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Theorem pexmidALTN 36099
Description: Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 36074. Lemma 3.3(2) in [Holland95] p. 215. In our proof, we use the variables  X,  M,  p,  q,  r in place of Hollands' l, m, P, Q, L respectively. TODO: should we make this obsolete? (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pexmidALT.a  |-  A  =  ( Atoms `  K )
pexmidALT.p  |-  .+  =  ( +P `  K
)
pexmidALT.o  |-  ._|_  =  ( _|_P `  K
)
Assertion
Ref Expression
pexmidALTN  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( X  .+  (  ._|_  `  X ) )  =  A )

Proof of Theorem pexmidALTN
StepHypRef Expression
1 id 22 . . . 4  |-  ( X  =  (/)  ->  X  =  (/) )
2 fveq2 5848 . . . 4  |-  ( X  =  (/)  ->  (  ._|_  `  X )  =  ( 
._|_  `  (/) ) )
31, 2oveq12d 6288 . . 3  |-  ( X  =  (/)  ->  ( X 
.+  (  ._|_  `  X
) )  =  (
(/)  .+  (  ._|_  `  (/) ) ) )
4 pexmidALT.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
5 pexmidALT.o . . . . . . . 8  |-  ._|_  =  ( _|_P `  K
)
64, 5pol0N 36030 . . . . . . 7  |-  ( K  e.  HL  ->  (  ._|_  `  (/) )  =  A )
7 eqimss 3541 . . . . . . 7  |-  ( ( 
._|_  `  (/) )  =  A  ->  (  ._|_  `  (/) )  C_  A )
86, 7syl 16 . . . . . 6  |-  ( K  e.  HL  ->  (  ._|_  `  (/) )  C_  A
)
9 pexmidALT.p . . . . . . 7  |-  .+  =  ( +P `  K
)
104, 9padd02 35933 . . . . . 6  |-  ( ( K  e.  HL  /\  (  ._|_  `  (/) )  C_  A )  ->  ( (/)  .+  (  ._|_  `  (/) ) )  =  (  ._|_  `  (/) ) )
118, 10mpdan 666 . . . . 5  |-  ( K  e.  HL  ->  ( (/)  .+  (  ._|_  `  (/) ) )  =  (  ._|_  `  (/) ) )
1211, 6eqtrd 2495 . . . 4  |-  ( K  e.  HL  ->  ( (/)  .+  (  ._|_  `  (/) ) )  =  A )
1312ad2antrr 723 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( (/)  .+  (  ._|_  `  (/) ) )  =  A )
143, 13sylan9eqr 2517 . 2  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X
) )  =  X )  /\  X  =  (/) )  ->  ( X 
.+  (  ._|_  `  X
) )  =  A )
154, 9, 5pexmidlem8N 36098 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( X  .+  (  ._|_  `  X ) )  =  A )
1615anassrs 646 . 2  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X
) )  =  X )  /\  X  =/=  (/) )  ->  ( X 
.+  (  ._|_  `  X
) )  =  A )
1714, 16pm2.61dane 2772 1  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( X  .+  (  ._|_  `  X ) )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649    C_ wss 3461   (/)c0 3783   ` cfv 5570  (class class class)co 6270   Atomscatm 35385   HLchlt 35472   +Pcpadd 35916   _|_PcpolN 36023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-riotaBAD 35081
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-undef 6994  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-p1 15869  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-psubsp 35624  df-pmap 35625  df-padd 35917  df-polarityN 36024  df-psubclN 36056
This theorem is referenced by: (None)
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