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Theorem occllem 26419
Description: Lemma for occl 26420. (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 2454 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
21cnfldhaus 21458 . . 3  |-  ( TopOpen ` fld )  e.  Haus
32a1i 11 . 2  |-  ( ph  ->  ( TopOpen ` fld )  e.  Haus )
4 occl.2 . . . . . . 7  |-  ( ph  ->  F  e.  Cauchy )
5 ax-hcompl 26317 . . . . . . 7  |-  ( F  e.  Cauchy  ->  E. x  e.  ~H  F  ~~>v  x )
6 hlimf 26353 . . . . . . . . . 10  |-  ~~>v  : dom  ~~>v  --> ~H
7 ffn 5713 . . . . . . . . . 10  |-  (  ~~>v  : dom  ~~>v  --> ~H  ->  ~~>v  Fn  dom  ~~>v  )
86, 7ax-mp 5 . . . . . . . . 9  |-  ~~>v  Fn  dom  ~~>v
9 fnbr 5665 . . . . . . . . 9  |-  ( ( 
~~>v  Fn  dom  ~~>v  /\  F  ~~>v  x )  ->  F  e.  dom  ~~>v  )
108, 9mpan 668 . . . . . . . 8  |-  ( F 
~~>v  x  ->  F  e.  dom 
~~>v  )
1110rexlimivw 2943 . . . . . . 7  |-  ( E. x  e.  ~H  F  ~~>v  x  ->  F  e.  dom  ~~>v  )
124, 5, 113syl 20 . . . . . 6  |-  ( ph  ->  F  e.  dom  ~~>v  )
13 ffun 5715 . . . . . . 7  |-  (  ~~>v  : dom  ~~>v  --> ~H  ->  Fun  ~~>v  )
14 funfvbrb 5976 . . . . . . 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 2454 . . . . . . . 8  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
18 eqid 2454 . . . . . . . . 9  |-  ( normh  o. 
-h  )  =  (
normh  o.  -h  )
1917, 18hhims 26287 . . . . . . . 8  |-  ( normh  o. 
-h  )  =  (
IndMet `  <. <.  +h  ,  .h  >. ,  normh >. )
20 eqid 2454 . . . . . . . 8  |-  ( MetOpen `  ( normh  o.  -h  )
)  =  ( MetOpen `  ( normh  o.  -h  )
)
2117, 19, 20hhlm 26314 . . . . . . 7  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( normh  o.  -h  ) ) )  |`  ( ~H  ^m  NN ) )
22 resss 5285 . . . . . . 7  |-  ( ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) )  |`  ( ~H  ^m  NN ) ) 
C_  ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) )
2321, 22eqsstri 3519 . . . . . 6  |-  ~~>v  C_  ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) )
2423ssbri 4481 . . . . 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 26310 . . . . . 6  |-  ( normh  o. 
-h  )  e.  ( *Met `  ~H )
2720mopntopon 21108 . . . . . 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 20328 . . . . 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 3490 . . . . . 6  |-  ( ph  ->  B  e.  ~H )
3328, 28, 32cnmptc 20329 . . . . 5  |-  ( ph  ->  ( x  e.  ~H  |->  B )  e.  ( ( MetOpen `  ( normh  o. 
-h  ) )  Cn  ( MetOpen `  ( normh  o. 
-h  ) ) ) )
3417hhnv 26280 . . . . . 6  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec
3517hhip 26292 . . . . . . 7  |-  .ih  =  ( .iOLD `  <. <.  +h  ,  .h  >. ,  normh >.
)
3635, 19, 20, 1dipcn 25831 . . . . . 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 20333 . . . 4  |-  ( ph  ->  ( x  e.  ~H  |->  ( x  .ih  B ) )  e.  ( (
MetOpen `  ( normh  o.  -h  ) )  Cn  ( TopOpen
` fld
) ) )
3925, 38lmcn 19973 . . 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 6007 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e.  ( _|_ `  A
) )
42 ocel 26397 . . . . . . . . . . . 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 463 . . . . . . . . . 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 457 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e. 
~H )
47 oveq1 6277 . . . . . . . . 9  |-  ( x  =  ( F `  k )  ->  (
x  .ih  B )  =  ( ( F `
 k )  .ih  B ) )
48 eqid 2454 . . . . . . . . 9  |-  ( x  e.  ~H  |->  ( x 
.ih  B ) )  =  ( x  e. 
~H  |->  ( x  .ih  B ) )
49 ovex 6298 . . . . . . . . 9  |-  ( ( F `  k ) 
.ih  B )  e. 
_V
5047, 48, 49fvmpt 5931 . . . . . . . 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 463 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  B  e.  A )
5345simprd 461 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  A. x  e.  A  ( ( F `  k )  .ih  x )  =  0 )
54 oveq2 6278 . . . . . . . . . 10  |-  ( x  =  B  ->  (
( F `  k
)  .ih  x )  =  ( ( F `
 k )  .ih  B ) )
5554eqeq1d 2456 . . . . . . . . 9  |-  ( x  =  B  ->  (
( ( F `  k )  .ih  x
)  =  0  <->  (
( F `  k
)  .ih  B )  =  0 ) )
5655rspcv 3203 . . . . . . . 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 2495 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( x  e.  ~H  |->  ( x  .ih  B ) ) `  ( F `
 k ) )  =  0 )
59 ocss 26401 . . . . . . . . 9  |-  ( A 
C_  ~H  ->  ( _|_ `  A )  C_  ~H )
6030, 59syl 16 . . . . . . . 8  |-  ( ph  ->  ( _|_ `  A
)  C_  ~H )
6140, 60fssd 5722 . . . . . . 7  |-  ( ph  ->  F : NN --> ~H )
62 fvco3 5925 . . . . . . 7  |-  ( ( F : NN --> ~H  /\  k  e.  NN )  ->  ( ( ( x  e.  ~H  |->  ( x 
.ih  B ) )  o.  F ) `  k )  =  ( ( x  e.  ~H  |->  ( x  .ih  B ) ) `  ( F `
 k ) ) )
6361, 62sylan 469 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( x  e.  ~H  |->  ( x  .ih  B ) )  o.  F ) `
 k )  =  ( ( x  e. 
~H  |->  ( x  .ih  B ) ) `  ( F `  k )
) )
64 c0ex 9579 . . . . . . . 8  |-  0  e.  _V
6564fvconst2 6103 . . . . . . 7  |-  ( k  e.  NN  ->  (
( NN  X.  {
0 } ) `  k )  =  0 )
6665adantl 464 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( NN  X.  { 0 } ) `  k
)  =  0 )
6758, 63, 663eqtr4d 2505 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( ( x  e.  ~H  |->  ( x  .ih  B ) )  o.  F ) `
 k )  =  ( ( NN  X.  { 0 } ) `
 k ) )
6867ralrimiva 2868 . . . 4  |-  ( ph  ->  A. k  e.  NN  ( ( ( x  e.  ~H  |->  ( x 
.ih  B ) )  o.  F ) `  k )  =  ( ( NN  X.  {
0 } ) `  k ) )
69 ovex 6298 . . . . . . 7  |-  ( x 
.ih  B )  e. 
_V
7069, 48fnmpti 5691 . . . . . 6  |-  ( x  e.  ~H  |->  ( x 
.ih  B ) )  Fn  ~H
71 fnfco 5732 . . . . . 6  |-  ( ( ( x  e.  ~H  |->  ( x  .ih  B ) )  Fn  ~H  /\  F : NN --> ~H )  ->  ( ( x  e. 
~H  |->  ( x  .ih  B ) )  o.  F
)  Fn  NN )
7270, 61, 71sylancr 661 . . . . 5  |-  ( ph  ->  ( ( x  e. 
~H  |->  ( x  .ih  B ) )  o.  F
)  Fn  NN )
7364fconst 5753 . . . . . 6  |-  ( NN 
X.  { 0 } ) : NN --> { 0 }
74 ffn 5713 . . . . . 6  |-  ( ( NN  X.  { 0 } ) : NN --> { 0 }  ->  ( NN  X.  { 0 } )  Fn  NN )
7573, 74ax-mp 5 . . . . 5  |-  ( NN 
X.  { 0 } )  Fn  NN
76 eqfnfv 5957 . . . . 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 ) ) )
7772, 75, 76sylancl 660 . . . 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 ) ) )
7868, 77mpbird 232 . . 3  |-  ( ph  ->  ( ( x  e. 
