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Theorem cdlemefr45e 30544
Description: Explicit expansion of cdlemefr45 30543. TODO: use to shorten cdlemefr45 30543 uses? TODO FIX COMMENT (Contributed by NM, 10-Apr-2013.)
Hypotheses
Ref Expression
cdlemef45.b  |-  B  =  ( Base `  K
)
cdlemef45.l  |-  .<_  =  ( le `  K )
cdlemef45.j  |-  .\/  =  ( join `  K )
cdlemef45.m  |-  ./\  =  ( meet `  K )
cdlemef45.a  |-  A  =  ( Atoms `  K )
cdlemef45.h  |-  H  =  ( LHyp `  K
)
cdlemef45.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdlemef45.d  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdlemef45.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
cdlemefr45e  |-  ( ( ( ( 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 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) ) ) )
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 cdlemefr45e
StepHypRef Expression
1 cdlemef45.b . . 3  |-  B  =  ( Base `  K
)
2 cdlemef45.l . . 3  |-  .<_  =  ( le `  K )
3 cdlemef45.j . . 3  |-  .\/  =  ( join `  K )
4 cdlemef45.m . . 3  |-  ./\  =  ( meet `  K )
5 cdlemef45.a . . 3  |-  A  =  ( Atoms `  K )
6 cdlemef45.h . . 3  |-  H  =  ( LHyp `  K
)
7 cdlemef45.u . . 3  |-  U  =  ( ( P  .\/  Q )  ./\  W )
8 cdlemef45.d . . 3  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
9 cdlemef45.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 ) )
101, 2, 3, 4, 5, 6, 7, 8, 9cdlemefr45 30543 . 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  /  t ]_ D )
11 simp2rl 1026 . . 3  |-  ( ( ( ( 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  e.  A )
12 eqid 2389 . . . 4  |-  ( ( R  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )  =  ( ( R  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
138, 12cdleme31sc 30500 . . 3  |-  ( R  e.  A  ->  [_ R  /  t ]_ D  =  ( ( R 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) ) ) )
1411, 13syl 16 . 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  /  t ]_ D  =  ( ( R 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) ) ) )
1510, 14eqtrd 2421 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 )  =  ( ( R 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2552   A.wral 2651   [_csb 3196   ifcif 3684   class class class wbr 4155    e. cmpt 4209   ` cfv 5396  (class class class)co 6022   iota_crio 6480   Basecbs 13398   lecple 13465   joincjn 14330   meetcmee 14331   Atomscatm 29380   HLchlt 29467   LHypclh 30100
This theorem is referenced by:  cdlemefs45eN  30547  cdleme46frvlpq  30620  cdleme50trn1  30665
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-poset 14332  df-plt 14344  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-p0 14397  df-p1 14398  df-lat 14404  df-clat 14466  df-oposet 29293  df-ol 29295  df-oml 29296  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468  df-lines 29617  df-psubsp 29619  df-pmap 29620  df-padd 29912  df-lhyp 30104
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