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Theorem cdlemg2k 33601
Description: cdleme42keg 33486 with simpler hypotheses. TODO: FIX COMMENT Todo: derive from cdlemg3a 33597, cdlemg2fv2 33600, cdlemg2jOLDN 33598, ltrnel 33137? (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 2402 . 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 2402 . 2  |-  ( ( p  .\/  q ) 
./\  W )  =  ( ( p  .\/  q )  ./\  W
)
9 eqid 2402 . 2  |-  ( ( t  .\/  ( ( p  .\/  q ) 
./\  W ) ) 
./\  ( q  .\/  ( ( p  .\/  t )  ./\  W
) ) )  =  ( ( t  .\/  ( ( p  .\/  q )  ./\  W
) )  ./\  (
q  .\/  ( (
p  .\/  t )  ./\  W ) ) )
10 eqid 2402 . 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 2402 . 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 33595 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2753   [_csb 3372   ifcif 3884   class class class wbr 4394    |-> cmpt 4452   ` cfv 5525   iota_crio 6195  (class class class)co 6234   Basecbs 14733   lecple 14808   joincjn 15789   meetcmee 15790   Atomscatm 32262   HLchlt 32349   LHypclh 32982   LTrncltrn 33099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-riotaBAD 31958
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-1st 6738  df-2nd 6739  df-undef 6959  df-map 7379  df-preset 15773  df-poset 15791  df-plt 15804  df-lub 15820  df-glb 15821  df-join 15822  df-meet 15823  df-p0 15885  df-p1 15886  df-lat 15892  df-clat 15954  df-oposet 32175  df-ol 32177  df-oml 32178  df-covers 32265  df-ats 32266  df-atl 32297  df-cvlat 32321  df-hlat 32350  df-llines 32496  df-lplanes 32497  df-lvols 32498  df-lines 32499  df-psubsp 32501  df-pmap 32502  df-padd 32794  df-lhyp 32986  df-laut 32987  df-ldil 33102  df-ltrn 33103  df-trl 33158
This theorem is referenced by:  cdlemg2kq  33602  cdlemg2l  33603  cdlemg2m  33604  cdlemg9b  33633  cdlemg10bALTN  33636  cdlemg12b  33644  cdlemg17e  33665
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