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Theorem cdlemefr44 31222
Description: Value of f(r) when r is an atom not under pq, using more compact hypotheses. TODO: eliminate and use cdlemefr45 instead? TODO FIX COMMENT (Contributed by NM, 31-Mar-2013.)
Hypotheses
Ref Expression
cdlemef44.b  |-  B  =  ( Base `  K
)
cdlemef44.l  |-  .<_  =  ( le `  K )
cdlemef44.j  |-  .\/  =  ( join `  K )
cdlemef44.m  |-  ./\  =  ( meet `  K )
cdlemef44.a  |-  A  =  ( Atoms `  K )
cdlemef44.h  |-  H  =  ( LHyp `  K
)
cdlemef44.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdlemef44.d  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdlemef44.o  |-  O  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( if ( s  .<_  ( P  .\/  Q ) ,  I ,  [_ s  /  t ]_ D
)  .\/  ( x  ./\ 
W ) ) ) )
cdlemef44.f  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
Assertion
Ref Expression
cdlemefr44  |-  ( ( ( ( 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 )
Distinct variable groups:    t, s, x, z, A    B, s,
t, x, z    x, D, z    H, s, t, x, z    x, I, z    .\/ , s, t, x, z    K, s, t, x, z    .<_ , s, t, x, z    ./\ , s, t, x, z    P, s, t, x, z    Q, s, t, x, z    R, s, t, x, z    U, s, t, x, z    W, s, t, x, z
Allowed substitution hints:    D( t, s)    F( x, z, t, s)    I( t, s)    O( x, z, t, s)

Proof of Theorem cdlemefr44
StepHypRef Expression
1 cdlemef44.b . . 3  |-  B  =  ( Base `  K
)
2 cdlemef44.l . . 3  |-  .<_  =  ( le `  K )
3 cdlemef44.j . . 3  |-  .\/  =  ( join `  K )
4 cdlemef44.m . . 3  |-  ./\  =  ( meet `  K )
5 cdlemef44.a . . 3  |-  A  =  ( Atoms `  K )
6 cdlemef44.h . . 3  |-  H  =  ( LHyp `  K
)
7 cdlemef44.u . . 3  |-  U  =  ( ( P  .\/  Q )  ./\  W )
8 eqid 2436 . . 3  |-  ( ( s  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
9 biid 228 . . . 4  |-  ( s 
.<_  ( P  .\/  Q
)  <->  s  .<_  ( P 
.\/  Q ) )
10 vex 2959 . . . . 5  |-  s  e. 
_V
11 cdlemef44.d . . . . . 6  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
1211, 8cdleme31sc 31181 . . . . 5  |-  ( s  e.  _V  ->  [_ s  /  t ]_ D  =  ( ( s 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s ) 
./\  W ) ) ) )
1310, 12ax-mp 8 . . . 4  |-  [_ s  /  t ]_ D  =  ( ( s 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s ) 
./\  W ) ) )
149, 13ifbieq2i 3758 . . 3  |-  if ( s  .<_  ( P  .\/  Q ) ,  I ,  [_ s  /  t ]_ D )  =  if ( s  .<_  ( P 
.\/  Q ) ,  I ,  ( ( s  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) ) )
15 cdlemef44.o . . 3  |-  O  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( if ( s  .<_  ( P  .\/  Q ) ,  I ,  [_ s  /  t ]_ D
)  .\/  ( x  ./\ 
W ) ) ) )
16 cdlemef44.f . . 3  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
17 eqid 2436 . . 3  |-  ( ( R  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )  =  ( ( R  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
181, 2, 3, 4, 5, 6, 7, 8, 14, 15, 16, 17cdlemefr31fv1 31208 . 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 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) ) ) )
19 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 )
2011, 17cdleme31sc 31181 . . 3  |-  ( R  e.  A  ->  [_ R  /  t ]_ D  =  ( ( R 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) ) ) )
2119, 20syl 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 ) ) ) )
2218, 21eqtr4d 2471 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  /  t ]_ D )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   _Vcvv 2956   [_csb 3251   ifcif 3739   class class class wbr 4212    e. cmpt 4266   ` cfv 5454  (class class class)co 6081   iota_crio 6542   Basecbs 13469   lecple 13536   joincjn 14401   meetcmee 14402   Atomscatm 30061   HLchlt 30148   LHypclh 30781
This theorem is referenced by:  cdlemefr45  31224
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-lines 30298  df-psubsp 30300  df-pmap 30301  df-padd 30593  df-lhyp 30785
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