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Theorem cdleme46fsvlpq 31476
Description: Show that  ( F `
 R ) is under  P  .\/  Q when  R is. (Contributed by NM, 1-Apr-2013.)
Hypotheses
Ref Expression
cdlemef46.b  |-  B  =  ( Base `  K
)
cdlemef46.l  |-  .<_  =  ( le `  K )
cdlemef46.j  |-  .\/  =  ( join `  K )
cdlemef46.m  |-  ./\  =  ( meet `  K )
cdlemef46.a  |-  A  =  ( Atoms `  K )
cdlemef46.h  |-  H  =  ( LHyp `  K
)
cdlemef46.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdlemef46.d  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdlemefs46.e  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
cdlemef46.f  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( if ( s  .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  E ) ) ,  [_ s  /  t ]_ D
)  .\/  ( x  ./\ 
W ) ) ) ) ,  x ) )
Assertion
Ref Expression
cdleme46fsvlpq  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  R  .<_  ( P  .\/  Q ) )  ->  ( F `  R )  .<_  ( P 
.\/  Q ) )
Distinct variable groups:    t, s, x, y, z, A    B, s, t, x, y, z    D, s, x, y, z   
x, E, y, z    H, s, t, x, y, z    .\/ , s, t, x, y, z    K, s, t, x, y, z    .<_ , s, t, x, y, z    ./\ , s, t, x, y, z    P, s, t, x, y, z    Q, s, t, x, y, z    R, s, t, x, y, z    U, s, t, x, y, z    W, s, t, x, y, z
Allowed substitution hints:    D( t)    E( t, s)    F( x, y, z, t, s)

Proof of Theorem cdleme46fsvlpq
StepHypRef Expression
1 cdlemef46.b . . 3  |-  B  =  ( Base `  K
)
2 cdlemef46.l . . 3  |-  .<_  =  ( le `  K )
3 cdlemef46.j . . 3  |-  .\/  =  ( join `  K )
4 cdlemef46.m . . 3  |-  ./\  =  ( meet `  K )
5 cdlemef46.a . . 3  |-  A  =  ( Atoms `  K )
6 cdlemef46.h . . 3  |-  H  =  ( LHyp `  K
)
7 cdlemef46.u . . 3  |-  U  =  ( ( P  .\/  Q )  ./\  W )
8 cdlemef46.d . . 3  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
9 cdlemefs46.e . . 3  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
10 eqid 2443 . . 3  |-  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  E ) )  =  (
iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  E ) )
11 eqid 2443 . . 3  |-  if ( s  .<_  ( P  .\/  Q ) ,  (
iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  E ) ) , 
[_ s  /  t ]_ D )  =  if ( s  .<_  ( P 
.\/  Q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  E ) ) , 
[_ s  /  t ]_ D )
12 eqid 2443 . . 3  |-  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  (
x  ./\  W )
)  =  x )  ->  z  =  ( if ( s  .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  E ) ) ,  [_ s  /  t ]_ D
)  .\/  ( x  ./\ 
W ) ) ) )  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  (
x  ./\  W )
)  =  x )  ->  z  =  ( if ( s  .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  E ) ) ,  [_ s  /  t ]_ D
)  .\/  ( x  ./\ 
W ) ) ) )
13 cdlemef46.f . . 3  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( if ( s  .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  E ) ) ,  [_ s  /  t ]_ D
)  .\/  ( x  ./\ 
W ) ) ) ) ,  x ) )
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13cdlemefs32fva1 31394 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  R  .<_  ( P  .\/  Q ) )  ->  ( F `  R )  =  [_ R  /  s ]_ if ( s  .<_  ( P 
.\/  Q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  E ) ) , 
[_ s  /  t ]_ D ) )
15 vex 2968 . . . 4  |-  s  e. 
_V
16 eqid 2443 . . . . 5  |-  ( ( s  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
178, 16cdleme31sc 31355 . . . 4  |-  ( s  e.  _V  ->  [_ s  /  t ]_ D  =  ( ( s 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s ) 
./\  W ) ) ) )
1815, 17ax-mp 5 . . 3  |-  [_ s  /  t ]_ D  =  ( ( s 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s ) 
./\  W ) ) )
19 eqid 2443 . . 3  |-  ( ( P  .\/  Q ) 
./\  ( D  .\/  ( ( R  .\/  t )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( R  .\/  t )  ./\  W
) ) )
20 eqid 2443 . . 3  |-  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  ( ( P  .\/  Q
)  ./\  ( D  .\/  ( ( R  .\/  t )  ./\  W
) ) ) ) )  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  ( ( P  .\/  Q
)  ./\  ( D  .\/  ( ( R  .\/  t )  ./\  W
) ) ) ) )
211, 2, 3, 4, 5, 6, 7, 18, 8, 9, 10, 11, 19, 20cdleme41sn3a 31404 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  R  .<_  ( P  .\/  Q ) )  ->  [_ R  / 
s ]_ if ( s 
.<_  ( P  .\/  Q
) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  E ) ) ,  [_ s  /  t ]_ D
)  .<_  ( P  .\/  Q ) )
2214, 21eqbrtrd 4263 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  R  .<_  ( P  .\/  Q ) )  ->  ( F `  R )  .<_  ( P 
.\/  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   _Vcvv 2965   [_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
This theorem is referenced by:  cdlemeg46rgv  31499  cdlemeg46req  31500  cdlemeg46gfv  31501
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-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
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