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Theorem lhpat2 29501
Description: Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 21-Nov-2012.)
Hypotheses
Ref Expression
lhpat.l  |-  .<_  =  ( le `  K )
lhpat.j  |-  .\/  =  ( join `  K )
lhpat.m  |-  ./\  =  ( meet `  K )
lhpat.a  |-  A  =  ( Atoms `  K )
lhpat.h  |-  H  =  ( LHyp `  K
)
lhpat2.r  |-  R  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
lhpat2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  R  e.  A
)

Proof of Theorem lhpat2
StepHypRef Expression
1 lhpat2.r . 2  |-  R  =  ( ( P  .\/  Q )  ./\  W )
2 lhpat.l . . 3  |-  .<_  =  ( le `  K )
3 lhpat.j . . 3  |-  .\/  =  ( join `  K )
4 lhpat.m . . 3  |-  ./\  =  ( meet `  K )
5 lhpat.a . . 3  |-  A  =  ( Atoms `  K )
6 lhpat.h . . 3  |-  H  =  ( LHyp `  K
)
72, 3, 4, 5, 6lhpat 29499 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  ( ( P 
.\/  Q )  ./\  W )  e.  A )
81, 7syl5eqel 2368 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  R  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685    =/= wne 2447   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   lecple 13209   joincjn 14072   meetcmee 14073   Atomscatm 28720   HLchlt 28807   LHypclh 29440
This theorem is referenced by:  lhpat3  29502  4atexlemu  29520  4atexlemv  29521  cdleme0a  29667  cdleme0dN  29672  cdleme0e  29673  cdleme02N  29678  cdleme0ex1N  29679  cdleme0moN  29681  cdleme3b  29685  cdleme3c  29686  cdleme3g  29690  cdleme3h  29691  cdleme3  29693  cdleme7aa  29698  cdleme7c  29701  cdleme7d  29702  cdleme7e  29703  cdleme7ga  29704  cdleme7  29705  cdleme9a  29707  cdleme16aN  29715  cdleme11a  29716  cdleme11c  29717  cdleme12  29727  cdleme16b  29735  cdleme16c  29736  cdleme16d  29737  cdleme20h  29772  cdleme20j  29774  cdleme20l2  29777  cdlemeg46rgv  29984  cdlemeg46req  29985
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 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-p1 14140  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-lhyp 29444
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