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Theorem restabs 19429
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 996 . . 3  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  J  e.  V )
2 simp3 998 . . 3  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  T  e.  W )
3 ssexg 4593 . . . 4  |-  ( ( S  C_  T  /\  T  e.  W )  ->  S  e.  _V )
433adant1 1014 . . 3  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  S  e.  _V )
5 restco 19428 . . 3  |-  ( ( J  e.  V  /\  T  e.  W  /\  S  e.  _V )  ->  ( ( Jt  T )t  S )  =  ( Jt  ( T  i^i  S ) ) )
61, 2, 4, 5syl3anc 1228 . 2  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  (
( Jt  T )t  S )  =  ( Jt  ( T  i^i  S
) ) )
7 simp2 997 . . . 4  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  S  C_  T )
8 dfss1 3703 . . . 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 6298 . 2  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  ( Jt  ( T  i^i  S ) )  =  ( Jt  S ) )
116, 10eqtrd 2508 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 973    = wceq 1379    e. wcel 1767   _Vcvv 3113    i^i cin 3475    C_ wss 3476  (class class class)co 6282   ↾t crest 14669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-rest 14671
This theorem is referenced by:  restcnrm  19626  fiuncmp  19667  subislly  19745  restnlly  19746  islly2  19748  llyrest  19749  nllyrest  19750  llyidm  19752  nllyidm  19753  cldllycmp  19759  txkgen  19885  rerest  21041  xrrest  21044  cnmpt2pc  21160  cnheiborlem  21186  pcoass  21256  limcres  22022  perfdvf  22039  dvreslem  22045  dvres2lem  22046  dvaddbr  22073  dvmulbr  22074  dvcnvrelem2  22151  psercn  22552  abelth  22567  cxpcn2  22845  cxpcn3  22847  lmlimxrge0  27563  pnfneige0  27566  cvmsss2  28356  cvmliftlem8  28374  cvmliftlem10  28376  cvmlift2lem9  28393  ivthALT  29728  tgiooss  31110  limcresiooub  31184  limcresioolb  31185  cncfuni  31225  cncfiooicclem1  31232  itgsubsticclem  31293  dirkercncflem4  31406  fourierdlem32  31439  fourierdlem33  31440  fourierdlem62  31469  fouriersw  31532
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