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Theorem pexmidN 30231
Description: Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 30215. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 30229. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pexmid.a  |-  A  =  ( Atoms `  K )
pexmid.p  |-  .+  =  ( + P `  K
)
pexmid.o  |-  ._|_  =  ( _|_ P `  K
)
Assertion
Ref Expression
pexmidN  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( X  .+  (  ._|_  `  X ) )  =  A )

Proof of Theorem pexmidN
StepHypRef Expression
1 simpll 730 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  ->  K  e.  HL )
2 simplr 731 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  ->  X  C_  A )
3 pexmid.a . . . . . . 7  |-  A  =  ( Atoms `  K )
4 pexmid.o . . . . . . 7  |-  ._|_  =  ( _|_ P `  K
)
53, 4polssatN 30170 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  C_  A )
65adantr 451 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  X )  C_  A )
7 pexmid.p . . . . . 6  |-  .+  =  ( + P `  K
)
83, 7, 4poldmj1N 30190 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A  /\  (  ._|_  `  X )  C_  A )  ->  (  ._|_  `  ( X  .+  (  ._|_  `  X )
) )  =  ( (  ._|_  `  X )  i^i  (  ._|_  `  (  ._|_  `  X ) ) ) )
91, 2, 6, 8syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  ( X  .+  (  ._|_  `  X
) ) )  =  ( (  ._|_  `  X
)  i^i  (  ._|_  `  (  ._|_  `  X ) ) ) )
103, 4pnonsingN 30195 . . . . 5  |-  ( ( K  e.  HL  /\  (  ._|_  `  X )  C_  A )  ->  (
(  ._|_  `  X )  i^i  (  ._|_  `  (  ._|_  `  X ) ) )  =  (/) )
111, 6, 10syl2anc 642 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( (  ._|_  `  X
)  i^i  (  ._|_  `  (  ._|_  `  X ) ) )  =  (/) )
129, 11eqtrd 2317 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  ( X  .+  (  ._|_  `  X
) ) )  =  (/) )
1312fveq2d 5531 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  (  ._|_  `  ( X  .+  (  ._|_  `  X ) ) ) )  =  ( 
._|_  `  (/) ) )
14 simpr 447 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  X )
15 eqid 2285 . . . . . . 7  |-  ( PSubCl `  K )  =  (
PSubCl `  K )
163, 4, 15ispsubclN 30199 . . . . . 6  |-  ( K  e.  HL  ->  ( X  e.  ( PSubCl `  K )  <->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
1716ad2antrr 706 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( X  e.  (
PSubCl `  K )  <->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
182, 14, 17mpbir2and 888 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  ->  X  e.  ( PSubCl `  K ) )
193, 4, 15polsubclN 30214 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  e.  ( PSubCl `  K )
)
2019adantr 451 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  X )  e.  ( PSubCl `  K )
)
213, 42polssN 30177 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  X  C_  (  ._|_  `  (  ._|_  `  X ) ) )
2221adantr 451 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  ->  X  C_  (  ._|_  `  (  ._|_  `  X ) ) )
237, 4, 15osumclN 30229 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  ( PSubCl `  K )  /\  (  ._|_  `  X )  e.  ( PSubCl `  K )
)  /\  X  C_  (  ._|_  `  (  ._|_  `  X
) ) )  -> 
( X  .+  (  ._|_  `  X ) )  e.  ( PSubCl `  K
) )
241, 18, 20, 22, 23syl31anc 1185 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( X  .+  (  ._|_  `  X ) )  e.  ( PSubCl `  K
) )
254, 15psubcli2N 30201 . . 3  |-  ( ( K  e.  HL  /\  ( X  .+  (  ._|_  `  X ) )  e.  ( PSubCl `  K )
)  ->  (  ._|_  `  (  ._|_  `  ( X 
.+  (  ._|_  `  X
) ) ) )  =  ( X  .+  (  ._|_  `  X )
) )
261, 24, 25syl2anc 642 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  (  ._|_  `  ( X  .+  (  ._|_  `  X ) ) ) )  =  ( X  .+  (  ._|_  `  X ) ) )
273, 4pol0N 30171 . . 3  |-  ( K  e.  HL  ->  (  ._|_  `  (/) )  =  A )
2827ad2antrr 706 . 2  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
(  ._|_  `  (/) )  =  A )
2913, 26, 283eqtr3d 2325 1  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  -> 
( X  .+  (  ._|_  `  X ) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1625    e. wcel 1686    i^i cin 3153    C_ wss 3154   (/)c0 3457   ` cfv 5257  (class class class)co 5860   Atomscatm 29526   HLchlt 29613   + Pcpadd 30057   _|_ PcpolN 30164   PSubClcpscN 30196
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  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 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-undef 6300  df-riota 6306  df-poset 14082  df-plt 14094  df-lub 14110  df-glb 14111  df-join 14112  df-meet 14113  df-p0 14147  df-p1 14148  df-lat 14154  df-clat 14216  df-oposet 29439  df-ol 29441  df-oml 29442  df-covers 29529  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614  df-psubsp 29765  df-pmap 29766  df-padd 30058  df-polarityN 30165  df-psubclN 30197
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