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Theorem restabs 18749
Description: Equivalence of being a subspace of a subspace and being a subspace of the original. (Contributed by Jeff Hankins, 11-Jul-2009.) (Proof shortened by Mario Carneiro, 1-May-2015.)
Assertion
Ref Expression
restabs  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  (
( Jt  T )t  S )  =  ( Jt  S ) )

Proof of Theorem restabs
StepHypRef Expression
1 simp1 988 . . 3  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  J  e.  V )
2 simp3 990 . . 3  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  T  e.  W )
3 ssexg 4433 . . . 4  |-  ( ( S  C_  T  /\  T  e.  W )  ->  S  e.  _V )
433adant1 1006 . . 3  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  S  e.  _V )
5 restco 18748 . . 3  |-  ( ( J  e.  V  /\  T  e.  W  /\  S  e.  _V )  ->  ( ( Jt  T )t  S )  =  ( Jt  ( T  i^i  S ) ) )
61, 2, 4, 5syl3anc 1218 . 2  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  (
( Jt  T )t  S )  =  ( Jt  ( T  i^i  S
) ) )
7 simp2 989 . . . 4  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  S  C_  T )
8 dfss1 3550 . . . 4  |-  ( S 
C_  T  <->  ( T  i^i  S )  =  S )
97, 8sylib 196 . . 3  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  ( T  i^i  S )  =  S )
109oveq2d 6102 . 2  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  ( Jt  ( T  i^i  S ) )  =  ( Jt  S ) )
116, 10eqtrd 2470 1  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  (
( Jt  T )t  S )  =  ( Jt  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2967    i^i cin 3322    C_ wss 3323  (class class class)co 6086   ↾t crest 14351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-rest 14353
This theorem is referenced by:  restcnrm  18946  fiuncmp  18987  subislly  19065  restnlly  19066  islly2  19068  llyrest  19069  nllyrest  19070  llyidm  19072  nllyidm  19073  cldllycmp  19079  txkgen  19205  rerest  20361  xrrest  20364  cnmpt2pc  20480  cnheiborlem  20506  pcoass  20576  limcres  21341  perfdvf  21358  dvreslem  21364  dvres2lem  21365  dvaddbr  21392  dvmulbr  21393  dvcnvrelem2  21470  psercn  21871  abelth  21886  cxpcn2  22164  cxpcn3  22166  lmlimxrge0  26347  pnfneige0  26350  cvmsss2  27132  cvmliftlem8  27150  cvmliftlem10  27152  cvmlift2lem9  27169  ivthALT  28501
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