~H  |->  ( x  .ih  B ) )  o.  F
)  =  ( NN 
X.  { 0 } ) )
79 fvex 5858 . . . . 5  |-  (  ~~>v  `  F )  e.  _V
8079hlimveci 26305 . . . 4  |-  ( F 
~~>v  (  ~~>v  `  F )  ->  (  ~~>v  `  F )  e.  ~H )
81 oveq1 6277 . . . . 5  |-  ( x  =  (  ~~>v  `  F
)  ->  ( x  .ih  B )  =  ( (  ~~>v  `  F )  .ih  B ) )
82 ovex 6298 . . . . 5  |-  ( ( 
~~>v  `  F )  .ih  B )  e.  _V
8381, 48, 82fvmpt 5931 . . . 4  |-  ( ( 
~~>v  `  F )  e. 
~H  ->  ( ( x  e.  ~H  |->  ( x 
.ih  B ) ) `
 (  ~~>v  `  F
) )  =  ( (  ~~>v  `  F )  .ih  B ) )
8416, 80, 833syl 20 . . 3  |-  ( ph  ->  ( ( x  e. 
~H  |->  ( x  .ih  B ) ) `  (  ~~>v 
`  F ) )  =  ( (  ~~>v  `  F )  .ih  B
) )
8539, 78, 843brtr3d 4468 . 2  |-  ( ph  ->  ( NN  X.  {
0 } ) ( ~~> t `  ( TopOpen ` fld )
) ( (  ~~>v  `  F )  .ih  B
) )
861cnfldtopon 21456 . . . 4  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
8786a1i 11 . . 3  |-  ( ph  ->  ( TopOpen ` fld )  e.  (TopOn `  CC ) )
88 0cnd 9578 . . 3  |-  ( ph  ->  0  e.  CC )
89 1zzd 10891 . . 3  |-  ( ph  ->  1  e.  ZZ )
90 nnuz 11117 . . . 4  |-  NN  =  ( ZZ>= `  1 )
9190lmconst 19929 . . 3  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  0  e.  CC  /\  1  e.  ZZ )  ->  ( NN  X.  { 0 } ) ( ~~> t `  ( TopOpen ` fld ) ) 0 )
9287, 88, 89, 91syl3anc 1226 . 2  |-  ( ph  ->  ( NN  X.  {
0 } ) ( ~~> t `  ( TopOpen ` fld )
) 0 )
933, 85, 92lmmo 20048 1  |-  ( ph  ->  ( (  ~~>v  `  F
)  .ih  B )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805    C_ wss 3461   {csn 4016   <.cop 4022   class class class wbr 4439    |-> cmpt 4497    X. cxp 4986   dom cdm 4988    |` cres 4990    o. ccom 4992   Fun wfun 5564    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270    ^m cmap 7412   CCcc 9479   0cc0 9481   1c1 9482   NNcn 10531   ZZcz 10860   TopOpenctopn 14911   *Metcxmt 18598   MetOpencmopn 18603  ℂfldccnfld 18615  TopOnctopon 19562    Cn ccn 19892   ~~> tclm 19894   Hauscha 19976    tX ctx 20227   NrmCVeccnv 25675   ~Hchil 26034    +h cva 26035    .h csm 26036    .ih csp 26037   normhcno 26038    -h cmv 26040   Cauchyccau 26041    ~~>v chli 26042   _|_cort 26045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561  ax-hilex 26114  ax-hfvadd 26115  ax-hvcom 26116  ax-hvass 26117  ax-hv0cl 26118  ax-hvaddid 26119  ax-hfvmul 26120  ax-hvmulid 26121  ax-hvmulass 26122  ax-hvdistr1 26123  ax-hvdistr2 26124  ax-hvmul0 26125  ax-hfi 26194  ax-his1 26197  ax-his2 26198  ax-his3 26199  ax-his4 26200  ax-hcompl 26317
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-icc 11539  df-fz 11676  df-fzo 11800  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-sum 13591  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-pt 14934  df-prds 14937  df-xrs 14991  df-qtop 14996  df-imas 14997  df-xps 14999  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-mulg 16259  df-cntz 16554  df-cmn 16999  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cn 19895  df-cnp 19896  df-lm 19897  df-haus 19983  df-tx 20229  df-hmeo 20422  df-xms 20989  df-ms 20990  df-tms 20991  df-grpo 25391  df-gid 25392  df-ginv 25393  df-gdiv 25394  df-ablo 25482  df-vc 25637  df-nv 25683  df-va 25686  df-ba 25687  df-sm 25688  df-0v 25689  df-vs 25690  df-nmcv 25691  df-ims 25692  df-dip 25809  df-hnorm 26083  df-hvsub 26086  df-hlim 26087  df-sh 26322  df-oc 26368
This theorem is referenced by:  occl  26420
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