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Theorem restabs 16896
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 955 . . 3  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  J  e.  V )
2 simp3 957 . . 3  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  T  e.  W )
3 ssexg 4160 . . . 4  |-  ( ( S  C_  T  /\  T  e.  W )  ->  S  e.  _V )
433adant1 973 . . 3  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  S  e.  _V )
5 restco 16895 . . 3  |-  ( ( J  e.  V  /\  T  e.  W  /\  S  e.  _V )  ->  ( ( Jt  T )t  S )  =  ( Jt  ( T  i^i  S ) ) )
61, 2, 4, 5syl3anc 1182 . 2  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  (
( Jt  T )t  S )  =  ( Jt  ( T  i^i  S
) ) )
7 simp2 956 . . . 4  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  S  C_  T )
8 dfss1 3373 . . . 4  |-  ( S 
C_  T  <->  ( T  i^i  S )  =  S )
97, 8sylib 188 . . 3  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  ( T  i^i  S )  =  S )
109oveq2d 5874 . 2  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  ( Jt  ( T  i^i  S ) )  =  ( Jt  S ) )
116, 10eqtrd 2315 1  |-  ( ( J  e.  V  /\  S  C_  T  /\  T  e.  W )  ->  (
( Jt  T )t  S )  =  ( Jt  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152  (class class class)co 5858   ↾t crest 13325
This theorem is referenced by:  restcnrm  17090  fiuncmp  17131  subislly  17207  restnlly  17208  islly2  17210  llyrest  17211  nllyrest  17212  llyidm  17214  nllyidm  17215  cldllycmp  17221  txkgen  17346  rerest  18310  xrrest  18313  cnmpt2pc  18426  cnheiborlem  18452  pcoass  18522  limcres  19236  perfdvf  19253  dvreslem  19259  dvres2lem  19260  dvaddbr  19287  dvmulbr  19288  dvcnvrelem2  19365  psercn  19802  abelth  19817  cxpcn2  20086  cxpcn3  20088  lmlimxrge0  23372  pnfneige0  23374  cvmsss2  23805  cvmliftlem8  23823  cvmliftlem10  23825  cvmlift2lem9  23842  ivthALT  26258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-rest 13327
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