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Theorem restcnrm 20455
Description: A subspace of a completely normal space is completely normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
restcnrm  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  ( Jt  A )  e. CNrm )

Proof of Theorem restcnrm
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2471 . . 3  |-  U. J  =  U. J
21restin 20259 . 2  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  ( Jt  A )  =  ( Jt  ( A  i^i  U. J ) ) )
3 simpll 768 . . . . . 6  |-  ( ( ( J  e. CNrm  /\  A  e.  V )  /\  x  e.  ~P ( A  i^i  U. J
) )  ->  J  e. CNrm )
4 elpwi 3951 . . . . . . 7  |-  ( x  e.  ~P ( A  i^i  U. J )  ->  x  C_  ( A  i^i  U. J ) )
54adantl 473 . . . . . 6  |-  ( ( ( J  e. CNrm  /\  A  e.  V )  /\  x  e.  ~P ( A  i^i  U. J
) )  ->  x  C_  ( A  i^i  U. J ) )
6 inex1g 4539 . . . . . . 7  |-  ( A  e.  V  ->  ( A  i^i  U. J )  e.  _V )
76ad2antlr 741 . . . . . 6  |-  ( ( ( J  e. CNrm  /\  A  e.  V )  /\  x  e.  ~P ( A  i^i  U. J
) )  ->  ( A  i^i  U. J )  e.  _V )
8 restabs 20258 . . . . . 6  |-  ( ( J  e. CNrm  /\  x  C_  ( A  i^i  U. J )  /\  ( A  i^i  U. J )  e.  _V )  -> 
( ( Jt  ( A  i^i  U. J ) )t  x )  =  ( Jt  x ) )
93, 5, 7, 8syl3anc 1292 . . . . 5  |-  ( ( ( J  e. CNrm  /\  A  e.  V )  /\  x  e.  ~P ( A  i^i  U. J
) )  ->  (
( Jt  ( A  i^i  U. J ) )t  x )  =  ( Jt  x ) )
10 cnrmi 20453 . . . . . 6  |-  ( ( J  e. CNrm  /\  x  e.  ~P ( A  i^i  U. J ) )  -> 
( Jt  x )  e.  Nrm )
1110adantlr 729 . . . . 5  |-  ( ( ( J  e. CNrm  /\  A  e.  V )  /\  x  e.  ~P ( A  i^i  U. J
) )  ->  ( Jt  x )  e.  Nrm )
129, 11eqeltrd 2549 . . . 4  |-  ( ( ( J  e. CNrm  /\  A  e.  V )  /\  x  e.  ~P ( A  i^i  U. J
) )  ->  (
( Jt  ( A  i^i  U. J ) )t  x )  e.  Nrm )
1312ralrimiva 2809 . . 3  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  A. x  e.  ~P  ( A  i^i  U. J ) ( ( Jt  ( A  i^i  U. J ) )t  x )  e.  Nrm )
14 cnrmtop 20430 . . . . . . 7  |-  ( J  e. CNrm  ->  J  e.  Top )
1514adantr 472 . . . . . 6  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  J  e.  Top )
161toptopon 20025 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
1715, 16sylib 201 . . . . 5  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  J  e.  (TopOn `  U. J ) )
18 inss2 3644 . . . . 5  |-  ( A  i^i  U. J ) 
C_  U. J
19 resttopon 20254 . . . . 5  |-  ( ( J  e.  (TopOn `  U. J )  /\  ( A  i^i  U. J ) 
C_  U. J )  -> 
( Jt  ( A  i^i  U. J ) )  e.  (TopOn `  ( A  i^i  U. J ) ) )
2017, 18, 19sylancl 675 . . . 4  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  ( Jt  ( A  i^i  U. J
) )  e.  (TopOn `  ( A  i^i  U. J ) ) )
21 iscnrm2 20431 . . . 4  |-  ( ( Jt  ( A  i^i  U. J ) )  e.  (TopOn `  ( A  i^i  U. J ) )  ->  ( ( Jt  ( A  i^i  U. J
) )  e. CNrm  <->  A. x  e.  ~P  ( A  i^i  U. J ) ( ( Jt  ( A  i^i  U. J ) )t  x )  e.  Nrm ) )
2220, 21syl 17 . . 3  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  (
( Jt  ( A  i^i  U. J ) )  e. CNrm  <->  A. x  e.  ~P  ( A  i^i  U. J ) ( ( Jt  ( A  i^i  U. J ) )t  x )  e.  Nrm ) )
2313, 22mpbird 240 . 2  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  ( Jt  ( A  i^i  U. J
) )  e. CNrm )
242, 23eqeltrd 2549 1  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  ( Jt  A )  e. CNrm )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   _Vcvv 3031    i^i cin 3389    C_ wss 3390   ~Pcpw 3942   U.cuni 4190   ` cfv 5589  (class class class)co 6308   ↾t crest 15397   Topctop 19994  TopOnctopon 19995   Nrmcnrm 20403  CNrmccnrm 20404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-oadd 7204  df-er 7381  df-en 7588  df-fin 7591  df-fi 7943  df-rest 15399  df-topgen 15420  df-top 19998  df-bases 19999  df-topon 20000  df-cnrm 20411
This theorem is referenced by: (None)
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