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Theorem restabs 19959
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 997 . . 3  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  J  e.  V )
2 simp3 999 . . 3  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  T  e.  W )
3 ssexg 4540 . . . 4  |-  ( ( S  C_  T  /\  T  e.  W )  ->  S  e.  _V )
433adant1 1015 . . 3  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  S  e.  _V )
5 restco 19958 . . 3  |-  ( ( J  e.  V  /\  T  e.  W  /\  S  e.  _V )  ->  ( ( Jt  T )t  S )  =  ( Jt  ( T  i^i  S ) ) )
61, 2, 4, 5syl3anc 1230 . 2  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  (
( Jt  T )t  S )  =  ( Jt  ( T  i^i  S
) ) )
7 simp2 998 . . . 4  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  S  C_  T )
8 dfss1 3644 . . . 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 6294 . 2  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  ( Jt  ( T  i^i  S ) )  =  ( Jt  S ) )
116, 10eqtrd 2443 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 974    = wceq 1405    e. wcel 1842   _Vcvv 3059    i^i cin 3413    C_ wss 3414  (class class class)co 6278   ↾t crest 15035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-rest 15037
This theorem is referenced by:  restcnrm  20156  fiuncmp  20197  subislly  20274  restnlly  20275  islly2  20277  llyrest  20278  nllyrest  20279  llyidm  20281  nllyidm  20282  cldllycmp  20288  txkgen  20445  rerest  21601  xrrest  21604  cnmpt2pc  21720  cnheiborlem  21746  pcoass  21816  limcres  22582  perfdvf  22599  dvreslem  22605  dvres2lem  22606  dvaddbr  22633  dvmulbr  22634  dvcnvrelem2  22711  psercn  23113  abelth  23128  cxpcn2  23416  cxpcn3  23418  lmlimxrge0  28383  pnfneige0  28386  cvmsss2  29571  cvmliftlem8  29589  cvmliftlem10  29591  cvmlift2lem9  29608  ivthALT  30563  limcresiooub  37016  limcresioolb  37017  cncfuni  37057  cncfiooicclem1  37064  itgsubsticclem  37142  dirkercncflem4  37256  fourierdlem32  37289  fourierdlem33  37290  fourierdlem62  37319  fouriersw  37382
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