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Theorem cdlemg2k 35397
Description: cdleme42keg 35282 with simpler hypotheses. TODO: FIX COMMENT Todo: derive from cdlemg3a 35393, cdlemg2fv2 35396, cdlemg2jOLDN 35394, ltrnel 34935? (Contributed by NM, 22-Apr-2013.)
Hypotheses
Ref Expression
cdlemg2inv.h  |-  H  =  ( LHyp `  K
)
cdlemg2inv.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg2j.l  |-  .<_  =  ( le `  K )
cdlemg2j.j  |-  .\/  =  ( join `  K )
cdlemg2j.a  |-  A  =  ( Atoms `  K )
cdlemg2j.m  |-  ./\  =  ( meet `  K )
cdlemg2j.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
cdlemg2k  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  ->  (
( F `  P
)  .\/  ( F `  Q ) )  =  ( ( F `  P )  .\/  U
) )

Proof of Theorem cdlemg2k
Dummy variables  q  p  s  t  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . 2  |-  ( Base `  K )  =  (
Base `  K )
2 cdlemg2j.l . 2  |-  .<_  =  ( le `  K )
3 cdlemg2j.j . 2  |-  .\/  =  ( join `  K )
4 cdlemg2j.m . 2  |-  ./\  =  ( meet `  K )
5 cdlemg2j.a . 2  |-  A  =  ( Atoms `  K )
6 cdlemg2inv.h . 2  |-  H  =  ( LHyp `  K
)
7 cdlemg2inv.t . 2  |-  T  =  ( ( LTrn `  K
) `  W )
8 eqid 2467 . 2  |-  ( ( p  .\/  q ) 
./\  W )  =  ( ( p  .\/  q )  ./\  W
)
9 eqid 2467 . 2  |-  ( ( t  .\/  ( ( p  .\/  q ) 
./\  W ) ) 
./\  ( q  .\/  ( ( p  .\/  t )  ./\  W
) ) )  =  ( ( t  .\/  ( ( p  .\/  q )  ./\  W
) )  ./\  (
q  .\/  ( (
p  .\/  t )  ./\  W ) ) )
10 eqid 2467 . 2  |-  ( ( p  .\/  q ) 
./\  ( ( ( t  .\/  ( ( p  .\/  q ) 
./\  W ) ) 
./\  ( q  .\/  ( ( p  .\/  t )  ./\  W
) ) )  .\/  ( ( s  .\/  t )  ./\  W
) ) )  =  ( ( p  .\/  q )  ./\  (
( ( t  .\/  ( ( p  .\/  q )  ./\  W
) )  ./\  (
q  .\/  ( (
p  .\/  t )  ./\  W ) ) ) 
.\/  ( ( s 
.\/  t )  ./\  W ) ) )
11 eqid 2467 . 2  |-  ( x  e.  ( Base `  K
)  |->  if ( ( p  =/=  q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  ( Base `  K
) A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  (
x  ./\  W )
)  =  x )  ->  z  =  ( if ( s  .<_  ( p  .\/  q ) ,  ( iota_ y  e.  ( Base `  K
) A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( p 
.\/  q ) )  ->  y  =  ( ( p  .\/  q
)  ./\  ( (
( t  .\/  (
( p  .\/  q
)  ./\  W )
)  ./\  ( q  .\/  ( ( p  .\/  t )  ./\  W
) ) )  .\/  ( ( s  .\/  t )  ./\  W
) ) ) ) ) ,  [_ s  /  t ]_ (
( t  .\/  (
( p  .\/  q
)  ./\  W )
)  ./\  ( q  .\/  ( ( p  .\/  t )  ./\  W
) ) ) ) 
.\/  ( x  ./\  W ) ) ) ) ,  x ) )  =  ( x  e.  ( Base `  K
)  |->  if ( ( p  =/=  q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  ( Base `  K
) A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  (
x  ./\  W )
)  =  x )  ->  z  =  ( if ( s  .<_  ( p  .\/  q ) ,  ( iota_ y  e.  ( Base `  K
) A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( p 
.\/  q ) )  ->  y  =  ( ( p  .\/  q
)  ./\  ( (
( t  .\/  (
( p  .\/  q
)  ./\  W )
)  ./\  ( q  .\/  ( ( p  .\/  t )  ./\  W
) ) )  .\/  ( ( s  .\/  t )  ./\  W
) ) ) ) ) ,  [_ s  /  t ]_ (
( t  .\/  (
( p  .\/  q
)  ./\  W )
)  ./\  ( q  .\/  ( ( p  .\/  t )  ./\  W
) ) ) ) 
.\/  ( x  ./\  W ) ) ) ) ,  x ) )
12 cdlemg2j.u . 2  |-  U  =  ( ( P  .\/  Q )  ./\  W )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cdlemg2klem 35391 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  ->  (
( F `  P
)  .\/  ( F `  Q ) )  =  ( ( F `  P )  .\/  U
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   [_csb 3435   ifcif 3939   class class class wbr 4447    |-> cmpt 4505   ` cfv 5586   iota_crio 6242  (class class class)co 6282   Basecbs 14486   lecple 14558   joincjn 15427   meetcmee 15428   Atomscatm 34060   HLchlt 34147   LHypclh 34780   LTrncltrn 34897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-riotaBAD 33756
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-undef 6999  df-map 7419  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-p1 15523  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294  df-lplanes 34295  df-lvols 34296  df-lines 34297  df-psubsp 34299  df-pmap 34300  df-padd 34592  df-lhyp 34784  df-laut 34785  df-ldil 34900  df-ltrn 34901  df-trl 34955
This theorem is referenced by:  cdlemg2kq  35398  cdlemg2l  35399  cdlemg2m  35400  cdlemg9b  35429  cdlemg10bALTN  35432  cdlemg12b  35440  cdlemg17e  35461
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