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Theorem atpsubN 35890
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 )

Proof of Theorem atpsubN
Dummy variables  q  p  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3436 . . 3  |-  A  C_  A
2 ax-1 6 . . . . 5  |-  ( r  e.  A  ->  (
r ( le `  K ) ( p ( join `  K
) q )  -> 
r  e.  A ) )
32rgen 2742 . . . 4  |-  A. r  e.  A  ( r
( le `  K
) ( p (
join `  K )
q )  ->  r  e.  A )
43rgen2w 2744 . . 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 453 . 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 2382 . . 3  |-  ( le
`  K )  =  ( le `  K
)
7 eqid 2382 . . 3  |-  ( join `  K )  =  (
join `  K )
8 atpsub.a . . 3  |-  A  =  ( Atoms `  K )
9 atpsub.s . . 3  |-  S  =  ( PSubSp `  K )
106, 7, 8, 9ispsubsp 35882 . 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 233 1  |-  ( K  e.  V  ->  A  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   A.wral 2732    C_ wss 3389   class class class wbr 4367   ` cfv 5496  (class class class)co 6196   lecple 14709   joincjn 15690   Atomscatm 35401   PSubSpcpsubsp 35633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-ov 6199  df-psubsp 35640
This theorem is referenced by:  pclvalN  36027  pclclN  36028
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