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Theorem cnrmi 17088
Description: A subspace of a completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
cnrmi  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  ( Jt  A )  e.  Nrm )

Proof of Theorem cnrmi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . 3  |-  U. J  =  U. J
21restin 16897 . 2  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  ( Jt  A )  =  ( Jt  ( A  i^i  U. J ) ) )
3 inss2 3390 . . . . 5  |-  ( A  i^i  U. J ) 
C_  U. J
4 inex1g 4157 . . . . . 6  |-  ( A  e.  V  ->  ( A  i^i  U. J )  e.  _V )
5 elpwg 3632 . . . . . 6  |-  ( ( A  i^i  U. J
)  e.  _V  ->  ( ( A  i^i  U. J )  e.  ~P U. J  <->  ( A  i^i  U. J )  C_  U. J
) )
64, 5syl 15 . . . . 5  |-  ( A  e.  V  ->  (
( A  i^i  U. J )  e.  ~P U. J  <->  ( A  i^i  U. J )  C_  U. J
) )
73, 6mpbiri 224 . . . 4  |-  ( A  e.  V  ->  ( A  i^i  U. J )  e.  ~P U. J
)
87adantl 452 . . 3  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  ( A  i^i  U. J )  e.  ~P U. J
)
91iscnrm 17051 . . . . 5  |-  ( J  e. CNrm 
<->  ( J  e.  Top  /\ 
A. x  e.  ~P  U. J ( Jt  x )  e.  Nrm ) )
109simprbi 450 . . . 4  |-  ( J  e. CNrm  ->  A. x  e.  ~P  U. J ( Jt  x )  e.  Nrm )
1110adantr 451 . . 3  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  A. x  e.  ~P  U. J ( Jt  x )  e.  Nrm )
12 oveq2 5866 . . . . 5  |-  ( x  =  ( A  i^i  U. J )  ->  ( Jt  x )  =  ( Jt  ( A  i^i  U. J ) ) )
1312eleq1d 2349 . . . 4  |-  ( x  =  ( A  i^i  U. J )  ->  (
( Jt  x )  e.  Nrm  <->  ( Jt  ( A  i^i  U. J
) )  e.  Nrm ) )
1413rspcv 2880 . . 3  |-  ( ( A  i^i  U. J
)  e.  ~P U. J  ->  ( A. x  e.  ~P  U. J ( Jt  x )  e.  Nrm  ->  ( Jt  ( A  i^i  U. J ) )  e. 
Nrm ) )
158, 11, 14sylc 56 . 2  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  ( Jt  ( A  i^i  U. J
) )  e.  Nrm )
162, 15eqeltrd 2357 1  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  ( Jt  A )  e.  Nrm )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   U.cuni 3827  (class class class)co 5858   ↾t crest 13325   Topctop 16631   Nrmcnrm 17038  CNrmccnrm 17039
This theorem is referenced by:  cnrmnrm  17089  restcnrm  17090
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-rest 13327  df-cnrm 17046
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