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Theorem cdlemg2jOLDN 30858
Description: TODO: Replace this with ltrnj 30392. (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 2366 . 2  |-  ( Base `  K )  =  (
Base `  K )
2 cdlemg2j.l . 2  |-  .<_  =  ( le `  K )
3 cdlemg2j.j . 2  |-  .\/  =  ( join `  K )
4 eqid 2366 . 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 2366 . 2  |-  ( ( p  .\/  q ) ( meet `  K
) W )  =  ( ( p  .\/  q ) ( meet `  K ) W )
9 eqid 2366 . 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 2366 . 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 2366 . 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 30853 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 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   A.wral 2628   [_csb 3167   ifcif 3654   class class class wbr 4125    e. cmpt 4179   ` cfv 5358  (class class class)co 5981   iota_crio 6439   Basecbs 13356   lecple 13423   joincjn 14288   meetcmee 14289   Atomscatm 29524   HLchlt 29611   LHypclh 30244   LTrncltrn 30361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-map 6917  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-p1 14356  df-lat 14362  df-clat 14424  df-oposet 29437  df-ol 29439  df-oml 29440  df-covers 29527  df-ats 29528  df-atl 29559  df-cvlat 29583  df-hlat 29612  df-llines 29758  df-lplanes 29759  df-lvols 29760  df-lines 29761  df-psubsp 29763  df-pmap 29764  df-padd 30056  df-lhyp 30248  df-laut 30249  df-ldil 30364  df-ltrn 30365  df-trl 30419
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