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Theorem pcl0bN 30657
Description: The projective subspace closure of the empty subspace. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
pcl0b.a  |-  A  =  ( Atoms `  K )
pcl0b.c  |-  U  =  ( PCl `  K
)
Assertion
Ref Expression
pcl0bN  |-  ( ( K  e.  HL  /\  P  C_  A )  -> 
( ( U `  P )  =  (/)  <->  P  =  (/) ) )

Proof of Theorem pcl0bN
StepHypRef Expression
1 pcl0b.a . . . . 5  |-  A  =  ( Atoms `  K )
2 pcl0b.c . . . . 5  |-  U  =  ( PCl `  K
)
31, 2pclssidN 30629 . . . 4  |-  ( ( K  e.  HL  /\  P  C_  A )  ->  P  C_  ( U `  P ) )
4 eqimss 3392 . . . 4  |-  ( ( U `  P )  =  (/)  ->  ( U `
 P )  C_  (/) )
53, 4sylan9ss 3353 . . 3  |-  ( ( ( K  e.  HL  /\  P  C_  A )  /\  ( U `  P
)  =  (/) )  ->  P  C_  (/) )
6 ss0 3650 . . 3  |-  ( P 
C_  (/)  ->  P  =  (/) )
75, 6syl 16 . 2  |-  ( ( ( K  e.  HL  /\  P  C_  A )  /\  ( U `  P
)  =  (/) )  ->  P  =  (/) )
8 fveq2 5720 . . . 4  |-  ( P  =  (/)  ->  ( U `
 P )  =  ( U `  (/) ) )
92pcl0N 30656 . . . 4  |-  ( K  e.  HL  ->  ( U `  (/) )  =  (/) )
108, 9sylan9eqr 2489 . . 3  |-  ( ( K  e.  HL  /\  P  =  (/) )  -> 
( U `  P
)  =  (/) )
1110adantlr 696 . 2  |-  ( ( ( K  e.  HL  /\  P  C_  A )  /\  P  =  (/) )  -> 
( U `  P
)  =  (/) )
127, 11impbida 806 1  |-  ( ( K  e.  HL  /\  P  C_  A )  -> 
( ( U `  P )  =  (/)  <->  P  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3312   (/)c0 3620   ` cfv 5446   Atomscatm 29998   HLchlt 30085   PClcpclN 30621
This theorem is referenced by:  pclfinclN  30684
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-psubsp 30237  df-pmap 30238  df-pclN 30622  df-polarityN 30637
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