Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pcl0bN Structured version   Unicode version

Theorem pcl0bN 35790
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 35762 . . . 4  |-  ( ( K  e.  HL  /\  P  C_  A )  ->  P  C_  ( U `  P ) )
4 eqimss 3551 . . . 4  |-  ( ( U `  P )  =  (/)  ->  ( U `
 P )  C_  (/) )
53, 4sylan9ss 3512 . . 3  |-  ( ( ( K  e.  HL  /\  P  C_  A )  /\  ( U `  P
)  =  (/) )  ->  P  C_  (/) )
6 ss0 3825 . . 3  |-  ( P 
C_  (/)  ->  P  =  (/) )
75, 6syl 16 . 2  |-  ( ( ( K  e.  HL  /\  P  C_  A )  /\  ( U `  P
)  =  (/) )  ->  P  =  (/) )
8 fveq2 5872 . . . 4  |-  ( P  =  (/)  ->  ( U `
 P )  =  ( U `  (/) ) )
92pcl0N 35789 . . . 4  |-  ( K  e.  HL  ->  ( U `  (/) )  =  (/) )
108, 9sylan9eqr 2520 . . 3  |-  ( ( K  e.  HL  /\  P  =  (/) )  -> 
( U `  P
)  =  (/) )
1110adantlr 714 . 2  |-  ( ( ( K  e.  HL  /\  P  C_  A )  /\  P  =  (/) )  -> 
( U `  P
)  =  (/) )
127, 11impbida 832 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 1395    e. wcel 1819    C_ wss 3471   (/)c0 3793   ` cfv 5594   Atomscatm 35131   HLchlt 35218   PClcpclN 35754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-riotaBAD 34827
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-undef 7020  df-preset 15684  df-poset 15702  df-plt 15715  df-lub 15731  df-glb 15732  df-join 15733  df-meet 15734  df-p0 15796  df-p1 15797  df-lat 15803  df-clat 15865  df-oposet 35044  df-ol 35046  df-oml 35047  df-covers 35134  df-ats 35135  df-atl 35166  df-cvlat 35190  df-hlat 35219  df-psubsp 35370  df-pmap 35371  df-pclN 35755  df-polarityN 35770
This theorem is referenced by:  pclfinclN  35817
  Copyright terms: Public domain W3C validator