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Theorem pexmidALTN 34649
Description: Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 34624. 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 5857 . . . 4  |-  ( X  =  (/)  ->  (  ._|_  `  X )  =  ( 
._|_  `  (/) ) )
31, 2oveq12d 6293 . . 3  |-  ( X  =  (/)  ->  ( X 
.+  (  ._|_  `  X
) )  =  (
(/)  .+  (  ._|_  `  (/) ) ) )
4 pexmidALT.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
5 pexmidALT.o . . . . . . . 8  |-  ._|_  =  ( _|_P `  K
)
64, 5pol0N 34580 . . . . . . 7  |-  ( K  e.  HL  ->  (  ._|_  `  (/) )  =  A )
7 eqimss 3549 . . . . . . 7  |-  ( ( 
._|_  `  (/) )  =  A  ->  (  ._|_  `  (/) )  C_  A )
86, 7syl 16 . . . . . 6  |-  ( K  e.  HL  ->  (  ._|_  `  (/) )  C_  A
)
9 pexmidALT.p . . . . . . 7  |-  .+  =  ( +P `  K
)
104, 9padd02 34483 . . . . . 6  |-  ( ( K  e.  HL  /\  (  ._|_  `  (/) )  C_  A )  ->  ( (/)  .+  (  ._|_  `  (/) ) )  =  (  ._|_  `  (/) ) )
118, 10mpdan 668 . . . . 5  |-  ( K  e.  HL  ->  ( (/)  .+  (  ._|_  `  (/) ) )  =  (  ._|_  `  (/) ) )
1211, 6eqtrd 2501 . . . 4  |-  ( K  e.  HL  ->  ( (/)  .+  (  ._|_  `  (/) ) )  =  A )
1312ad2antrr 725 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( (/)  .+  (  ._|_  `  (/) ) )  =  A )
143, 13sylan9eqr 2523 . 2  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X
) )  =  X )  /\  X  =  (/) )  ->  ( X 
.+  (  ._|_  `  X
) )  =  A )
154, 9, 5pexmidlem8N 34648 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( X  .+  (  ._|_  `  X ) )  =  A )
1615anassrs 648 . 2  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X
) )  =  X )  /\  X  =/=  (/) )  ->  ( X 
.+  (  ._|_  `  X
) )  =  A )
1714, 16pm2.61dane 2778 1  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( X  .+  (  ._|_  `  X ) )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655    C_ wss 3469   (/)c0 3778   ` cfv 5579  (class class class)co 6275   Atomscatm 33935   HLchlt 34022   +Pcpadd 34466   _|_PcpolN 34573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-riotaBAD 33631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-undef 6992  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-p1 15516  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-psubsp 34174  df-pmap 34175  df-padd 34467  df-polarityN 34574  df-psubclN 34606
This theorem is referenced by: (None)
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