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Theorem cnrmi 17386
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 2412 . . 3  |-  U. J  =  U. J
21restin 17192 . 2  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  ( Jt  A )  =  ( Jt  ( A  i^i  U. J ) ) )
3 inss2 3530 . . . . 5  |-  ( A  i^i  U. J ) 
C_  U. J
4 inex1g 4314 . . . . . 6  |-  ( A  e.  V  ->  ( A  i^i  U. J )  e.  _V )
5 elpwg 3774 . . . . . 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 225 . . . 4  |-  ( A  e.  V  ->  ( A  i^i  U. J )  e.  ~P U. J
)
87adantl 453 . . 3  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  ( A  i^i  U. J )  e.  ~P U. J
)
91iscnrm 17349 . . . . 5  |-  ( J  e. CNrm 
<->  ( J  e.  Top  /\ 
A. x  e.  ~P  U. J ( Jt  x )  e.  Nrm ) )
109simprbi 451 . . . 4  |-  ( J  e. CNrm  ->  A. x  e.  ~P  U. J ( Jt  x )  e.  Nrm )
1110adantr 452 . . 3  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  A. x  e.  ~P  U. J ( Jt  x )  e.  Nrm )
12 oveq2 6056 . . . . 5  |-  ( x  =  ( A  i^i  U. J )  ->  ( Jt  x )  =  ( Jt  ( A  i^i  U. J ) ) )
1312eleq1d 2478 . . . 4  |-  ( x  =  ( A  i^i  U. J )  ->  (
( Jt  x )  e.  Nrm  <->  ( Jt  ( A  i^i  U. J
) )  e.  Nrm ) )
1413rspcv 3016 . . 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 58 . 2  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  ( Jt  ( A  i^i  U. J
) )  e.  Nrm )
162, 15eqeltrd 2486 1  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  ( Jt  A )  e.  Nrm )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674   _Vcvv 2924    i^i cin 3287    C_ wss 3288   ~Pcpw 3767   U.cuni 3983  (class class class)co 6048   ↾t crest 13611   Topctop 16921   Nrmcnrm 17336  CNrmccnrm 17337
This theorem is referenced by:  cnrmnrm  17387  restcnrm  17388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-rest 13613  df-cnrm 17344
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