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Theorem occllem 22807
Description: Lemma for occl 22808. (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 18821 . . 3  |-  ( TopOpen ` fld )  e.  Haus
32a1i 11 . 2  |-  ( ph  ->  ( TopOpen ` fld )  e.  Haus )
4 occl.2 . . . . . . 7  |-  ( ph  ->  F  e.  Cauchy )
5 ax-hcompl 22706 . . . . . . 7  |-  ( F  e.  Cauchy  ->  E. x  e.  ~H  F  ~~>v  x )
6 hlimf 22742 . . . . . . . . . 10  |-  ~~>v  : dom  ~~>v  --> ~H
7 ffn 5593 . . . . . . . . . 10  |-  (  ~~>v  : dom  ~~>v  --> ~H  ->  ~~>v  Fn  dom  ~~>v  )
86, 7ax-mp 8 . . . . . . . . 9  |-  ~~>v  Fn  dom  ~~>v
9 fnbr 5549 . . . . . . . . 9  |-  ( ( 
~~>v  Fn  dom  ~~>v  /\  F  ~~>v  x )  ->  F  e.  dom  ~~>v  )
108, 9mpan 653 . . . . . . . 8  |-  ( F 
~~>v  x  ->  F  e.  dom 
~~>v  )
1110rexlimivw 2828 . . . . . . 7  |-  ( E. x  e.  ~H  F  ~~>v  x  ->  F  e.  dom  ~~>v  )
124, 5, 113syl 19 . . . . . 6  |-  ( ph  ->  F  e.  dom  ~~>v  )
13 ffun 5595 . . . . . . 7  |-  (  ~~>v  : dom  ~~>v  --> ~H  ->  Fun  ~~>v  )
14 funfvbrb 5845 . . . . . . 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 190 . . . . 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 22676 . . . . . . . 8  |-  ( normh  o. 
-h  )  =  (
IndMet `  <. <.  +h  ,  .h  >. ,  normh >. )
20 eqid 2438 . . . . . . . 8  |-  ( MetOpen `  ( normh  o.  -h  )
)  =  ( MetOpen `  ( normh  o.  -h  )
)
2117, 19, 20hhlm 22703 . . . . . . 7  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( normh  o.  -h  ) ) )  |`  ( ~H  ^m  NN ) )
22 resss 5172 . . . . . . 7  |-  ( ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) )  |`  ( ~H  ^m  NN ) ) 
C_  ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) )
2321, 22eqsstri 3380 . . . . . 6  |-  ~~>v  C_  ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) )
2423ssbri 4256 . . . . 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 22699 . . . . . 6  |-  ( normh  o. 
-h  )  e.  ( * Met `  ~H )
2720mopntopon 18471 . . . . . 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 17695 . . . . 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 3351 . . . . . 6  |-  ( ph  ->  B  e.  ~H )
3328, 28, 32cnmptc 17696 . . . . 5  |-  ( ph  ->  ( x  e.  ~H  |->  B )  e.  ( ( MetOpen `  ( normh  o. 
-h  ) )  Cn  ( MetOpen `  ( normh  o. 
-h  ) ) ) )
3417hhnv 22669 . . . . . 6  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec
3517hhip 22681 . . . . . . 7  |-  .ih  =  ( .i OLD `  <. <.  +h  ,  .h  >. ,  normh >.
)
3635, 19, 20, 1dipcn 22221 . . . . . 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 17700 . . . 4  |-  ( ph  ->  ( x  e.  ~H  |->  ( x  .ih  B ) )  e.  ( (
MetOpen `  ( normh  o.  -h  ) )  Cn  ( TopOpen
` fld
) ) )
3925, 38lmcn 17371 . . 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 5872 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e.  ( _|_ `  A
) )
42 ocel 22785 . . . . . . . . . . . 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 453 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( F `  k )  e.  ( _|_ `  A
)  <->  ( ( F `
 k )  e. 
~H  /\  A. x  e.  A  ( ( F `  k )  .ih  x )  =  0 ) ) )
4541, 44mpbid 203 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( F `  k )  e.  ~H  /\  A. x  e.  A  (
( F `  k
)  .ih  x )  =  0 ) )
4645simpld 447 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e. 
