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Theorem occllem 24529
Description: Lemma for occl 24530. (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 2433 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
21cnfldhaus 20206 . . 3  |-  ( TopOpen ` fld )  e.  Haus
32a1i 11 . 2  |-  ( ph  ->  ( TopOpen ` fld )  e.  Haus )
4 occl.2 . . . . . . 7  |-  ( ph  ->  F  e.  Cauchy )
5 ax-hcompl 24427 . . . . . . 7  |-  ( F  e.  Cauchy  ->  E. x  e.  ~H  F  ~~>v  x )
6 hlimf 24463 . . . . . . . . . 10  |-  ~~>v  : dom  ~~>v  --> ~H
7 ffn 5547 . . . . . . . . . 10  |-  (  ~~>v  : dom  ~~>v  --> ~H  ->  ~~>v  Fn  dom  ~~>v  )
86, 7ax-mp 5 . . . . . . . . 9  |-  ~~>v  Fn  dom  ~~>v
9 fnbr 5501 . . . . . . . . 9  |-  ( ( 
~~>v  Fn  dom  ~~>v  /\  F  ~~>v  x )  ->  F  e.  dom  ~~>v  )
108, 9mpan 663 . . . . . . . 8  |-  ( F 
~~>v  x  ->  F  e.  dom 
~~>v  )
1110rexlimivw 2827 . . . . . . 7  |-  ( E. x  e.  ~H  F  ~~>v  x  ->  F  e.  dom  ~~>v  )
124, 5, 113syl 20 . . . . . 6  |-  ( ph  ->  F  e.  dom  ~~>v  )
13 ffun 5549 . . . . . . 7  |-  (  ~~>v  : dom  ~~>v  --> ~H  ->  Fun  ~~>v  )
14 funfvbrb 5804 . . . . . . 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 2433 . . . . . . . 8  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
18 eqid 2433 . . . . . . . . 9  |-  ( normh  o. 
-h  )  =  (
normh  o.  -h  )
1917, 18hhims 24397 . . . . . . . 8  |-  ( normh  o. 
-h  )  =  (
IndMet `  <. <.  +h  ,  .h  >. ,  normh >. )
20 eqid 2433 . . . . . . . 8  |-  ( MetOpen `  ( normh  o.  -h  )
)  =  ( MetOpen `  ( normh  o.  -h  )
)
2117, 19, 20hhlm 24424 . . . . . . 7  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( normh  o.  -h  ) ) )  |`  ( ~H  ^m  NN ) )
22 resss 5122 . . . . . . 7  |-  ( ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) )  |`  ( ~H  ^m  NN ) ) 
C_  ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) )
2321, 22eqsstri 3374 . . . . . 6  |-  ~~>v  C_  ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) )
2423ssbri 4322 . . . . 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 24420 . . . . . 6  |-  ( normh  o. 
-h  )  e.  ( *Met `  ~H )
2720mopntopon 19856 . . . . . 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 19076 . . . . 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 3345 . . . . . 6  |-  ( ph  ->  B  e.  ~H )
3328, 28, 32cnmptc 19077 . . . . 5  |-  ( ph  ->  ( x  e.  ~H  |->  B )  e.  ( ( MetOpen `  ( normh  o. 
-h  ) )  Cn  ( MetOpen `  ( normh  o. 
-h  ) ) ) )
3417hhnv 24390 . . . . . 6  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec
3517hhip 24402 . . . . . . 7  |-  .ih  =  ( .iOLD `  <. <.  +h  ,  .h  >. ,  normh >.
)
3635, 19, 20, 1dipcn 23941 . . . . . 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 19081 . . . 4  |-  ( ph  ->  ( x  e.  ~H  |->  ( x  .ih  B ) )  e.  ( (
MetOpen `  ( normh  o.  -h  ) )  Cn  ( TopOpen
` fld
) ) )
3925, 38lmcn 18751 . . 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 5831 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e.  ( _|_ `  A
) )
42 ocel 24507 . . . . . . . . . . . 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 462 . . . . . . . . . 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 456 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e. 
