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Theorem cnrmi 17188
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 2358 . . 3  |-  U. J  =  U. J
21restin 16997 . 2  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  ( Jt  A )  =  ( Jt  ( A  i^i  U. J ) ) )
3 inss2 3466 . . . . 5  |-  ( A  i^i  U. J ) 
C_  U. J
4 inex1g 4236 . . . . . 6  |-  ( A  e.  V  ->  ( A  i^i  U. J )  e.  _V )
5 elpwg 3708 . . . . . 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 17151 . . . . 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 5950 . . . . 5  |-  ( x  =  ( A  i^i  U. J )  ->  ( Jt  x )  =  ( Jt  ( A  i^i  U. J ) ) )
1312eleq1d 2424 . . . 4  |-  ( x  =  ( A  i^i  U. J )  ->  (
( Jt  x )  e.  Nrm  <->  ( Jt  ( A  i^i  U. J
) )  e.  Nrm ) )
1413rspcv 2956 . . 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 2432 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 1642    e. wcel 1710   A.wral 2619   _Vcvv 2864    i^i cin 3227    C_ wss 3228   ~Pcpw 3701   U.cuni 3906  (class class class)co 5942   ↾t crest 13418   Topctop 16731   Nrmcnrm 17138  CNrmccnrm 17139
This theorem is referenced by:  cnrmnrm  17189  restcnrm  17190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-rest 13420  df-cnrm 17146
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