Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  0psubN Unicode version

Theorem 0psubN 29205
Description: The empty set is a projective subspace. Remark below Definition 15.1 of [MaedaMaeda] p. 61. (Contributed by NM, 13-Oct-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
0psub.s  |-  S  =  ( PSubSp `  K )
Assertion
Ref Expression
0psubN  |-  ( K  e.  V  ->  (/)  e.  S
)

Proof of Theorem 0psubN
StepHypRef Expression
1 0ss 3484 . . 3  |-  (/)  C_  ( Atoms `  K )
2 ral0 3559 . . 3  |-  A. p  e.  (/)  A. q  e.  (/)  A. r  e.  (
Atoms `  K ) ( r ( le `  K ) ( p ( join `  K
) q )  -> 
r  e.  (/) )
31, 2pm3.2i 443 . 2  |-  ( (/)  C_  ( Atoms `  K )  /\  A. p  e.  (/)  A. q  e.  (/)  A. r  e.  ( Atoms `  K )
( r ( le
`  K ) ( p ( join `  K
) q )  -> 
r  e.  (/) ) )
4 eqid 2284 . . 3  |-  ( le
`  K )  =  ( le `  K
)
5 eqid 2284 . . 3  |-  ( join `  K )  =  (
join `  K )
6 eqid 2284 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
7 0psub.s . . 3  |-  S  =  ( PSubSp `  K )
84, 5, 6, 7ispsubsp 29201 . 2  |-  ( K  e.  V  ->  ( (/) 
e.  S  <->  ( (/)  C_  ( Atoms `  K )  /\  A. p  e.  (/)  A. q  e.  (/)  A. r  e.  ( Atoms `  K )
( r ( le
`  K ) ( p ( join `  K
) q )  -> 
r  e.  (/) ) ) ) )
93, 8mpbiri 226 1  |-  ( K  e.  V  ->  (/)  e.  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1628    e. wcel 1688   A.wral 2544    C_ wss 3153   (/)c0 3456   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   lecple 13209   joincjn 14072   Atomscatm 28720   PSubSpcpsubsp 28952
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fv 5229  df-ov 5822  df-psubsp 28959
  Copyright terms: Public domain W3C validator