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Theorem cdleme25c 31214
Description: Transform cdleme25b 31213. (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
cdleme25c  |-  ( ( ( ( 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 cdleme25c
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, 9cdleme25b 31213 . 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. u  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) )  ->  u  =  N ) )
11 simp11l 1069 . . . 4  |-  ( ( ( ( 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 ) ) )  ->  K  e.  HL )
12 simp11r 1070 . . . 4  |-  ( ( ( ( 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 ) ) )  ->  W  e.  H )
13 simp12 989 . . . 4  |-  ( ( ( ( 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 ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
14 simp13 990 . . . 4  |-  ( ( ( ( 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 ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
15 simp3l 986 . . . 4  |-  ( ( ( ( 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 ) ) )  ->  P  =/=  Q )
162, 3, 5, 6cdlemb2 30900 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  E. s  e.  A  ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) ) )
1711, 12, 13, 14, 15, 16syl221anc 1196 . . 3  |-  ( ( ( ( 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 ) ) )
18 reusv1 4725 . . 3  |-  ( E. s  e.  A  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) )  ->  ( E! u  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) )  ->  u  =  N )  <->  E. u  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q
) )  ->  u  =  N ) ) )
1917, 18syl 16 . 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! u  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q
) )  ->  u  =  N )  <->  E. u  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) )  ->  u  =  N ) ) )
2010, 19mpbird 225 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    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   E!wreu 2709   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   meetcmee 14404   Atomscatm 30123   HLchlt 30210   LHypclh 30843
This theorem is referenced by:  cdleme25dN  31215  cdleme25cl  31216
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-llines 30357  df-lplanes 30358  df-lvols 30359  df-lines 30360  df-psubsp 30362  df-pmap 30363  df-padd 30655  df-lhyp 30847
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