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Theorem cdlemg2jOLDN 31092
Description: TODO: Replace this with ltrnj 30626. (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 2412 . 2  |-  ( Base `  K )  =  (
Base `  K )
2 cdlemg2j.l . 2  |-  .<_  =  ( le `  K )
3 cdlemg2j.j . 2  |-  .\/  =  ( join `  K )
4 eqid 2412 . 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 2412 . 2  |-  ( ( p  .\/  q ) ( meet `  K
) W )  =  ( ( p  .\/  q ) ( meet `  K ) W )
9 eqid 2412 . 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 2412 . 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 2412 . 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 31087 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   A.wral 2674   [_csb 3219   ifcif 3707   class class class wbr 4180    e. cmpt 4234   ` cfv 5421  (class class class)co 6048   iota_crio 6509   Basecbs 13432   lecple 13499   joincjn 14364   meetcmee 14365   Atomscatm 29758   HLchlt 29845   LHypclh 30478   LTrncltrn 30595
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-map 6987  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-p1 14432  df-lat 14438  df-clat 14500  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-llines 29992  df-lplanes 29993  df-lvols 29994  df-lines 29995  df-psubsp 29997  df-pmap 29998  df-padd 30290  df-lhyp 30482  df-laut 30483  df-ldil 30598  df-ltrn 30599  df-trl 30653
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