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Theorem atpsubN 29315
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 3197 . . 3  |-  A  C_  A
2 ax-1 5 . . . . 5  |-  ( r  e.  A  ->  (
r ( le `  K ) ( p ( join `  K
) q )  -> 
r  e.  A ) )
32rgen 2608 . . . 4  |-  A. r  e.  A  ( r
( le `  K
) ( p (
join `  K )
q )  ->  r  e.  A )
43rgen2w 2611 . . 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 441 . 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 2283 . . 3  |-  ( le
`  K )  =  ( le `  K
)
7 eqid 2283 . . 3  |-  ( join `  K )  =  (
join `  K )
8 atpsub.a . . 3  |-  A  =  ( Atoms `  K )
9 atpsub.s . . 3  |-  S  =  ( PSubSp `  K )
106, 7, 8, 9ispsubsp 29307 . 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 224 1  |-  ( K  e.  V  ->  A  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   Atomscatm 28826   PSubSpcpsubsp 29058
This theorem is referenced by:  pclvalN  29452  pclclN  29453
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-psubsp 29065
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