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Theorem pexmidN 29437
Description: Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 29421. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 29435. (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 730 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  ->  K  e.  HL )
2 simplr 731 . . . . 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 29376 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  C_  A )
65adantr 451 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  X )  C_  A )
7 pexmid.p . . . . . 6  |-  .+  =  ( + P `  K
)
83, 7, 4poldmj1N 29396 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A  /\  (  ._|_  `  X )  C_  A )  ->  (  ._|_  `  ( X  .+  (  ._|_  `  X )
) )  =  ( (  ._|_  `  X )  i^i  (  ._|_  `  (  ._|_  `  X ) ) ) )
91, 2, 6, 8syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  ( X  .+  (  ._|_  `  X
) ) )  =  ( (  ._|_  `  X
)  i^i  (  ._|_  `  (  ._|_  `  X ) ) ) )
103, 4pnonsingN 29401 . . . . 5  |-  ( ( K  e.  HL  /\  (  ._|_  `  X )  C_  A )  ->  (
(  ._|_  `  X )  i^i  (  ._|_  `  (  ._|_  `  X ) ) )  =  (/) )
111, 6, 10syl2anc 642 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( (  ._|_  `  X
)  i^i  (  ._|_  `  (  ._|_  `  X ) ) )  =  (/) )
129, 11eqtrd 2316 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  ( X  .+  (  ._|_  `  X
) ) )  =  (/) )
1312fveq2d 5490 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  (  ._|_  `  ( X  .+  (  ._|_  `  X ) ) ) )  =  ( 
._|_  `  (/) ) )
14 simpr 447 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  X )
15 eqid 2284 . . . . . . 7  |-  ( PSubCl `  K )  =  (
PSubCl `  K )
163, 4, 15ispsubclN 29405 . . . . . 6  |-  ( K  e.  HL  ->  ( X  e.  ( PSubCl `  K )  <->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
1716ad2antrr 706 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( X  e.  (
PSubCl `  K )  <->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
182, 14, 17mpbir2and 888 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  ->  X  e.  ( PSubCl `  K ) )
193, 4, 15polsubclN 29420 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  e.  ( PSubCl `  K )
)
2019adantr 451 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  X )  e.  ( PSubCl `  K )
)
213, 42polssN 29383 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  X  C_  (  ._|_  `  (  ._|_  `  X ) ) )
2221adantr 451 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  ->  X  C_  (  ._|_  `  (  ._|_  `  X ) ) )
237, 4, 15osumclN 29435 . . . 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 1185 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( X  .+  (  ._|_  `  X ) )  e.  ( PSubCl `  K
) )
254, 15psubcli2N 29407 . . 3  |-  ( ( K  e.  HL  /\  ( X  .+  (  ._|_  `  X ) )  e.  ( PSubCl `  K )
)  ->  (  ._|_  `  (  ._|_  `  ( X 
.+  (  ._|_  `  X
) ) ) )  =  ( X  .+  (  ._|_  `  X )
) )
261, 24, 25syl2anc 642 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  (  ._|_  `  ( X  .+  (  ._|_  `  X ) ) ) )  =  ( X  .+  (  ._|_  `  X ) ) )
273, 4pol0N 29377 . . 3  |-  ( K  e.  HL  ->  (  ._|_  `  (/) )  =  A )
2827ad2antrr 706 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  (/) )  =  A )
2913, 26, 283eqtr3d 2324 1  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( X  .+  (  ._|_  `  X ) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1685    i^i cin 3152    C_ wss 3153   (/)c0 3456   ` cfv 5221  (class class class)co 5820   Atomscatm 28732   HLchlt 28819   + Pcpadd 29263   _|_ PcpolN 29370   PSubClcpscN 29402
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-poset 14076  df-plt 14088  df-lub 14104  df-glb 14105  df-join 14106  df-meet 14107  df-p0 14141  df-p1 14142  df-lat 14148  df-clat 14210  df-oposet 28645  df-ol 28647  df-oml 28648  df-covers 28735  df-ats 28736  df-atl 28767  df-cvlat 28791  df-hlat 28820  df-psubsp 28971  df-pmap 28972  df-padd 29264  df-polarityN 29371  df-psubclN 29403
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