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Theorem pclidN 30631
Description: The projective subspace closure of a projective subspace is itself. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
pclid.s  |-  S  =  ( PSubSp `  K )
pclid.c  |-  U  =  ( PCl `  K
)
Assertion
Ref Expression
pclidN  |-  ( ( K  e.  V  /\  X  e.  S )  ->  ( U `  X
)  =  X )

Proof of Theorem pclidN
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqid 2436 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
2 pclid.s . . . 4  |-  S  =  ( PSubSp `  K )
31, 2psubssat 30489 . . 3  |-  ( ( K  e.  V  /\  X  e.  S )  ->  X  C_  ( Atoms `  K ) )
4 pclid.c . . . 4  |-  U  =  ( PCl `  K
)
51, 2, 4pclvalN 30625 . . 3  |-  ( ( K  e.  V  /\  X  C_  ( Atoms `  K
) )  ->  ( U `  X )  =  |^| { y  e.  S  |  X  C_  y } )
63, 5syldan 457 . 2  |-  ( ( K  e.  V  /\  X  e.  S )  ->  ( U `  X
)  =  |^| { y  e.  S  |  X  C_  y } )
7 intmin 4063 . . 3  |-  ( X  e.  S  ->  |^| { y  e.  S  |  X  C_  y }  =  X )
87adantl 453 . 2  |-  ( ( K  e.  V  /\  X  e.  S )  ->  |^| { y  e.  S  |  X  C_  y }  =  X
)
96, 8eqtrd 2468 1  |-  ( ( K  e.  V  /\  X  e.  S )  ->  ( U `  X
)  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2702    C_ wss 3313   |^|cint 4043   ` cfv 5447   Atomscatm 29999   PSubSpcpsubsp 30231   PClcpclN 30622
This theorem is referenced by:  pclbtwnN  30632  pclunN  30633  pclun2N  30634  pclfinN  30635  pclss2polN  30656  pclfinclN  30685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-psubsp 30238  df-pclN 30623
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