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Theorem pexmidN 36145
Description: Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 36129. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 36143. (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 751 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  ->  K  e.  HL )
2 simplr 753 . . . . 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 36084 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  C_  A )
65adantr 463 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  X )  C_  A )
7 pexmid.p . . . . . 6  |-  .+  =  ( +P `  K
)
83, 7, 4poldmj1N 36104 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A  /\  (  ._|_  `  X )  C_  A )  ->  (  ._|_  `  ( X  .+  (  ._|_  `  X )
) )  =  ( (  ._|_  `  X )  i^i  (  ._|_  `  (  ._|_  `  X ) ) ) )
91, 2, 6, 8syl3anc 1226 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  ( X  .+  (  ._|_  `  X
) ) )  =  ( (  ._|_  `  X
)  i^i  (  ._|_  `  (  ._|_  `  X ) ) ) )
103, 4pnonsingN 36109 . . . . 5  |-  ( ( K  e.  HL  /\  (  ._|_  `  X )  C_  A )  ->  (
(  ._|_  `  X )  i^i  (  ._|_  `  (  ._|_  `  X ) ) )  =  (/) )
111, 6, 10syl2anc 659 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( (  ._|_  `  X
)  i^i  (  ._|_  `  (  ._|_  `  X ) ) )  =  (/) )
129, 11eqtrd 2437 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  ( X  .+  (  ._|_  `  X
) ) )  =  (/) )
1312fveq2d 5795 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  (  ._|_  `  ( X  .+  (  ._|_  `  X ) ) ) )  =  ( 
._|_  `  (/) ) )
14 simpr 459 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  X )
15 eqid 2396 . . . . . . 7  |-  ( PSubCl `  K )  =  (
PSubCl `  K )
163, 4, 15ispsubclN 36113 . . . . . 6  |-  ( K  e.  HL  ->  ( X  e.  ( PSubCl `  K )  <->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
1716ad2antrr 723 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( X  e.  (
PSubCl `  K )  <->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
182, 14, 17mpbir2and 920 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  ->  X  e.  ( PSubCl `  K ) )
193, 4, 15polsubclN 36128 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  e.  ( PSubCl `  K )
)
2019adantr 463 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  X )  e.  ( PSubCl `  K )
)
213, 42polssN 36091 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  X  C_  (  ._|_  `  (  ._|_  `  X ) ) )
2221adantr 463 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  ->  X  C_  (  ._|_  `  (  ._|_  `  X ) ) )
237, 4, 15osumclN 36143 . . . 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 1229 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( X  .+  (  ._|_  `  X ) )  e.  ( PSubCl `  K
) )
254, 15psubcli2N 36115 . . 3  |-  ( ( K  e.  HL  /\  ( X  .+  (  ._|_  `  X ) )  e.  ( PSubCl `  K )
)  ->  (  ._|_  `  (  ._|_  `  ( X 
.+  (  ._|_  `  X
) ) ) )  =  ( X  .+  (  ._|_  `  X )
) )
261, 24, 25syl2anc 659 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  (  ._|_  `  ( X  .+  (  ._|_  `  X ) ) ) )  =  ( X  .+  (  ._|_  `  X ) ) )
273, 4pol0N 36085 . . 3  |-  ( K  e.  HL  ->  (  ._|_  `  (/) )  =  A )
2827ad2antrr 723 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  (/) )  =  A )
2913, 26, 283eqtr3d 2445 1  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( X  .+  (  ._|_  `  X ) )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1836    i^i cin 3405    C_ wss 3406   (/)c0 3728   ` cfv 5513  (class class class)co 6218   Atomscatm 35440   HLchlt 35527   +Pcpadd 35971   _|_PcpolN 36078   PSubClcpscN 36110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-riotaBAD 35136
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rmo 2754  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-iin 4263  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-1st 6721  df-2nd 6722  df-undef 6942  df-preset 15697  df-poset 15715  df-plt 15728  df-lub 15744  df-glb 15745  df-join 15746  df-meet 15747  df-p0 15809  df-p1 15810  df-lat 15816  df-clat 15878  df-oposet 35353  df-ol 35355  df-oml 35356  df-covers 35443  df-ats 35444  df-atl 35475  df-cvlat 35499  df-hlat 35528  df-psubsp 35679  df-pmap 35680  df-padd 35972  df-polarityN 36079  df-psubclN 36111
This theorem is referenced by: (None)
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