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Theorem pexmidALTN 28856
Description: Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 28831. 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 5377 . . . 4  |-  ( X  =  (/)  ->  (  ._|_  `  X )  =  ( 
._|_  `  (/) ) )
31, 2oveq12d 5728 . . 3  |-  ( X  =  (/)  ->  ( X 
.+  (  ._|_  `  X
) )  =  (
(/)  .+  (  ._|_  `  (/) ) ) )
4 pexmidALT.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
5 pexmidALT.o . . . . . . . 8  |-  ._|_  =  ( _|_ P `  K
)
64, 5pol0N 28787 . . . . . . 7  |-  ( K  e.  HL  ->  (  ._|_  `  (/) )  =  A )
7 eqimss 3151 . . . . . . 7  |-  ( ( 
._|_  `  (/) )  =  A  ->  (  ._|_  `  (/) )  C_  A )
86, 7syl 17 . . . . . 6  |-  ( K  e.  HL  ->  (  ._|_  `  (/) )  C_  A
)
9 pexmidALT.p . . . . . . 7  |-  .+  =  ( + P `  K
)
104, 9padd02 28690 . . . . . 6  |-  ( ( K  e.  HL  /\  (  ._|_  `  (/) )  C_  A )  ->  ( (/)  .+  (  ._|_  `  (/) ) )  =  (  ._|_  `  (/) ) )
118, 10mpdan 652 . . . . 5  |-  ( K  e.  HL  ->  ( (/)  .+  (  ._|_  `  (/) ) )  =  (  ._|_  `  (/) ) )
1211, 6eqtrd 2285 . . . 4  |-  ( K  e.  HL  ->  ( (/)  .+  (  ._|_  `  (/) ) )  =  A )
1312ad2antrr 709 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( (/)  .+  (  ._|_  `  (/) ) )  =  A )
143, 13sylan9eqr 2307 . 2  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X
) )  =  X )  /\  X  =  (/) )  ->  ( X 
.+  (  ._|_  `  X
) )  =  A )
154, 9, 5pexmidlem8N 28855 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  X  =/=  (/) ) )  -> 
( X  .+  (  ._|_  `  X ) )  =  A )
1615anassrs 632 . 2  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X
) )  =  X )  /\  X  =/=  (/) )  ->  ( X 
.+  (  ._|_  `  X
) )  =  A )
1714, 16pm2.61dane 2490 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 1619    e. wcel 1621    =/= wne 2412    C_ wss 3078   (/)c0 3362   ` cfv 4592  (class class class)co 5710   Atomscatm 28142   HLchlt 28229   + Pcpadd 28673   _|_ PcpolN 28780
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-psubsp 28381  df-pmap 28382  df-padd 28674  df-polarityN 28781  df-psubclN 28813
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