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Theorem cdleme29b 30857
Description: Transform cdleme28 30855. (Compare cdleme25b 30836.) TODO: FIX COMMENT (Contributed by NM, 7-Feb-2013.)
Hypotheses
Ref Expression
cdleme26.b  |-  B  =  ( Base `  K
)
cdleme26.l  |-  .<_  =  ( le `  K )
cdleme26.j  |-  .\/  =  ( join `  K )
cdleme26.m  |-  ./\  =  ( meet `  K )
cdleme26.a  |-  A  =  ( Atoms `  K )
cdleme26.h  |-  H  =  ( LHyp `  K
)
cdleme27.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme27.f  |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
cdleme27.z  |-  Z  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )
cdleme27.n  |-  N  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( s  .\/  z )  ./\  W
) ) )
cdleme27.d  |-  D  =  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  N ) )
cdleme27.c  |-  C  =  if ( s  .<_  ( P  .\/  Q ) ,  D ,  F
)
Assertion
Ref Expression
cdleme29b  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  E. v  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  -> 
v  =  ( C 
.\/  ( X  ./\  W ) ) ) )
Distinct variable groups:    u, s,
z, A    B, s, u, z    u, F    H, s, z    .\/ , s, u, z    K, s, z    .<_ , s, u, z    ./\ , s, u, z    u, N    P, s, u, z    Q, s, u, z    U, s, u, z    W, s, u, z    X, s   
v, A    v, B    v, 
.\/    v,  .<_    v,  ./\    v, P    v, Q    v, U    v, W    v, C    v, s, Z, u    z, v, X
Allowed substitution hints:    C( z, u, s)    D( z, v, u, s)    F( z, v, s)    H( v, u)    K( v, u)    N( z, v, s)    X( u)    Z( z)

Proof of Theorem cdleme29b
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 cdleme26.b . . 3  |-  B  =  ( Base `  K
)
2 cdleme26.l . . 3  |-  .<_  =  ( le `  K )
3 cdleme26.j . . 3  |-  .\/  =  ( join `  K )
4 cdleme26.m . . 3  |-  ./\  =  ( meet `  K )
5 cdleme26.a . . 3  |-  A  =  ( Atoms `  K )
6 cdleme26.h . . 3  |-  H  =  ( LHyp `  K
)
7 cdleme27.u . . 3  |-  U  =  ( ( P  .\/  Q )  ./\  W )
8 cdleme27.f . . 3  |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
9 cdleme27.z . . 3  |-  Z  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )
10 cdleme27.n . . 3  |-  N  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( s  .\/  z )  ./\  W
) ) )
11 cdleme27.d . . 3  |-  D  =  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  N ) )
12 cdleme27.c . . 3  |-  C  =  if ( s  .<_  ( P  .\/  Q ) ,  D ,  F
)
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cdleme29ex 30856 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  E. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  /\  ( C  .\/  ( X 
./\  W ) )  e.  B ) )
14 eqid 2404 . . 3  |-  ( ( t  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
15 eqid 2404 . . 3  |-  ( ( P  .\/  Q ) 
./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) )
16 eqid 2404 . . 3  |-  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P 
.\/  Q ) )  ->  u  =  ( ( P  .\/  Q
)  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) ) ) )  =  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P 
.\/  Q ) )  ->  u  =  ( ( P  .\/  Q
)  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) ) ) )
17 eqid 2404 . . 3  |-  if ( t  .<_  ( P  .\/  Q ) ,  (
iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  ( ( P 
.\/  Q )  ./\  ( Z  .\/  ( ( t  .\/  z ) 
./\  W ) ) ) ) ) ,  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) ) )  =  if ( t 
.<_  ( P  .\/  Q
) ,  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P 
.\/  Q ) )  ->  u  =  ( ( P  .\/  Q
)  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) ) ) ) ,  ( ( t  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) ) )
181, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17cdleme28 30855 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  A. s  e.  A  A. t  e.  A  ( (
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  /\  ( -.  t  .<_  W  /\  ( t 
.\/  ( X  ./\  W ) )  =  X ) )  ->  ( C  .\/  ( X  ./\  W ) )  =  ( if ( t  .<_  ( P  .\/  Q ) ,  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P 
.\/  Q ) )  ->  u  =  ( ( P  .\/  Q
)  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) ) ) ) ,  ( ( t  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) ) ) 
.\/  ( X  ./\  W ) ) ) )
19 breq1 4175 . . . . . 6  |-  ( s  =  t  ->  (
s  .<_  W  <->  t  .<_  W ) )
2019notbid 286 . . . . 5  |-  ( s  =  t  ->  ( -.  s  .<_  W  <->  -.  t  .<_  W ) )
21 oveq1 6047 . . . . . 6  |-  ( s  =  t  ->  (
s  .\/  ( X  ./\ 
W ) )  =  ( t  .\/  ( X  ./\  W ) ) )
2221eqeq1d 2412 . . . . 5  |-  ( s  =  t  ->  (
( s  .\/  ( X  ./\  W ) )  =  X  <->  ( t  .\/  ( X  ./\  W
) )  =  X ) )
2320, 22anbi12d 692 . . . 4  |-  ( s  =  t  ->  (
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  <-> 
( -.  t  .<_  W  /\  ( t  .\/  ( X  ./\  W ) )  =  X ) ) )
2412oveq1i 6050 . . . . 5  |-  ( C 
.\/  ( X  ./\  W ) )  =  ( if ( s  .<_  ( P  .\/  Q ) ,  D ,  F
)  .\/  ( X  ./\ 
W ) )
25 breq1 4175 . . . . . . 7  |-  ( s  =  t  ->  (
s  .<_  ( P  .\/  Q )  <->  t  .<_  ( P 
.\/  Q ) ) )
26 oveq1 6047 . . . . . . . . . . . . . . . 16  |-  ( s  =  t  ->  (
s  .\/  z )  =  ( t  .\/  z ) )
2726oveq1d 6055 . . . . . . . . . . . . . . 15  |-  ( s  =  t  ->  (
( s  .\/  z
)  ./\  W )  =  ( ( t 
.\/  z )  ./\  W ) )
2827oveq2d 6056 . . . . . . . . . . . . . 14  |-  ( s  =  t  ->  ( Z  .\/  ( ( s 
.\/  z )  ./\  W ) )  =  ( Z  .\/  ( ( t  .\/  z ) 
./\  W ) ) )
2928oveq2d 6056 . . . . . . . . . . . . 13  |-  ( s  =  t  ->  (
( P  .\/  Q
)  ./\  ( Z  .\/  ( ( s  .\/  z )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) ) )
3010, 29syl5eq 2448 . . . . . . . . . . . 12  |-  ( s  =  t  ->  N  =  ( ( P 
.\/  Q )  ./\  ( Z  .\/  ( ( t  .\/  z ) 
./\  W ) ) ) )
3130eqeq2d 2415 . . . . . . . . . . 11  |-  ( s  =  t  ->  (
u  =  N  <->  u  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) ) ) )
3231imbi2d 308 . . . . . . . . . 10  |-  ( s  =  t  ->  (
( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  N )  <->  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P 
.\/  Q ) )  ->  u  =  ( ( P  .\/  Q
)  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) ) ) ) )
3332ralbidv 2686 . . . . . . . . 9  |-  ( s  =  t  ->  ( A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  N )  <->  A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P 
.\/  Q ) )  ->  u  =  ( ( P  .\/  Q
)  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) ) ) ) )
3433riotabidv 6510 . . . . . . . 8  |-  ( s  =  t  ->  ( iota_ u  e.  B A. z  e.  A  (
( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q ) )  ->  u  =  N ) )  =  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  ( ( P 
.\/  Q )  ./\  ( Z  .\/  ( ( t  .\/  z ) 
./\  W ) ) ) ) ) )
3511, 34syl5eq 2448 . . . . . . 7  |-  ( s  =  t  ->  D  =  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P 
.\/  Q ) )  ->  u  =  ( ( P  .\/  Q
)  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) ) ) ) )
36 oveq1 6047 . . . . . . . . 9  |-  ( s  =  t  ->  (
s  .\/  U )  =  ( t  .\/  U ) )
37 oveq2 6048 . . . . . . . . . . 11  |-  ( s  =  t  ->  ( P  .\/  s )  =  ( P  .\/  t
) )
3837oveq1d 6055 . . . . . . . . . 10  |-  ( s  =  t  ->  (
