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Theorem pexmidALTN 29435
Description: Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 29410. 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 21 . . . 4  |-  ( X  =  (/)  ->  X  =  (/) )
2 fveq2 5486 . . . 4  |-  ( X  =  (/)  ->  (  ._|_  `  X )  =  ( 
._|_  `  (/) ) )
31, 2oveq12d 5838 . . 3  |-  ( X  =  (/)  ->  ( X 
.+  (  ._|_  `  X
) )  =  (
(/)  .+  (  ._|_  `  (/) ) ) )
4 pexmidALT.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
5 pexmidALT.o . . . . . . . 8  |-  ._|_  =  ( _|_ P `  K
)
64, 5pol0N 29366 . . . . . . 7  |-  ( K  e.  HL  ->  (  ._|_  `  (/) )  =  A )
7 eqimss 3232 . . . . . . 7  |-  ( ( 
._|_  `  (/) )  =  A  ->  (  ._|_  `  (/) )  C_  A )
86, 7syl 17 . . . . . 6  |-  ( K  e.  HL  ->  (  ._|_  `  (/) )  C_  A
)
9 pexmidALT.p . . . . . . 7  |-  .+  =  ( + P `  K
)
104, 9padd02 29269 . . . . . 6  |-  ( ( K  e.  HL  /\  (  ._|_  `  (/) )  C_  A )  ->  ( (/)  .+  (  ._|_  `  (/) ) )  =  (  ._|_  `  (/) ) )
118, 10mpdan 651 . . . . 5  |-  ( K  e.  HL  ->  ( (/)  .+  (  ._|_  `  (/) ) )  =  (  ._|_  `  (/) ) )
1211, 6eqtrd 2317 . . . 4  |-  ( K  e.  HL  ->  ( (/)  .+  (  ._|_  `  (/) ) )  =  A )
1312ad2antrr 708 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( (/)  .+  (  ._|_  `  (/) ) )  =  A )
143, 13sylan9eqr 2339 . 2  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X
) )  =  X )  /\  X  =  (/) )  ->  ( X 
.+  (  ._|_  `  X
) )  =  A )
154, 9, 5pexmidlem8N 29434 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( X  .+  (  ._|_  `  X ) )  =  A )
1615anassrs 631 . 2  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X
) )  =  X )  /\  X  =/=  (/) )  ->  ( X 
.+  (  ._|_  `  X
) )  =  A )
1714, 16pm2.61dane 2526 1  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( X  .+  (  ._|_  `  X ) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685    =/= wne 2448    C_ wss 3154   (/)c0 3457   ` cfv 5222  (class class class)co 5820   Atomscatm 28721   HLchlt 28808   + Pcpadd 29252   _|_ PcpolN 29359
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-poset 14075  df-plt 14087  df-lub 14103  df-glb 14104  df-join 14105  df-meet 14106  df-p0 14140  df-p1 14141  df-lat 14147  df-clat 14209  df-oposet 28634  df-ol 28636  df-oml 28637  df-covers 28724  df-ats 28725  df-atl 28756  df-cvlat 28780  df-hlat 28809  df-psubsp 28960  df-pmap 28961  df-padd 29253  df-polarityN 29360  df-psubclN 29392
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