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Theorem cdleme25b 30470
Description: Transform cdleme24 30468. TODO get rid of $d's on  U,  N (Contributed by NM, 1-Jan-2013.)
Hypotheses
Ref Expression
cdleme24.b  |-  B  =  ( Base `  K
)
cdleme24.l  |-  .<_  =  ( le `  K )
cdleme24.j  |-  .\/  =  ( join `  K )
cdleme24.m  |-  ./\  =  ( meet `  K )
cdleme24.a  |-  A  =  ( Atoms `  K )
cdleme24.h  |-  H  =  ( LHyp `  K
)
cdleme24.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme24.f  |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
cdleme24.n  |-  N  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  s )  ./\  W
) ) )
Assertion
Ref Expression
cdleme25b  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  E. u  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) )  ->  u  =  N ) )
Distinct variable groups:    u, s, A    B, s, u    H, s    .\/ , s, u    K, s   
.<_ , s, u    ./\ , s, u    P, s, u    Q, s, u    R, s, u    W, s, u    u, N    U, s, u
Allowed substitution hints:    F( u, s)    H( u)    K( u)    N( s)

Proof of Theorem cdleme25b
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 cdleme24.b . . 3  |-  B  =  ( Base `  K
)
2 cdleme24.l . . 3  |-  .<_  =  ( le `  K )
3 cdleme24.j . . 3  |-  .\/  =  ( join `  K )
4 cdleme24.m . . 3  |-  ./\  =  ( meet `  K )
5 cdleme24.a . . 3  |-  A  =  ( Atoms `  K )
6 cdleme24.h . . 3  |-  H  =  ( LHyp `  K
)
7 cdleme24.u . . 3  |-  U  =  ( ( P  .\/  Q )  ./\  W )
8 cdleme24.f . . 3  |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
9 cdleme24.n . . 3  |-  N  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  s )  ./\  W
) ) )
101, 2, 3, 4, 5, 6, 7, 8, 9cdleme25a 30469 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  E. s  e.  A  ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) )  /\  N  e.  B
) )
11 eqid 2389 . . 3  |-  ( ( t  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
12 eqid 2389 . . 3  |-  ( ( P  .\/  Q ) 
./\  ( ( ( t  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )  .\/  ( ( R  .\/  t )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( (
( t  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )  .\/  ( ( R  .\/  t )  ./\  W
) ) )
131, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12cdleme24 30468 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  A. s  e.  A  A. t  e.  A  ( (
( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) )  /\  ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) ) )  ->  N  =  ( ( P  .\/  Q )  ./\  ( (
( t  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )  .\/  ( ( R  .\/  t )  ./\  W
) ) ) ) )
14 breq1 4158 . . . . . 6  |-  ( s  =  t  ->  (
s  .<_  W  <->  t  .<_  W ) )
1514notbid 286 . . . . 5  |-  ( s  =  t  ->  ( -.  s  .<_  W  <->  -.  t  .<_  W ) )
16 breq1 4158 . . . . . 6  |-  ( s  =  t  ->  (
s  .<_  ( P  .\/  Q )  <->  t  .<_  ( P 
.\/  Q ) ) )
1716notbid 286 . . . . 5  |-  ( s  =  t  ->  ( -.  s  .<_  ( P 
.\/  Q )  <->  -.  t  .<_  ( P  .\/  Q
) ) )
1815, 17anbi12d 692 . . . 4  |-  ( s  =  t  ->  (
( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) )  <->  ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) ) ) )
19 oveq1 6029 . . . . . . . . 9  |-  ( s  =  t  ->  (
s  .\/  U )  =  ( t  .\/  U ) )
20 oveq2 6030 . . . . . . . . . . 11  |-  ( s  =  t  ->  ( P  .\/  s )  =  ( P  .\/  t
) )
2120oveq1d 6037 . . . . . . . . . 10  |-  ( s  =  t  ->  (
( P  .\/  s
)  ./\  W )  =  ( ( P 
.\/  t )  ./\  W ) )
2221oveq2d 6038 . . . . . . . . 9  |-  ( s  =  t  ->  ( Q  .\/  ( ( P 
.\/  s )  ./\  W ) )  =  ( Q  .\/  ( ( P  .\/  t ) 
./\  W ) ) )
2319, 22oveq12d 6040 . . . . . . . 8  |-  ( s  =  t  ->  (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) ) )
248, 23syl5eq 2433 . . . . . . 7  |-  ( s  =  t  ->  F  =  ( ( t 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t ) 
./\  W ) ) ) )
25 oveq2 6030 . . . . . . . 8  |-  ( s  =  t  ->  ( R  .\/  s )  =  ( R  .\/  t
) )
2625oveq1d 6037 . . . . . . 7  |-  ( s  =  t  ->  (
( R  .\/  s
)  ./\  W )  =  ( ( R 
.\/  t )  ./\  W ) )
2724, 26oveq12d 6040 . . . . . 6  |-  ( s  =  t  ->  ( F  .\/  ( ( R 
.\/  s )  ./\  W ) )  =  ( ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )  .\/  ( ( R  .\/  t )  ./\  W
) ) )
2827oveq2d 6038 . . . . 5  |-  ( s  =  t  ->  (
( P  .\/  Q
)  ./\  ( F  .\/  ( ( R  .\/  s )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( (
( t  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )  .\/  ( ( R  .\/  t )  ./\  W
) ) ) )
299, 28syl5eq 2433 . . . 4  |-  ( s  =  t  ->  N  =  ( ( P 
.\/  Q )  ./\  ( ( ( t 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t ) 
./\  W ) ) )  .\/  ( ( R  .\/  t ) 
./\  W ) ) ) )
3018, 29reusv3 4673 . . 3  |-  ( E. s  e.  A  ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) )  /\  N  e.  B )  ->  ( A. s  e.  A  A. t  e.  A  ( ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) )  /\  ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) ) )  ->  N  =  ( ( P  .\/  Q )  ./\  ( ( ( t 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t ) 
./\  W ) ) )  .\/  ( ( R  .\/  t ) 
./\  W ) ) ) )  <->  E. u  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) )  ->  u  =  N ) ) )
3130biimpd 199 . 2  |-  ( E. s  e.  A  ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) )  /\  N  e.  B )  ->  ( A. s  e.  A  A. t  e.  A  ( ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) )  /\  ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) ) )  ->  N  =  ( ( P  .\/  Q )  ./\  ( ( ( t 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t ) 
./\  W ) ) )  .\/  ( ( R  .\/  t ) 
./\  W ) ) ) )  ->  E. u  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) )  ->  u  =  N ) ) )
3210, 13, 31sylc 58 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  E. u  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) )  ->  u  =  N ) )
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   E.wrex 2652   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   Basecbs 13398   lecple 13465   joincjn 14330   meetcmee 14331   Atomscatm 29380   HLchlt 29467   LHypclh 30100
This theorem is referenced by:  cdleme25c  30471
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-3or 937  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-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-llines 29614  df-lplanes 29615  df-lvols 29616  df-lines 29617  df-psubsp 29619  df-pmap 29620  df-padd 29912  df-lhyp 30104
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