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Theorem atpsubN 29209
Description: The set of all atoms is a projective subspace. Remark below Definition 15.1 of [MaedaMaeda] p. 61. (Contributed by NM, 13-Oct-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
atpsub.a  |-  A  =  ( Atoms `  K )
atpsub.s  |-  S  =  ( PSubSp `  K )
Assertion
Ref Expression
atpsubN  |-  ( K  e.  V  ->  A  e.  S )
Dummy variables  q  p  r are mutually distinct and distinct from all other variables.

Proof of Theorem atpsubN
StepHypRef Expression
1 ssid 3198 . . 3  |-  A  C_  A
2 ax-1 7 . . . . 5  |-  ( r  e.  A  ->  (
r ( le `  K ) ( p ( join `  K
) q )  -> 
r  e.  A ) )
32rgen 2609 . . . 4  |-  A. r  e.  A  ( r
( le `  K
) ( p (
join `  K )
q )  ->  r  e.  A )
43rgen2w 2612 . . 3  |-  A. p  e.  A  A. q  e.  A  A. r  e.  A  ( r
( le `  K
) ( p (
join `  K )
q )  ->  r  e.  A )
51, 4pm3.2i 443 . 2  |-  ( A 
C_  A  /\  A. p  e.  A  A. q  e.  A  A. r  e.  A  (
r ( le `  K ) ( p ( join `  K
) q )  -> 
r  e.  A ) )
6 eqid 2284 . . 3  |-  ( le
`  K )  =  ( le `  K
)
7 eqid 2284 . . 3  |-  ( join `  K )  =  (
join `  K )
8 atpsub.a . . 3  |-  A  =  ( Atoms `  K )
9 atpsub.s . . 3  |-  S  =  ( PSubSp `  K )
106, 7, 8, 9ispsubsp 29201 . 2  |-  ( K  e.  V  ->  ( A  e.  S  <->  ( A  C_  A  /\  A. p  e.  A  A. q  e.  A  A. r  e.  A  ( r
( le `  K
) ( p (
join `  K )
q )  ->  r  e.  A ) ) ) )
115, 10mpbiri 226 1  |-  ( K  e.  V  ->  A  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685   A.wral 2544    C_ wss 3153   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   lecple 13209   joincjn 14072   Atomscatm 28720   PSubSpcpsubsp 28952
This theorem is referenced by:  pclvalN  29346  pclclN  29347
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  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 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  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
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