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Theorem cdleme29b 31186
Description: Transform cdleme28 31184. (Compare cdleme25b 31165.) 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 31185 . 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 2296 . . 3  |-  ( ( t  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
15 eqid 2296 . . 3  |-  ( ( P  .\/  Q ) 
./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) )
16 eqid 2296 . . 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 2296 . . 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 31184 . 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 4042 . . . . . 6  |-  ( s  =  t  ->  (
s  .<_  W  <->  t  .<_  W ) )
2019notbid 285 . . . . 5  |-  ( s  =  t  ->  ( -.  s  .<_  W  <->  -.  t  .<_  W ) )
21 oveq1 5881 . . . . . 6  |-  ( s  =  t  ->  (
s  .\/  ( X  ./\ 
W ) )  =  ( t  .\/  ( X  ./\  W ) ) )
2221eqeq1d 2304 . . . . 5  |-  ( s  =  t  ->  (
( s  .\/  ( X  ./\  W ) )  =  X  <->  ( t  .\/  ( X  ./\  W
) )  =  X ) )
2320, 22anbi12d 691 . . . 4  |-  ( s  =  t  ->  (
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  <-> 
( -.  t  .<_  W  /\  ( t  .\/  ( X  ./\  W ) )  =  X ) ) )
2412oveq1i 5884 . . . . 5  |-  ( C 
.\/  ( X  ./\  W ) )  =  ( if ( s  .<_  ( P  .\/  Q ) ,  D ,  F
)  .\/  ( X  ./\ 
W ) )
25 breq1 4042 . . . . . . 7  |-  ( s  =  t  ->  (
s  .<_  ( P  .\/  Q )  <->  t  .<_  ( P 
.\/  Q ) ) )
26 oveq1 5881 . . . . . . . . . . . . . . . 16  |-  ( s  =  t  ->  (
s  .\/  z )  =  ( t  .\/  z ) )
2726oveq1d 5889 . . . . . . . . . . . . . . 15  |-  ( s  =  t  ->  (
( s  .\/  z
)  ./\  W )  =  ( ( t 
.\/  z )  ./\  W ) )
2827oveq2d 5890 . . . . . . . . . . . . . 14  |-  ( s  =  t  ->  ( Z  .\/  ( ( s 
.\/  z )  ./\  W ) )  =  ( Z  .\/  ( ( t  .\/  z ) 
./\  W ) ) )
2928oveq2d 5890 . . . . . . . . . . . . 13  |-  ( s  =  t  ->  (
( P  .\/  Q
)  ./\  ( Z  .\/  ( ( s  .\/  z )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) ) )
3010, 29syl5eq 2340 . . . . . . . . . . . 12  |-  ( s  =  t  ->  N  =  ( ( P 
.\/  Q )  ./\  ( Z  .\/  ( ( t  .\/  z ) 
./\  W ) ) ) )
3130eqeq2d 2307 . . . . . . . . . . 11  |-  ( s  =  t  ->  (
u  =  N  <->  u  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) ) ) )
3231imbi2d 307 . . . . . . . . . 10  |-  ( s  =  t  ->  (
( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  N )  <->  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P 
.\/  Q ) )  ->  u  =  ( ( P  .\/  Q
)  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) ) ) ) )
3332ralbidv 2576 . . . . . . . . 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 6322 . . . . . . . 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 2340 . . . . . . 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 5881 . . . . . . . . 9  |-  ( s  =  t  ->  (
s  .\/  U )  =  ( t  .\/  U ) )
37 oveq2 5882 . . . . . . . . . . 11  |-  ( s  =  t  ->  ( P  .\/  s )  =  ( P  .\/  t
) )
3837oveq1d 5889 . . . . . . . . . 10  |-  ( s  =  t  ->  (
( P  .\/  s
)  ./\  W )  =  ( ( P 
.\/  t )  ./\  W ) )
3938oveq2d 5890 . . . . . . . . 9  |-  ( s  =  t  ->  ( Q  .\/  ( ( P 
.\/  s )  ./\  W ) )  =  ( Q  .\/  ( ( P  .\/  t ) 
./\  W ) ) )
4036, 39oveq12d 5892 . . . . . . . 8  |-  ( s  =  t  ->  (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) ) )
418, 40syl5eq 2340 . . . . . . 7  |-  ( s  =  t  ->  F  =  ( ( t 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t ) 
./\  W ) ) ) )
4225, 35, 41ifbieq12d 3600 . . . . . 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 5889 . . . . 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 2340 . . . 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 4558 . . 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 198 . 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 56 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   ifcif 3578   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   iota_crio 6313   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Atomscatm 30075   HLchlt 30162   LHypclh 30795
This theorem is referenced by:  cdleme29c  31187
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799
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