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Theorem cdleme25b 31088
Description: Transform cdleme24 31086. 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 31087 . 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 2435 . . 3  |-  ( ( t  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
12 eqid 2435 . . 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 31086 . 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 4207 . . . . . 6  |-  ( s  =  t  ->  (
s  .<_  W  <->  t  .<_  W ) )
1514notbid 286 . . . . 5  |-  ( s  =  t  ->  ( -.  s  .<_  W  <->  -.  t  .<_  W ) )
16 breq1 4207 . . . . . 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 6080 . . . . . . . . 9  |-  ( s  =  t  ->  (
s  .\/  U )  =  ( t  .\/  U ) )
20 oveq2 6081 . . . . . . . . . . 11  |-  ( s  =  t  ->  ( P  .\/  s )  =  ( P  .\/  t
) )
2120oveq1d 6088 . . . . . . . . . 10  |-  ( s  =  t  ->  (
( P  .\/  s
)  ./\  W )  =  ( ( P 
.\/  t )  ./\  W ) )
2221oveq2d 6089 . . . . . . . . 9  |-  ( s  =  t  ->  ( Q  .\/  ( ( P 
.\/  s )  ./\  W ) )  =  ( Q  .\/  ( ( P  .\/  t ) 
./\  W ) ) )
2319, 22oveq12d 6091 . . . . . . . 8  |-  ( s  =  t  ->  (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) ) )
248, 23syl5eq 2479 . . . . . . 7  |-  ( s  =  t  ->  F  =  ( ( t 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t ) 
./\  W ) ) ) )
25 oveq2 6081 . . . . . . . 8  |-  ( s  =  t  ->  ( R  .\/  s )  =  ( R  .\/  t
) )
2625oveq1d 6088 . . . . . . 7  |-  ( s  =  t  ->  (
( R  .\/  s
)  ./\  W )  =  ( ( R 
.\/  t )  ./\  W ) )
2724, 26oveq12d 6091 . . . . . 6  |-  ( s  =  t  ->  ( F  .\/  ( ( R 
.\/  s )  ./\  W ) )  =  ( ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )  .\/  ( ( R  .\/  t )  ./\  W
) ) )
2827oveq2d 6089 . . . . 5  |-  ( s  =  t  ->  (
( P  .\/  Q
)  ./\  ( F  .\/  ( ( R  .\/  s )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( (
( t  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )  .\/  ( ( R  .\/  t )  ./\  W
) ) ) )
299, 28syl5eq 2479 . . . 4  |-  ( s  =  t  ->  N  =  ( ( P 
.\/  Q )  ./\  ( ( ( t 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t ) 
./\  W ) ) )  .\/  ( ( R  .\/  t ) 
./\  W ) ) ) )
3018, 29reusv3 4723 . . 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   meetcmee 14394   Atomscatm 29998   HLchlt 30085   LHypclh 30718
This theorem is referenced by:  cdleme25c  31089
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lvols 30234  df-lines 30235  df-psubsp 30237  df-pmap 30238  df-padd 30530  df-lhyp 30722
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