Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme25c Unicode version

Theorem cdleme25c 31166
Description: Transform cdleme25b 31165. (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 31165 . 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 1066 . . . 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 1067 . . . 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 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  e.  A  /\  -.  P  .<_  W ) )
14 simp13 987 . . . 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 983 . . . 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 30852 . . . 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 1193 . . 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 4550 . . 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 15 . 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 223 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   E!wreu 2558   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Atomscatm 30075   HLchlt 30162   LHypclh 30795
This theorem is referenced by:  cdleme25dN  31167  cdleme25cl  31168
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
  Copyright terms: Public domain W3C validator