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Theorem elpcliN 29900
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 2316 . . . . . . . . 9  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 elpcli.s . . . . . . . . 9  |-  S  =  ( PSubSp `  K )
53, 4psubssat 29761 . . . . . . . 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 3223 . . . . . 6  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  X  C_  ( Atoms `  K )
)
8 elpcli.c . . . . . . 7  |-  U  =  ( PCl `  K
)
93, 4, 8pclvalN 29897 . . . . . 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 2383 . . . 4  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  ( Q  e.  ( U `  X )  <->  Q  e.  |^|
{ z  e.  S  |  X  C_  z } ) )
12 elintrabg 3912 . . . . 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 3234 . . . . . . . 8  |-  ( z  =  Y  ->  ( X  C_  z  <->  X  C_  Y
) )
16 eleq2 2377 . . . . . . . 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 2915 . . . . . 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 1633    e. wcel 1701   A.wral 2577   {crab 2581    C_ wss 3186   |^|cint 3899   ` cfv 5292   Atomscatm 29271   PSubSpcpsubsp 29503   PClcpclN 29894
This theorem is referenced by:  pclfinclN  29957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-psubsp 29510  df-pclN 29895
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