~H )
47 oveq1 6090 . . . . . . . . 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 6108 . . . . . . . . 9  |-  ( ( F `  k ) 
.ih  B )  e. 
_V
5047, 48, 49fvmpt 5808 . . . . . . . 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 453 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  B  e.  A )
5345simprd 451 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  A. x  e.  A  ( ( F `  k )  .ih  x )  =  0 )
54 oveq2 6091 . . . . . . . . . 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 3050 . . . . . . . 8  |-  ( B  e.  A  ->  ( A. x  e.  A  ( ( F `  k )  .ih  x
)  =  0  -> 
( ( F `  k )  .ih  B
)  =  0 ) )
5752, 53, 56sylc 59 . . . . . . 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 22789 . . . . . . . . 9  |-  ( A 
C_  ~H  ->  ( _|_ `  A )  C_  ~H )
6030, 59syl 16 . . . . . . . 8  |-  ( ph  ->  ( _|_ `  A
)  C_  ~H )
61 fss 5601 . . . . . . . 8  |-  ( ( F : NN --> ( _|_ `  A )  /\  ( _|_ `  A )  C_  ~H )  ->  F : NN
--> ~H )
6240, 60, 61syl2anc 644 . . . . . . 7  |-  ( ph  ->  F : NN --> ~H )
63 fvco3 5802 . . . . . . 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 459 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( x  e.  ~H  |->  ( x  .ih  B ) )  o.  F ) `
 k )  =  ( ( x  e. 
~H  |->  ( x  .ih  B ) ) `  ( F `  k )
) )
65 c0ex 9087 . . . . . . . 8  |-  0  e.  _V
6665fvconst2 5949 . . . . . . 7  |-  ( k  e.  NN  ->  (
( NN  X.  {
0 } ) `  k )  =  0 )
6766adantl 454 . . . . . 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 2791 . . . 4  |-  ( ph  ->  A. k  e.  NN  ( ( ( x  e.  ~H  |->  ( x 
.ih  B ) )  o.  F ) `  k )  =  ( ( NN  X.  {
0 } ) `  k ) )
70 ovex 6108 . . . . . . . 8  |-  ( x 
.ih  B )  e. 
_V
7170, 48fnmpti 5575 . . . . . . 7  |-  ( x  e.  ~H  |->  ( x 
.ih  B ) )  Fn  ~H
7271a1i 11 . . . . . 6  |-  ( ph  ->  ( x  e.  ~H  |->  ( x  .ih  B ) )  Fn  ~H )
73 fnfco 5611 . . . . . 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 644 . . . . 5  |-  ( ph  ->  ( ( x  e. 
~H  |->  ( x  .ih  B ) )  o.  F
)  Fn  NN )
7565fconst 5631 . . . . . 6  |-  ( NN 
X.  { 0 } ) : NN --> { 0 }
76 ffn 5593 . . . . . 6  |-  ( ( NN  X.  { 0 } ) : NN --> { 0 }  ->  ( NN  X.  { 0 } )  Fn  NN )
7775, 76ax-mp 8 . . . . 5  |-  ( NN 
X.  { 0 } )  Fn  NN
78 eqfnfv 5829 . . . . 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 645 . . . 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 225 . . 3  |-  ( ph  ->  ( ( x  e. 
~H  |->  ( x  .ih  B ) )  o.  F
)  =  ( NN 
X.  { 0 } ) )
81 fvex 5744 . . . . 5  |-  (  ~~>v  `  F )  e.  _V
8281hlimveci 22694 . . . 4  |-  ( F 
~~>v  (  ~~>v  `  F )  ->  (  ~~>v  `  F )  e.  ~H )
83 oveq1 6090 . . . . 5  |-  ( x  =  (  ~~>v  `  F
)  ->  ( x  .ih  B )  =  ( (  ~~>v  `  F )  .ih  B ) )
84 ovex 6108 . . . . 5  |-  ( ( 
~~>v  `  F )  .ih  B )  e.  _V
8583, 48, 84fvmpt 5808 . . . 4  |-  ( ( 
~~>v  `  F )  e. 
