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Theorem cdlemg2k 30790
Description: cdleme42keg 30675 with simpler hypotheses. TODO: FIX COMMENT Todo: derive from cdlemg3a 30786, cdlemg2fv2 30789, cdlemg2jOLDN 30787, ltrnel 30328? (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 2283 . 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 2283 . 2  |-  ( ( p  .\/  q ) 
./\  W )  =  ( ( p  .\/  q )  ./\  W
)
9 eqid 2283 . 2  |-  ( ( t  .\/  ( ( p  .\/  q ) 
./\  W ) ) 
./\  ( q  .\/  ( ( p  .\/  t )  ./\  W
) ) )  =  ( ( t  .\/  ( ( p  .\/  q )  ./\  W
) )  ./\  (
q  .\/  ( (
p  .\/  t )  ./\  W ) ) )
10 eqid 2283 . 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 2283 . 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 30784 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   [_csb 3081   ifcif 3565   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290
This theorem is referenced by:  cdlemg2kq  30791  cdlemg2l  30792  cdlemg2m  30793  cdlemg9b  30822  cdlemg10bALTN  30825  cdlemg12b  30833  cdlemg17e  30854
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348
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