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Theorem atpsubN 30389
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 3359 . . 3  |-  A  C_  A
2 ax-1 5 . . . . 5  |-  ( r  e.  A  ->  (
r ( le `  K ) ( p ( join `  K
) q )  -> 
r  e.  A ) )
32rgen 2763 . . . 4  |-  A. r  e.  A  ( r
( le `  K
) ( p (
join `  K )
q )  ->  r  e.  A )
43rgen2w 2766 . . 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 442 . 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 2435 . . 3  |-  ( le
`  K )  =  ( le `  K
)
7 eqid 2435 . . 3  |-  ( join `  K )  =  (
join `  K )
8 atpsub.a . . 3  |-  A  =  ( Atoms `  K )
9 atpsub.s . . 3  |-  S  =  ( PSubSp `  K )
106, 7, 8, 9ispsubsp 30381 . 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 225 1  |-  ( K  e.  V  ->  A  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   class class class wbr 4204   ` cfv 5445  (class class class)co 6072   lecple 13524   joincjn 14389   Atomscatm 29900   PSubSpcpsubsp 30132
This theorem is referenced by:  pclvalN  30526  pclclN  30527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-iota 5409  df-fun 5447  df-fv 5453  df-ov 6075  df-psubsp 30139
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