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Theorem cdlemg2jOLDN 31569
Description: TODO: Replace this with ltrnj 31103. (Contributed by NM, 22-Apr-2013.) (New usage is discouraged.)
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 )
Assertion
Ref Expression
cdlemg2jOLDN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  ->  ( F `  ( P  .\/  Q ) )  =  ( ( F `  P )  .\/  ( F `  Q )
) )

Proof of Theorem cdlemg2jOLDN
Dummy variables  q  p  s  t  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2443 . 2  |-  ( Base `  K )  =  (
Base `  K )
2 cdlemg2j.l . 2  |-  .<_  =  ( le `  K )
3 cdlemg2j.j . 2  |-  .\/  =  ( join `  K )
4 eqid 2443 . 2  |-  ( meet `  K )  =  (
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 2443 . 2  |-  ( ( p  .\/  q ) ( meet `  K
) W )  =  ( ( p  .\/  q ) ( meet `  K ) W )
9 eqid 2443 . 2  |-  ( ( t  .\/  ( ( p  .\/  q ) ( meet `  K
) W ) ) ( meet `  K
) ( q  .\/  ( ( p  .\/  t ) ( meet `  K ) W ) ) )  =  ( ( t  .\/  (
( p  .\/  q
) ( meet `  K
) W ) ) ( meet `  K
) ( q  .\/  ( ( p  .\/  t ) ( meet `  K ) W ) ) )
10 eqid 2443 . 2  |-  ( ( p  .\/  q ) ( meet `  K
) ( ( ( t  .\/  ( ( p  .\/  q ) ( meet `  K
) W ) ) ( meet `  K
) ( q  .\/  ( ( p  .\/  t ) ( meet `  K ) W ) ) )  .\/  (
( s  .\/  t
) ( meet `  K
) W ) ) )  =  ( ( p  .\/  q ) ( meet `  K
) ( ( ( t  .\/  ( ( p  .\/  q ) ( meet `  K
) W ) ) ( meet `  K
) ( q  .\/  ( ( p  .\/  t ) ( meet `  K ) W ) ) )  .\/  (
( s  .\/  t
) ( meet `  K
) W ) ) )
11 eqid 2443 . 2  |-  ( x  e.  ( Base `  K
)  |->  if ( ( p  =/=  q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  ( Base `  K
) A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  (
x ( meet `  K
) W ) )  =  x )  -> 
z  =  ( if ( s  .<_  ( p 
.\/  q ) ,  ( iota_ y  e.  (
Base `  K ) A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( p  .\/  q
) )  ->  y  =  ( ( p 
.\/  q ) (
meet `  K )
( ( ( t 
.\/  ( ( p 
.\/  q ) (
meet `  K ) W ) ) (
meet `  K )
( q  .\/  (
( p  .\/  t
) ( meet `  K
) W ) ) )  .\/  ( ( s  .\/  t ) ( meet `  K
) W ) ) ) ) ) , 
[_ s  /  t ]_ ( ( t  .\/  ( ( p  .\/  q ) ( meet `  K ) W ) ) ( meet `  K
) ( q  .\/  ( ( p  .\/  t ) ( meet `  K ) W ) ) ) )  .\/  ( x ( meet `  K ) W ) ) ) ) ,  x ) )  =  ( x  e.  (
Base `  K )  |->  if ( ( p  =/=  q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  (
Base `  K ) A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x (
meet `  K ) W ) )  =  x )  ->  z  =  ( if ( s  .<_  ( p  .\/  q ) ,  (
iota_ y  e.  ( Base `  K ) A. t  e.  A  (
( -.  t  .<_  W  /\  -.  t  .<_  ( p  .\/  q ) )  ->  y  =  ( ( p  .\/  q ) ( meet `  K ) ( ( ( t  .\/  (
( p  .\/  q
) ( meet `  K
) W ) ) ( meet `  K
) ( q  .\/  ( ( p  .\/  t ) ( meet `  K ) W ) ) )  .\/  (
( s  .\/  t
) ( meet `  K
) W ) ) ) ) ) , 
[_ s  /  t ]_ ( ( t  .\/  ( ( p  .\/  q ) ( meet `  K ) W ) ) ( meet `  K
) ( q  .\/  ( ( p  .\/  t ) ( meet `  K ) W ) ) ) )  .\/  ( x ( meet `  K ) W ) ) ) ) ,  x ) )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11cdlemg2jlemOLDN 31564 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  ->  ( F `  ( P  .\/  Q ) )  =  ( ( F `  P )  .\/  ( F `  Q )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1654    e. wcel 1728    =/= wne 2606   A.wral 2712   [_csb 3270   ifcif 3767   class class class wbr 4243    e. cmpt 4297   ` cfv 5489  (class class class)co 6117   iota_crio 6578   Basecbs 13507   lecple 13574   joincjn 14439   meetcmee 14440   Atomscatm 30235   HLchlt 30322   LHypclh 30955   LTrncltrn 31072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rmo 2720  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-iun 4124  df-iin 4125  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-1st 6385  df-2nd 6386  df-undef 6579  df-riota 6585  df-map 7056  df-poset 14441  df-plt 14453  df-lub 14469  df-glb 14470  df-join 14471  df-meet 14472  df-p0 14506  df-p1 14507  df-lat 14513  df-clat 14575  df-oposet 30148  df-ol 30150  df-oml 30151  df-covers 30238  df-ats 30239  df-atl 30270  df-cvlat 30294  df-hlat 30323  df-llines 30469  df-lplanes 30470  df-lvols 30471  df-lines 30472  df-psubsp 30474  df-pmap 30475  df-padd 30767  df-lhyp 30959  df-laut 30960  df-ldil 31075  df-ltrn 31076  df-trl 31130
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