Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pexmidN Unicode version

Theorem pexmidN 29288
Description: Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 29272. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 29286. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pexmid.a  |-  A  =  ( Atoms `  K )
pexmid.p  |-  .+  =  ( + P `  K
)
pexmid.o  |-  ._|_  =  ( _|_ P `  K
)
Assertion
Ref Expression
pexmidN  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( X  .+  (  ._|_  `  X ) )  =  A )

Proof of Theorem pexmidN
StepHypRef Expression
1 simpll 733 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  ->  K  e.  HL )
2 simplr 734 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  ->  X  C_  A )
3 pexmid.a . . . . . . 7  |-  A  =  ( Atoms `  K )
4 pexmid.o . . . . . . 7  |-  ._|_  =  ( _|_ P `  K
)
53, 4polssatN 29227 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  C_  A )
65adantr 453 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  X )  C_  A )
7 pexmid.p . . . . . 6  |-  .+  =  ( + P `  K
)
83, 7, 4poldmj1N 29247 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A  /\  (  ._|_  `  X )  C_  A )  ->  (  ._|_  `  ( X  .+  (  ._|_  `  X )
) )  =  ( (  ._|_  `  X )  i^i  (  ._|_  `  (  ._|_  `  X ) ) ) )
91, 2, 6, 8syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  ( X  .+  (  ._|_  `  X
) ) )  =  ( (  ._|_  `  X
)  i^i  (  ._|_  `  (  ._|_  `  X ) ) ) )
103, 4pnonsingN 29252 . . . . 5  |-  ( ( K  e.  HL  /\  (  ._|_  `  X )  C_  A )  ->  (
(  ._|_  `  X )  i^i  (  ._|_  `  (  ._|_  `  X ) ) )  =  (/) )
111, 6, 10syl2anc 645 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( (  ._|_  `  X
)  i^i  (  ._|_  `  (  ._|_  `  X ) ) )  =  (/) )
129, 11eqtrd 2288 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  ( X  .+  (  ._|_  `  X
) ) )  =  (/) )
1312fveq2d 5427 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  (  ._|_  `  ( X  .+  (  ._|_  `  X ) ) ) )  =  ( 
._|_  `  (/) ) )
14 simpr 449 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  X )
15 eqid 2256 . . . . . . 7  |-  ( PSubCl `  K )  =  (
PSubCl `  K )
163, 4, 15ispsubclN 29256 . . . . . 6  |-  ( K  e.  HL  ->  ( X  e.  ( PSubCl `  K )  <->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
1716ad2antrr 709 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( X  e.  (
PSubCl `  K )  <->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
182, 14, 17mpbir2and 893 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  ->  X  e.  ( PSubCl `  K ) )
193, 4, 15polsubclN 29271 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  e.  ( PSubCl `  K )
)
2019adantr 453 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  X )  e.  ( PSubCl `  K )
)
213, 42polssN 29234 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  X  C_  (  ._|_  `  (  ._|_  `  X ) ) )
2221adantr 453 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  ->  X  C_  (  ._|_  `  (  ._|_  `  X ) ) )
237, 4, 15osumclN 29286 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  ( PSubCl `  K )  /\  (  ._|_  `  X )  e.  ( PSubCl `  K )
)  /\  X  C_  (  ._|_  `  (  ._|_  `  X
) ) )  -> 
( X  .+  (  ._|_  `  X ) )  e.  ( PSubCl `  K
) )
241, 18, 20, 22, 23syl31anc 1190 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( X  .+  (  ._|_  `  X ) )  e.  ( PSubCl `  K
) )
254, 15psubcli2N 29258 . . 3  |-  ( ( K  e.  HL  /\  ( X  .+  (  ._|_  `  X ) )  e.  ( PSubCl `  K )
)  ->  (  ._|_  `  (  ._|_  `  ( X 
.+  (  ._|_  `  X
) ) ) )  =  ( X  .+  (  ._|_  `  X )
) )
261, 24, 25syl2anc 645 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  (  ._|_  `  ( X  .+  (  ._|_  `  X ) ) ) )  =  ( X  .+  (  ._|_  `  X ) ) )
273, 4pol0N 29228 . . 3  |-  ( K  e.  HL  ->  (  ._|_  `  (/) )  =  A )
2827ad2antrr 709 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  (/) )  =  A )
2913, 26, 283eqtr3d 2296 1  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( X  .+  (  ._|_  `  X ) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    i^i cin 3093    C_ wss 3094   (/)c0 3397   ` cfv 4638  (class class class)co 5757   Atomscatm 28583   HLchlt 28670   + Pcpadd 29114   _|_ PcpolN 29221   PSubClcpscN 29253
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 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-undef 6229  df-riota 6237  df-poset 14007  df-plt 14019  df-lub 14035  df-glb 14036  df-join 14037  df-meet 14038  df-p0 14072  df-p1 14073  df-lat 14079  df-clat 14141  df-oposet 28496  df-ol 28498  df-oml 28499  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-psubsp 28822  df-pmap 28823  df-padd 29115  df-polarityN 29222  df-psubclN 29254
  Copyright terms: Public domain W3C validator