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Theorem elpcliN 30082
Description: Implication of membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
elpcli.s  |-  S  =  ( PSubSp `  K )
elpcli.c  |-  U  =  ( PCl `  K
)
Assertion
Ref Expression
elpcliN  |-  ( ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  /\  Q  e.  ( U `  X )
)  ->  Q  e.  Y )

Proof of Theorem elpcliN
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 simp1 955 . . . . . 6  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  K  e.  V )
2 simp2 956 . . . . . . 7  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  X  C_  Y )
3 eqid 2283 . . . . . . . . 9  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 elpcli.s . . . . . . . . 9  |-  S  =  ( PSubSp `  K )
53, 4psubssat 29943 . . . . . . . 8  |-  ( ( K  e.  V  /\  Y  e.  S )  ->  Y  C_  ( Atoms `  K ) )
653adant2 974 . . . . . . 7  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  Y  C_  ( Atoms `  K )
)
72, 6sstrd 3189 . . . . . 6  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  X  C_  ( Atoms `  K )
)
8 elpcli.c . . . . . . 7  |-  U  =  ( PCl `  K
)
93, 4, 8pclvalN 30079 . . . . . 6  |-  ( ( K  e.  V  /\  X  C_  ( Atoms `  K
) )  ->  ( U `  X )  =  |^| { z  e.  S  |  X  C_  z } )
101, 7, 9syl2anc 642 . . . . 5  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  ( U `  X )  =  |^| { z  e.  S  |  X  C_  z } )
1110eleq2d 2350 . . . 4  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  ( Q  e.  ( U `  X )  <->  Q  e.  |^|
{ z  e.  S  |  X  C_  z } ) )
12 elintrabg 3875 . . . . 5  |-  ( Q  e.  |^| { z  e.  S  |  X  C_  z }  ->  ( Q  e.  |^| { z  e.  S  |  X  C_  z }  <->  A. z  e.  S  ( X  C_  z  ->  Q  e.  z )
) )
1312ibi 232 . . . 4  |-  ( Q  e.  |^| { z  e.  S  |  X  C_  z }  ->  A. z  e.  S  ( X  C_  z  ->  Q  e.  z ) )
1411, 13syl6bi 219 . . 3  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  ( Q  e.  ( U `  X )  ->  A. z  e.  S  ( X  C_  z  ->  Q  e.  z ) ) )
15 sseq2 3200 . . . . . . . 8  |-  ( z  =  Y  ->  ( X  C_  z  <->  X  C_  Y
) )
16 eleq2 2344 . . . . . . . 8  |-  ( z  =  Y  ->  ( Q  e.  z  <->  Q  e.  Y ) )
1715, 16imbi12d 311 . . . . . . 7  |-  ( z  =  Y  ->  (
( X  C_  z  ->  Q  e.  z )  <-> 
( X  C_  Y  ->  Q  e.  Y ) ) )
1817rspccv 2881 . . . . . 6  |-  ( A. z  e.  S  ( X  C_  z  ->  Q  e.  z )  ->  ( Y  e.  S  ->  ( X  C_  Y  ->  Q  e.  Y ) ) )
1918com13 74 . . . . 5  |-  ( X 
C_  Y  ->  ( Y  e.  S  ->  ( A. z  e.  S  ( X  C_  z  ->  Q  e.  z )  ->  Q  e.  Y ) ) )
2019imp 418 . . . 4  |-  ( ( X  C_  Y  /\  Y  e.  S )  ->  ( A. z  e.  S  ( X  C_  z  ->  Q  e.  z )  ->  Q  e.  Y ) )
21203adant1 973 . . 3  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  ( A. z  e.  S  ( X  C_  z  ->  Q  e.  z )  ->  Q  e.  Y ) )
2214, 21syld 40 . 2  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  ( Q  e.  ( U `  X )  ->  Q  e.  Y ) )
2322imp 418 1  |-  ( ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  /\  Q  e.  ( U `  X )
)  ->  Q  e.  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547    C_ wss 3152   |^|cint 3862   ` cfv 5255   Atomscatm 29453   PSubSpcpsubsp 29685   PClcpclN 30076
This theorem is referenced by:  pclfinclN  30139
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-pow 4188  ax-pr 4214
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-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  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-psubsp 29692  df-pclN 30077
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