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Theorem pexmidALTN 30789
Description: Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 30764. 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 19 . . . 4  |-  ( X  =  (/)  ->  X  =  (/) )
2 fveq2 5541 . . . 4  |-  ( X  =  (/)  ->  (  ._|_  `  X )  =  ( 
._|_  `  (/) ) )
31, 2oveq12d 5892 . . 3  |-  ( X  =  (/)  ->  ( X 
.+  (  ._|_  `  X
) )  =  (
(/)  .+  (  ._|_  `  (/) ) ) )
4 pexmidALT.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
5 pexmidALT.o . . . . . . . 8  |-  ._|_  =  ( _|_ P `  K
)
64, 5pol0N 30720 . . . . . . 7  |-  ( K  e.  HL  ->  (  ._|_  `  (/) )  =  A )
7 eqimss 3243 . . . . . . 7  |-  ( ( 
._|_  `  (/) )  =  A  ->  (  ._|_  `  (/) )  C_  A )
86, 7syl 15 . . . . . 6  |-  ( K  e.  HL  ->  (  ._|_  `  (/) )  C_  A
)
9 pexmidALT.p . . . . . . 7  |-  .+  =  ( + P `  K
)
104, 9padd02 30623 . . . . . 6  |-  ( ( K  e.  HL  /\  (  ._|_  `  (/) )  C_  A )  ->  ( (/)  .+  (  ._|_  `  (/) ) )  =  (  ._|_  `  (/) ) )
118, 10mpdan 649 . . . . 5  |-  ( K  e.  HL  ->  ( (/)  .+  (  ._|_  `  (/) ) )  =  (  ._|_  `  (/) ) )
1211, 6eqtrd 2328 . . . 4  |-  ( K  e.  HL  ->  ( (/)  .+  (  ._|_  `  (/) ) )  =  A )
1312ad2antrr 706 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( (/)  .+  (  ._|_  `  (/) ) )  =  A )
143, 13sylan9eqr 2350 . 2  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X
) )  =  X )  /\  X  =  (/) )  ->  ( X 
.+  (  ._|_  `  X
) )  =  A )
154, 9, 5pexmidlem8N 30788 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( X  .+  (  ._|_  `  X ) )  =  A )
1615anassrs 629 . 2  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X
) )  =  X )  /\  X  =/=  (/) )  ->  ( X 
.+  (  ._|_  `  X
) )  =  A )
1714, 16pm2.61dane 2537 1  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( X  .+  (  ._|_  `  X ) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    C_ wss 3165   (/)c0 3468   ` cfv 5271  (class class class)co 5874   Atomscatm 30075   HLchlt 30162   + Pcpadd 30606   _|_ PcpolN 30713
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-polarityN 30714  df-psubclN 30746
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