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Theorem cnrmi 17429
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 2438 . . 3  |-  U. J  =  U. J
21restin 17235 . 2  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  ( Jt  A )  =  ( Jt  ( A  i^i  U. J ) ) )
3 inss2 3564 . . . . 5  |-  ( A  i^i  U. J ) 
C_  U. J
4 inex1g 4349 . . . . . 6  |-  ( A  e.  V  ->  ( A  i^i  U. J )  e.  _V )
5 elpwg 3808 . . . . . 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 16 . . . . 5  |-  ( A  e.  V  ->  (
( A  i^i  U. J )  e.  ~P U. J  <->  ( A  i^i  U. J )  C_  U. J
) )
73, 6mpbiri 226 . . . 4  |-  ( A  e.  V  ->  ( A  i^i  U. J )  e.  ~P U. J
)
87adantl 454 . . 3  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  ( A  i^i  U. J )  e.  ~P U. J
)
91iscnrm 17392 . . . . 5  |-  ( J  e. CNrm 
<->  ( J  e.  Top  /\ 
A. x  e.  ~P  U. J ( Jt  x )  e.  Nrm ) )
109simprbi 452 . . . 4  |-  ( J  e. CNrm  ->  A. x  e.  ~P  U. J ( Jt  x )  e.  Nrm )
1110adantr 453 . . 3  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  A. x  e.  ~P  U. J ( Jt  x )  e.  Nrm )
12 oveq2 6092 . . . . 5  |-  ( x  =  ( A  i^i  U. J )  ->  ( Jt  x )  =  ( Jt  ( A  i^i  U. J ) ) )
1312eleq1d 2504 . . . 4  |-  ( x  =  ( A  i^i  U. J )  ->  (
( Jt  x )  e.  Nrm  <->  ( Jt  ( A  i^i  U. J
) )  e.  Nrm ) )
1413rspcv 3050 . . 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 59 . 2  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  ( Jt  ( A  i^i  U. J
) )  e.  Nrm )
162, 15eqeltrd 2512 1  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  ( Jt  A )  e.  Nrm )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    i^i cin 3321    C_ wss 3322   ~Pcpw 3801   U.cuni 4017  (class class class)co 6084   ↾t crest 13653   Topctop 16963   Nrmcnrm 17379  CNrmccnrm 17380
This theorem is referenced by:  cnrmnrm  17430  restcnrm  17431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  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-ral 2712  df-rex 2713  df-reu 2714  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-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-rest 13655  df-cnrm 17387
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