MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  restcnrm Structured version   Unicode version

Theorem restcnrm 19949
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 2382 . . 3  |-  U. J  =  U. J
21restin 19753 . 2  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  ( Jt  A )  =  ( Jt  ( A  i^i  U. J ) ) )
3 simpll 751 . . . . . 6  |-  ( ( ( J  e. CNrm  /\  A  e.  V )  /\  x  e.  ~P ( A  i^i  U. J
) )  ->  J  e. CNrm )
4 elpwi 3936 . . . . . . 7  |-  ( x  e.  ~P ( A  i^i  U. J )  ->  x  C_  ( A  i^i  U. J ) )
54adantl 464 . . . . . 6  |-  ( ( ( J  e. CNrm  /\  A  e.  V )  /\  x  e.  ~P ( A  i^i  U. J
) )  ->  x  C_  ( A  i^i  U. J ) )
6 inex1g 4508 . . . . . . 7  |-  ( A  e.  V  ->  ( A  i^i  U. J )  e.  _V )
76ad2antlr 724 . . . . . 6  |-  ( ( ( J  e. CNrm  /\  A  e.  V )  /\  x  e.  ~P ( A  i^i  U. J
) )  ->  ( A  i^i  U. J )  e.  _V )
8 restabs 19752 . . . . . 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 1226 . . . . 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 19947 . . . . . 6  |-  ( ( J  e. CNrm  /\  x  e.  ~P ( A  i^i  U. J ) )  -> 
( Jt  x )  e.  Nrm )
1110adantlr 712 . . . . 5  |-  ( ( ( J  e. CNrm  /\  A  e.  V )  /\  x  e.  ~P ( A  i^i  U. J
) )  ->  ( Jt  x )  e.  Nrm )
129, 11eqeltrd 2470 . . . 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 2796 . . 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 19924 . . . . . . 7  |-  ( J  e. CNrm  ->  J  e.  Top )
1514adantr 463 . . . . . 6  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  J  e.  Top )
161toptopon 19519 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
1715, 16sylib 196 . . . . 5  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  J  e.  (TopOn `  U. J ) )
18 inss2 3633 . . . . 5  |-  ( A  i^i  U. J ) 
C_  U. J
19 resttopon 19748 . . . . 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 660 . . . 4  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  ( Jt  ( A  i^i  U. J
) )  e.  (TopOn `  ( A  i^i  U. J ) ) )
21 iscnrm2 19925 . . . 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 16 . . 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 232 . 2  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  ( Jt  ( A  i^i  U. J
) )  e. CNrm )
242, 23eqeltrd 2470 1  |-  ( ( J  e. CNrm  /\  A  e.  V )  ->  ( Jt  A )  e. CNrm )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   A.wral 2732   _Vcvv 3034    i^i cin 3388    C_ wss 3389   ~Pcpw 3927   U.cuni 4163   ` cfv 5496  (class class class)co 6196   ↾t crest 14828   Topctop 19479  TopOnctopon 19480   Nrmcnrm 19897  CNrmccnrm 19898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-oadd 7052  df-er 7229  df-en 7436  df-fin 7439  df-fi 7786  df-rest 14830  df-topgen 14851  df-top 19484  df-bases 19486  df-topon 19487  df-cnrm 19905
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator