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Theorem atpsubN 28743
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
StepHypRef Expression
1 ssid 3118 . . 3  |-  A  C_  A
2 ax-1 7 . . . . 5  |-  ( r  e.  A  ->  (
r ( le `  K ) ( p ( join `  K
) q )  -> 
r  e.  A ) )
32rgen 2570 . . . 4  |-  A. r  e.  A  ( r
( le `  K
) ( p (
join `  K )
q )  ->  r  e.  A )
43rgen2w 2573 . . 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 2253 . . 3  |-  ( le
`  K )  =  ( le `  K
)
7 eqid 2253 . . 3  |-  ( join `  K )  =  (
join `  K )
8 atpsub.a . . 3  |-  A  =  ( Atoms `  K )
9 atpsub.s . . 3  |-  S  =  ( PSubSp `  K )
106, 7, 8, 9ispsubsp 28735 . 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 1619    e. wcel 1621   A.wral 2509    C_ wss 3078   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   lecple 13089   joincjn 13922   Atomscatm 28254   PSubSpcpsubsp 28486
This theorem is referenced by:  pclvalN  28880  pclclN  28881
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fv 4608  df-ov 5713  df-psubsp 28493
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