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Theorem pcl0bN 33559
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 33531 . . . 4  |-  ( ( K  e.  HL  /\  P  C_  A )  ->  P  C_  ( U `  P ) )
4 eqimss 3470 . . . 4  |-  ( ( U `  P )  =  (/)  ->  ( U `
 P )  C_  (/) )
53, 4sylan9ss 3431 . . 3  |-  ( ( ( K  e.  HL  /\  P  C_  A )  /\  ( U `  P
)  =  (/) )  ->  P  C_  (/) )
6 ss0 3768 . . 3  |-  ( P 
C_  (/)  ->  P  =  (/) )
75, 6syl 17 . 2  |-  ( ( ( K  e.  HL  /\  P  C_  A )  /\  ( U `  P
)  =  (/) )  ->  P  =  (/) )
8 fveq2 5879 . . . 4  |-  ( P  =  (/)  ->  ( U `
 P )  =  ( U `  (/) ) )
92pcl0N 33558 . . . 4  |-  ( K  e.  HL  ->  ( U `  (/) )  =  (/) )
108, 9sylan9eqr 2527 . . 3  |-  ( ( K  e.  HL  /\  P  =  (/) )  -> 
( U `  P
)  =  (/) )
1110adantlr 729 . 2  |-  ( ( ( K  e.  HL  /\  P  C_  A )  /\  P  =  (/) )  -> 
( U `  P
)  =  (/) )
127, 11impbida 850 1  |-  ( ( K  e.  HL  /\  P  C_  A )  -> 
( ( U `  P )  =  (/)  <->  P  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    C_ wss 3390   (/)c0 3722   ` cfv 5589   Atomscatm 32900   HLchlt 32987   PClcpclN 33523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-riotaBAD 32589
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-undef 7038  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-p1 16364  df-lat 16370  df-clat 16432  df-oposet 32813  df-ol 32815  df-oml 32816  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-psubsp 33139  df-pmap 33140  df-pclN 33524  df-polarityN 33539
This theorem is referenced by:  pclfinclN  33586
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