~H  ->  ( ( x  e.  ~H  |->  ( x 
.ih  B ) ) `
 (  ~~>v  `  F
) )  =  ( (  ~~>v  `  F )  .ih  B ) )
8616, 82, 853syl 19 . . 3  |-  ( ph  ->  ( ( x  e. 
~H  |->  ( x  .ih  B ) ) `  (  ~~>v 
`  F ) )  =  ( (  ~~>v  `  F )  .ih  B
) )
8739, 80, 863brtr3d 4243 . 2  |-  ( ph  ->  ( NN  X.  {
0 } ) ( ~~> t `  ( TopOpen ` fld )
) ( (  ~~>v  `  F )  .ih  B
) )
881cnfldtopon 18819 . . . 4  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
8988a1i 11 . . 3  |-  ( ph  ->  ( TopOpen ` fld )  e.  (TopOn `  CC ) )
90 0cn 9086 . . . 4  |-  0  e.  CC
9190a1i 11 . . 3  |-  ( ph  ->  0  e.  CC )
92 1z 10313 . . . 4  |-  1  e.  ZZ
9392a1i 11 . . 3  |-  ( ph  ->  1  e.  ZZ )
94 nnuz 10523 . . . 4  |-  NN  =  ( ZZ>= `  1 )
9594lmconst 17327 . . 3  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  0  e.  CC  /\  1  e.  ZZ )  ->  ( NN  X.  { 0 } ) ( ~~> t `  ( TopOpen ` fld ) ) 0 )
9689, 91, 93, 95syl3anc 1185 . 2  |-  ( ph  ->  ( NN  X.  {
0 } ) ( ~~> t `  ( TopOpen ` fld )
) 0 )
973, 87, 96lmmo 17446 1  |-  ( ph  ->  ( (  ~~>v  `  F
)  .ih  B )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708    C_ wss 3322   {csn 3816   <.cop 3819   class class class wbr 4214    e. cmpt 4268    X. cxp 4878   dom cdm 4880    |` cres 4882    o. ccom 4884   Fun wfun 5450    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083    ^m cmap 7020   CCcc 8990   0cc0 8992   1c1 8993   NNcn 10002   ZZcz 10284   TopOpenctopn 13651   * Metcxmt 16688   MetOpencmopn 16693  ℂfldccnfld 16705  TopOnctopon 16961    Cn ccn 17290   ~~> tclm 17292   Hauscha 17374    tX ctx 17594   NrmCVeccnv 22065   ~Hchil 22424    +h cva 22425    .h csm 22426    .ih csp 22427   normhcno 22428    -h cmv 22430   Cauchyccau 22431    ~~>v chli 22432   _|_cort 22435
This theorem is referenced by:  occl  22808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072  ax-hilex 22504  ax-hfvadd 22505  ax-hvcom 22506  ax-hvass 22507  ax-hv0cl 22508  ax-hvaddid 22509  ax-hfvmul 22510  ax-hvmulid 22511  ax-hvmulass 22512  ax-hvdistr1 22513  ax-hvdistr2 22514  ax-hvmul0 22515  ax-hfi 22583  ax-his1 22586  ax-his2 22587  ax-his3 22588  ax-his4 22589  ax-hcompl 22706
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-ioo 10922  df-icc 10925  df-fz 11046  df-fzo 11138  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-sum 12482  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-hom 13555  df-cco 13556  df-rest 13652  df-topn 13653  df-topgen 13669  df-pt 13670  df-prds 13673  df-xrs 13728  df-0g 13729  df-gsum 13730  df-qtop 13735  df-imas 13736  df-xps 13738  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-submnd 14741  df-mulg 14817  df-cntz 15118  df-cmn 15416  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-cnfld 16706  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-cn 17293  df-cnp 17294  df-lm 17295  df-haus 17381  df-tx 17596  df-hmeo 17789  df-xms 18352  df-ms 18353  df-tms 18354  df-grpo 21781  df-gid 21782  df-ginv 21783  df-gdiv 21784  df-ablo 21872  df-vc 22027  df-nv 22073  df-va 22076  df-ba 22077  df-sm 22078  df-0v 22079  df-vs 22080  df-nmcv 22081  df-ims 22082  df-dip 22199  df-hnorm 22473  df-hvsub 22476  df-hlim 22477  df-sh 22711  df-oc 22756
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