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Theorem pcl0bN 33567
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 33539 . . . 4  |-  ( ( K  e.  HL  /\  P  C_  A )  ->  P  C_  ( U `  P ) )
4 eqimss 3408 . . . 4  |-  ( ( U `  P )  =  (/)  ->  ( U `
 P )  C_  (/) )
53, 4sylan9ss 3369 . . 3  |-  ( ( ( K  e.  HL  /\  P  C_  A )  /\  ( U `  P
)  =  (/) )  ->  P  C_  (/) )
6 ss0 3668 . . 3  |-  ( P 
C_  (/)  ->  P  =  (/) )
75, 6syl 16 . 2  |-  ( ( ( K  e.  HL  /\  P  C_  A )  /\  ( U `  P
)  =  (/) )  ->  P  =  (/) )
8 fveq2 5691 . . . 4  |-  ( P  =  (/)  ->  ( U `
 P )  =  ( U `  (/) ) )
92pcl0N 33566 . . . 4  |-  ( K  e.  HL  ->  ( U `  (/) )  =  (/) )
108, 9sylan9eqr 2497 . . 3  |-  ( ( K  e.  HL  /\  P  =  (/) )  -> 
( U `  P
)  =  (/) )
1110adantlr 714 . 2  |-  ( ( ( K  e.  HL  /\  P  C_  A )  /\  P  =  (/) )  -> 
( U `  P
)  =  (/) )
127, 11impbida 828 1  |-  ( ( K  e.  HL  /\  P  C_  A )  -> 
( ( U `  P )  =  (/)  <->  P  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3328   (/)c0 3637   ` cfv 5418   Atomscatm 32908   HLchlt 32995   PClcpclN 33531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-riotaBAD 32604
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-undef 6792  df-poset 15116  df-plt 15128  df-lub 15144  df-glb 15145  df-join 15146  df-meet 15147  df-p0 15209  df-p1 15210  df-lat 15216  df-clat 15278  df-oposet 32821  df-ol 32823  df-oml 32824  df-covers 32911  df-ats 32912  df-atl 32943  df-cvlat 32967  df-hlat 32996  df-psubsp 33147  df-pmap 33148  df-pclN 33532  df-polarityN 33547
This theorem is referenced by:  pclfinclN  33594
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