /\  ( NN  X.  { 0 } )  Fn  NN )  ->  ( ( ( x  e.  ~H  |->  ( x  .ih  B ) )  o.  F )  =  ( NN  X.  { 0 } )  <->  A. k  e.  NN  ( ( ( x  e.  ~H  |->  ( x 
.ih  B ) )  o.  F ) `  k )  =  ( ( NN  X.  {
0 } ) `  k ) ) )
7974, 77, 78sylancl 662 . . . 4  |-  ( ph  ->  ( ( ( x  e.  ~H  |->  ( x 
.ih  B ) )  o.  F )  =  ( NN  X.  {
0 } )  <->  A. k  e.  NN  ( ( ( x  e.  ~H  |->  ( x  .ih  B ) )  o.  F ) `
 k )  =  ( ( NN  X.  { 0 } ) `
 k ) ) )
8069, 79mpbird 232 . . 3  |-  ( ph  ->  ( ( x  e. 
~H  |->  ( x  .ih  B ) )  o.  F
)  =  ( NN 
X.  { 0 } ) )
81 fvex 5696 . . . . 5  |-  (  ~~>v  `  F )  e.  _V
8281hlimveci 24543 . . . 4  |-  ( F 
~~>v  (  ~~>v  `  F )  ->  (  ~~>v  `  F )  e.  ~H )
83 oveq1 6093 . . . . 5  |-  ( x  =  (  ~~>v  `  F
)  ->  ( x  .ih  B )  =  ( (  ~~>v  `  F )  .ih  B ) )
84 ovex 6111 . . . . 5  |-  ( ( 
~~>v  `  F )  .ih  B )  e.  _V
8583, 48, 84fvmpt 5769 . . . 4  |-  ( ( 
~~>v  `  F )  e. 
~H  ->  ( ( x  e.  ~H  |->  ( x 
.ih  B ) ) `
 (  ~~>v  `  F
) )  =  ( (  ~~>v  `  F )  .ih  B ) )
8616, 82, 853syl 20 . . 3  |-  ( ph  ->  ( ( x  e. 
~H  |->  ( x  .ih  B ) ) `  (  ~~>v 
`  F ) )  =  ( (  ~~>v  `  F )  .ih  B
) )
8739, 80, 863brtr3d 4316 . 2  |-  ( ph  ->  ( NN  X.  {
0 } ) ( ~~> t `  ( TopOpen ` fld )
) ( (  ~~>v  `  F )  .ih  B
) )
881cnfldtopon 20337 . . . 4  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
8988a1i 11 . . 3  |-  ( ph  ->  ( TopOpen ` fld )  e.  (TopOn `  CC ) )
90 0cnd 9371 . . 3  |-  ( ph  ->  0  e.  CC )
91 1zzd 10669 . . 3  |-  ( ph  ->  1  e.  ZZ )
92 nnuz 10888 . . . 4  |-  NN  =  ( ZZ>= `  1 )
9392lmconst 18840 . . 3  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  0  e.  CC  /\  1  e.  ZZ )  ->  ( NN  X.  { 0 } ) ( ~~> t `  ( TopOpen ` fld ) ) 0 )
9489, 90, 91, 93syl3anc 1218 . 2  |-  ( ph  ->  ( NN  X.  {
0 } ) ( ~~> t `  ( TopOpen ` fld )
) 0 )
953, 87, 94lmmo 18959 1  |-  ( ph  ->  ( (  ~~>v  `  F
)  .ih  B )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2710   E.wrex 2711    C_ wss 3323   {csn 3872   <.cop 3878   class class class wbr 4287    e. cmpt 4345    X. cxp 4833   dom cdm 4835    |` cres 4837    o. ccom 4839   Fun wfun 5407    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086    ^m cmap 7206   CCcc 9272   0cc0 9274   1c1 9275   NNcn 10314   ZZcz 10638   TopOpenctopn 14352   *Metcxmt 17776   MetOpencmopn 17781  ℂfldccnfld 17793  TopOnctopon 18474    Cn ccn 18803   ~~> tclm 18805   Hauscha 18887    tX ctx 19108   NrmCVeccnv 23913   ~Hchil 24272    +h cva 24273    .h csm 24274    .ih csp 24275   normhcno 24276    -h cmv 24278   Cauchyccau 24279    ~~>v chli 24280   _|_cort 24283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354  ax-hilex 24352  ax-hfvadd 24353  ax-hvcom 24354  ax-hvass 24355  ax-hv0cl 24356  ax-hvaddid 24357  ax-hfvmul 24358  ax-hvmulid 24359  ax-hvmulass 24360  ax-hvdistr1 24361  ax-hvdistr2 24362  ax-hvmul0 24363  ax-hfi 24432  ax-his1 24435  ax-his2 24436  ax-his3 24437  ax-his4 24438  ax-hcompl 24555
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-icc 11299  df-fz 11430  df-fzo 11541  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-sum 13156  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cn 18806  df-cnp 18807  df-lm 18808  df-haus 18894  df-tx 19110  df-hmeo 19303  df-xms 19870  df-ms 19871  df-tms 19872  df-grpo 23629  df-gid 23630  df-ginv 23631  df-gdiv 23632  df-ablo 23720  df-vc 23875  df-nv 23921  df-va 23924  df-ba 23925  df-sm 23926  df-0v 23927  df-vs 23928  df-nmcv 23929  df-ims 23930  df-dip 24047  df-hnorm 24321  df-hvsub 24324  df-hlim 24325  df-sh 24560  df-oc 24606
This theorem is referenced by:  occl  24658
  Copyright terms: Public domain W3C validator