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Theorem pexmidN 30451
Description: Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 30435. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 30449. (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 731 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  ->  K  e.  HL )
2 simplr 732 . . . . 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 30390 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  C_  A )
65adantr 452 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  X )  C_  A )
7 pexmid.p . . . . . 6  |-  .+  =  ( + P `  K
)
83, 7, 4poldmj1N 30410 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A  /\  (  ._|_  `  X )  C_  A )  ->  (  ._|_  `  ( X  .+  (  ._|_  `  X )
) )  =  ( (  ._|_  `  X )  i^i  (  ._|_  `  (  ._|_  `  X ) ) ) )
91, 2, 6, 8syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  ( X  .+  (  ._|_  `  X
) ) )  =  ( (  ._|_  `  X
)  i^i  (  ._|_  `  (  ._|_  `  X ) ) ) )
103, 4pnonsingN 30415 . . . . 5  |-  ( ( K  e.  HL  /\  (  ._|_  `  X )  C_  A )  ->  (
(  ._|_  `  X )  i^i  (  ._|_  `  (  ._|_  `  X ) ) )  =  (/) )
111, 6, 10syl2anc 643 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( (  ._|_  `  X
)  i^i  (  ._|_  `  (  ._|_  `  X ) ) )  =  (/) )
129, 11eqtrd 2436 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  ( X  .+  (  ._|_  `  X
) ) )  =  (/) )
1312fveq2d 5691 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  (  ._|_  `  ( X  .+  (  ._|_  `  X ) ) ) )  =  ( 
._|_  `  (/) ) )
14 simpr 448 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  X )
15 eqid 2404 . . . . . . 7  |-  ( PSubCl `  K )  =  (
PSubCl `  K )
163, 4, 15ispsubclN 30419 . . . . . 6  |-  ( K  e.  HL  ->  ( X  e.  ( PSubCl `  K )  <->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
1716ad2antrr 707 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( X  e.  (
PSubCl `  K )  <->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
182, 14, 17mpbir2and 889 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  ->  X  e.  ( PSubCl `  K ) )
193, 4, 15polsubclN 30434 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  e.  ( PSubCl `  K )
)
2019adantr 452 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  X )  e.  ( PSubCl `  K )
)
213, 42polssN 30397 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  X  C_  (  ._|_  `  (  ._|_  `  X ) ) )
2221adantr 452 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  ->  X  C_  (  ._|_  `  (  ._|_  `  X ) ) )
237, 4, 15osumclN 30449 . . . 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 1187 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( X  .+  (  ._|_  `  X ) )  e.  ( PSubCl `  K
) )
254, 15psubcli2N 30421 . . 3  |-  ( ( K  e.  HL  /\  ( X  .+  (  ._|_  `  X ) )  e.  ( PSubCl `  K )
)  ->  (  ._|_  `  (  ._|_  `  ( X 
.+  (  ._|_  `  X
) ) ) )  =  ( X  .+  (  ._|_  `  X )
) )
261, 24, 25syl2anc 643 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  (  ._|_  `  ( X  .+  (  ._|_  `  X ) ) ) )  =  ( X  .+  (  ._|_  `  X ) ) )
273, 4pol0N 30391 . . 3  |-  ( K  e.  HL  ->  (  ._|_  `  (/) )  =  A )
2827ad2antrr 707 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  (/) )  =  A )
2913, 26, 283eqtr3d 2444 1  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( X  .+  (  ._|_  `  X ) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    i^i cin 3279    C_ wss 3280   (/)c0 3588   ` cfv 5413  (class class class)co 6040   Atomscatm 29746   HLchlt 29833   + Pcpadd 30277   _|_ PcpolN 30384   PSubClcpscN 30416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-polarityN 30385  df-psubclN 30417
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