( P  .\/  s
)  ./\  W )  =  ( ( P 
.\/  t )  ./\  W ) )
3938oveq2d 6056 . . . . . . . . 9  |-  ( s  =  t  ->  ( Q  .\/  ( ( P 
.\/  s )  ./\  W ) )  =  ( Q  .\/  ( ( P  .\/  t ) 
./\  W ) ) )
4036, 39oveq12d 6058 . . . . . . . 8  |-  ( s  =  t  ->  (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) ) )
418, 40syl5eq 2448 . . . . . . 7  |-  ( s  =  t  ->  F  =  ( ( t 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t ) 
./\  W ) ) ) )
4225, 35, 41ifbieq12d 3721 . . . . . 6  |-  ( s  =  t  ->  if ( s  .<_  ( P 
.\/  Q ) ,  D ,  F )  =  if ( t 
.<_  ( P  .\/  Q
) ,  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P 
.\/  Q ) )  ->  u  =  ( ( P  .\/  Q
)  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) ) ) ) ,  ( ( t  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) ) ) )
4342oveq1d 6055 . . . . 5  |-  ( s  =  t  ->  ( if ( s  .<_  ( P 
.\/  Q ) ,  D ,  F ) 
.\/  ( X  ./\  W ) )  =  ( if ( t  .<_  ( P  .\/  Q ) ,  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P 
.\/  Q ) )  ->  u  =  ( ( P  .\/  Q
)  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) ) ) ) ,  ( ( t  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) ) ) 
.\/  ( X  ./\  W ) ) )
4424, 43syl5eq 2448 . . . 4  |-  ( s  =  t  ->  ( C  .\/  ( X  ./\  W ) )  =  ( if ( t  .<_  ( P  .\/  Q ) ,  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P 
.\/  Q ) )  ->  u  =  ( ( P  .\/  Q
)  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) ) ) ) ,  ( ( t  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) ) ) 
.\/  ( X  ./\  W ) ) )
4523, 44reusv3 4690 . . 3  |-  ( E. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  /\  ( C  .\/  ( X  ./\  W ) )  e.  B )  ->  ( A. s  e.  A  A. t  e.  A  ( (
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  /\  ( -.  t  .<_  W  /\  ( t 
.\/  ( X  ./\  W ) )  =  X ) )  ->  ( C  .\/  ( X  ./\  W ) )  =  ( if ( t  .<_  ( P  .\/  Q ) ,  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P 
.\/  Q ) )  ->  u  =  ( ( P  .\/  Q
)  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) ) ) ) ,  ( ( t  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) ) ) 
.\/  ( X  ./\  W ) ) )  <->  E. v  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  -> 
v  =  ( C 
.\/  ( X  ./\  W ) ) ) ) )
4645biimpd 199 . 2  |-  ( E. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  /\  ( C  .\/  ( X  ./\  W ) )  e.  B )  ->  ( A. s  e.  A  A. t  e.  A  ( (
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  /\  ( -.  t  .<_  W  /\  ( t 
.\/  ( X  ./\  W ) )  =  X ) )  ->  ( C  .\/  ( X  ./\  W ) )  =  ( if ( t  .<_  ( P  .\/  Q ) ,  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P 
.\/  Q ) )  ->  u  =  ( ( P  .\/  Q
)  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) ) ) ) ,  ( ( t  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) ) ) 
.\/  ( X  ./\  W ) ) )  ->  E. v  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( X  ./\  W ) )  =  X )  ->  v  =  ( C  .\/  ( X 
./\  W ) ) ) ) )
4713, 18, 46sylc 58 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  E. v  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  -> 
v  =  ( C 
.\/  ( X  ./\  W ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   ifcif 3699   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   iota_crio 6501   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   HLchlt 29833   LHypclh 30466
This theorem is referenced by:  cdleme29c  30858
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470
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