~H )
47 oveq1 6087 . . . . . . . . 9  |-  ( x  =  ( F `  k )  ->  (
x  .ih  B )  =  ( ( F `
 k )  .ih  B ) )
48 eqid 2433 . . . . . . . . 9  |-  ( x  e.  ~H  |->  ( x 
.ih  B ) )  =  ( x  e. 
~H  |->  ( x  .ih  B ) )
49 ovex 6105 . . . . . . . . 9  |-  ( ( F `  k ) 
.ih  B )  e. 
_V
5047, 48, 49fvmpt 5762 . . . . . . . 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 462 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  B  e.  A )
5345simprd 460 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  A. x  e.  A  ( ( F `  k )  .ih  x )  =  0 )
54 oveq2 6088 . . . . . . . . . 10  |-  ( x  =  B  ->  (
( F `  k
)  .ih  x )  =  ( ( F `
 k )  .ih  B ) )
5554eqeq1d 2441 . . . . . . . . 9  |-  ( x  =  B  ->  (
( ( F `  k )  .ih  x
)  =  0  <->  (
( F `  k
)  .ih  B )  =  0 ) )
5655rspcv 3058 . . . . . . . 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 2465 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( x  e.  ~H  |->  ( x  .ih  B ) ) `  ( F `
 k ) )  =  0 )
59 ocss 24511 . . . . . . . . 9  |-  ( A 
C_  ~H  ->  ( _|_ `  A )  C_  ~H )
6030, 59syl 16 . . . . . . . 8  |-  ( ph  ->  ( _|_ `  A
)  C_  ~H )
61 fss 5555 . . . . . . . 8  |-  ( ( F : NN --> ( _|_ `  A )  /\  ( _|_ `  A )  C_  ~H )  ->  F : NN
--> ~H )
6240, 60, 61syl2anc 654 . . . . . . 7  |-  ( ph  ->  F : NN --> ~H )
63 fvco3 5756 . . . . . . 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 468 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( x  e.  ~H  |->  ( x  .ih  B ) )  o.  F ) `
 k )  =  ( ( x  e. 
~H  |->  ( x  .ih  B ) ) `  ( F `  k )
) )
65 c0ex 9368 . . . . . . . 8  |-  0  e.  _V
6665fvconst2 5920 . . . . . . 7  |-  ( k  e.  NN  ->  (
( NN  X.  {
0 } ) `  k )  =  0 )
6766adantl 463 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( NN  X.  { 0 } ) `  k
)  =  0 )
6858, 64, 673eqtr4d 2475 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( x  e.  ~H  |->  ( x  .ih  B ) )  o.  F ) `
 k )  =  ( ( NN  X.  { 0 } ) `
 k ) )
6968ralrimiva 2789 . . . 4  |-  ( ph  ->  A. k  e.  NN  ( ( ( x  e.  ~H  |->  ( x 
.ih  B ) )  o.  F ) `  k )  =  ( ( NN  X.  {
0 } ) `  k ) )
70 ovex 6105 . . . . . . . 8  |-  ( x 
.ih  B )  e. 
_V
7170, 48fnmpti 5527 . . . . . . 7  |-  ( x  e.  ~H  |->  ( x 
.ih  B ) )  Fn  ~H
7271a1i 11 . . . . . 6  |-  ( ph  ->  ( x  e.  ~H  |->  ( x  .ih  B ) )  Fn  ~H )
73 fnfco 5565 . . . . . 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 654 . . . . 5  |-  ( ph  ->  ( ( x  e. 
~H  |->  ( x  .ih  B ) )  o.  F
)  Fn  NN )
7565fconst 5584 . . . . . 6  |-  ( NN 
X.  { 0 } ) : NN --> { 0 }
76 ffn 5547 . . . . . 6  |-  ( ( NN  X.  { 0 } ) : NN --> { 0 }  ->  ( NN  X.  { 0 } )  Fn  NN )
7775, 76ax-mp 5 . . . . 5  |-  ( NN 
X.  { 0 } )  Fn  NN
78 eqfnfv 5785 . . . . 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 655 . . . 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 5689 . . . . 5  |-  (  ~~>v  `  F )  e.  _V
8281hlimveci 24415 . . . 4  |-  ( F 
~~>v  (  ~~>v  `  F )  ->  (  ~~>v  `  F )  e.  ~H )
83 oveq1 6087 . . . . 5  |-  ( x  =  (  ~~>v  `  F
)  ->  ( x  .ih  B )  =  ( (  ~~>v  `  F )  .ih  B ) )
84 ovex 6105 . . . . 5  |-  ( ( 
~~>v  `  F )  .ih  B )  e.  _V
8583, 48, 84fvmpt 5762 . . . 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 4309 . 2  |-  ( ph  ->  ( NN  X.  {
0 } ) ( ~~> t `  ( TopOpen ` fld )
) ( (  ~~>v  `  F )  .ih  B
) )
881cnfldtopon 20204 . . . 4  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
8988a1i 11 . . 3  |-  ( ph  ->  ( TopOpen ` fld )  e.  (TopOn `  CC ) )
90 0cnd 9367 . . 3  |-  ( ph  ->  0  e.  CC )
91 1zzd 10665 . . 3  |-  ( ph  ->  1  e.  ZZ )
92 nnuz 10884 . . . 4  |-  NN  =  ( ZZ>= `  1 )
9392lmconst 18707 . . 3  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  0  e.  CC  /\  1  e.  ZZ )  ->  ( NN  X.  { 0 } ) ( ~~> t `  ( TopOpen ` fld ) ) 0 )
9489, 90, 91, 93syl3anc 1211 . 2  |-  ( ph  ->  ( NN  X.  {
0 } ) ( ~~> t `  ( TopOpen ` fld )
) 0 )
953, 87, 94lmmo 18826 1  |-  ( ph  ->  ( (  ~~>v  `  F
)  .ih  B )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1362    e. wcel 1755   A.wral 2705   E.wrex 2706    C_ wss 3316   {csn 3865   <.cop 3871   class class class wbr 4280    e. cmpt 4338    X. cxp 4825   dom cdm 4827    |` cres 4829    o. ccom 4831   Fun wfun 5400    Fn wfn 5401   -->wf 5402   ` cfv 5406  (class class class)co 6080    ^m cmap 7202   CCcc 9268   0cc0 9270   1c1 9271   NNcn 10310   ZZcz 10634   TopOpenctopn 14343   *Metcxmt 17645   MetOpencmopn 17650  ℂfldccnfld 17662  TopOnctopon 18341    Cn ccn 18670   ~~> tclm 18672   Hauscha 18754    tX ctx 18975   NrmCVeccnv 23785   ~Hchil 24144    +h cva 24145    .h csm 24146    .ih csp 24147   normhcno 24148    -h cmv 24150   Cauchyccau 24151    ~~>v chli 24152   _|_cort 24155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350  ax-hilex 24224  ax-hfvadd 24225  ax-hvcom 24226  ax-hvass 24227  ax-hv0cl 24228  ax-hvaddid 24229  ax-hfvmul 24230  ax-hvmulid 24231  ax-hvmulass 24232  ax-hvdistr1 24233  ax-hvdistr2 24234  ax-hvmul0 24235  ax-hfi 24304  ax-his1 24307  ax-his2 24308  ax-his3 24309  ax-his4 24310  ax-hcompl 24427
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-icc 11295  df-fz 11425  df-fzo 11533  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-clim 12950  df-sum 13148  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cn 18673  df-cnp 18674  df-lm 18675  df-haus 18761  df-tx 18977  df-hmeo 19170  df-xms 19737  df-ms 19738  df-tms 19739  df-grpo 23501  df-gid 23502  df-ginv 23503  df-gdiv 23504  df-ablo 23592  df-vc 23747  df-nv 23793  df-va 23796  df-ba 23797  df-sm 23798  df-0v 23799  df-vs 23800  df-nmcv 23801  df-ims 23802  df-dip 23919  df-hnorm 24193  df-hvsub 24196  df-hlim 24197  df-sh 24432  df-oc 24478
This theorem is referenced by:  occl  24530
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