Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemg2klem Unicode version

Theorem cdlemg2klem 30710
Description: cdleme42keg 30601 with simpler hypotheses. TODO: FIX COMMENT (Contributed by NM, 22-Apr-2013.)
Hypotheses
Ref Expression
cdlemg2.b  |-  B  =  ( Base `  K
)
cdlemg2.l  |-  .<_  =  ( le `  K )
cdlemg2.j  |-  .\/  =  ( join `  K )
cdlemg2.m  |-  ./\  =  ( meet `  K )
cdlemg2.a  |-  A  =  ( Atoms `  K )
cdlemg2.h  |-  H  =  ( LHyp `  K
)
cdlemg2.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg2ex.u  |-  U  =  ( ( p  .\/  q )  ./\  W
)
cdlemg2ex.d  |-  D  =  ( ( t  .\/  U )  ./\  ( q  .\/  ( ( p  .\/  t )  ./\  W
) ) )
cdlemg2ex.e  |-  E  =  ( ( p  .\/  q )  ./\  ( D  .\/  ( ( s 
.\/  t )  ./\  W ) ) )
cdlemg2ex.g  |-  G  =  ( x  e.  B  |->  if ( ( p  =/=  q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( if ( s  .<_  ( p  .\/  q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( p 
.\/  q ) )  ->  y  =  E ) ) ,  [_ s  /  t ]_ D
)  .\/  ( x  ./\ 
W ) ) ) ) ,  x ) )
cdlemg2klem.v  |-  V  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
cdlemg2klem  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  ->  (
( F `  P
)  .\/  ( F `  Q ) )  =  ( ( F `  P )  .\/  V
) )
Distinct variable groups:    t, s, x, y, z, A    B, s, t, x, y, z    D, s, x, y, z   
x, E, y, z    H, s, t, x, y, z    .\/ , s, t, x, y, z    K, s, t, x, y, z    .<_ , s, t, x, y, z    ./\ , s, t, x, y, z    P, s, t, x, y, z    Q, s, t, x, y, z    U, s, t, x, y, z    W, s, t, x, y, z   
q, p, A    F, p, q    H, p, q    K, p, q    .<_ , p, q    T, p, q    W, p, q, s, t, x, y, z    .\/ , p, q    P, p, q    Q, p, q    B, p, q    ./\ , p, q    V, p, q, s, t, x, z
Allowed substitution hints:    D( t, q, p)    T( x, y, z, t, s)    U( q, p)    E( t, s, q, p)    F( x, y, z, t, s)    G( x, y, z, t, s, q, p)    V( y)

Proof of Theorem cdlemg2klem
StepHypRef Expression
1 cdlemg2.b . . 3  |-  B  =  ( Base `  K
)
2 cdlemg2.l . . 3  |-  .<_  =  ( le `  K )
3 cdlemg2.j . . 3  |-  .\/  =  ( join `  K )
4 cdlemg2.m . . 3  |-  ./\  =  ( meet `  K )
5 cdlemg2.a . . 3  |-  A  =  ( Atoms `  K )
6 cdlemg2.h . . 3  |-  H  =  ( LHyp `  K
)
7 cdlemg2.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
8 cdlemg2ex.u . . 3  |-  U  =  ( ( p  .\/  q )  ./\  W
)
9 cdlemg2ex.d . . 3  |-  D  =  ( ( t  .\/  U )  ./\  ( q  .\/  ( ( p  .\/  t )  ./\  W
) ) )
10 cdlemg2ex.e . . 3  |-  E  =  ( ( p  .\/  q )  ./\  ( D  .\/  ( ( s 
.\/  t )  ./\  W ) ) )
11 cdlemg2ex.g . . 3  |-  G  =  ( x  e.  B  |->  if ( ( p  =/=  q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( if ( s  .<_  ( p  .\/  q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( p 
.\/  q ) )  ->  y  =  E ) ) ,  [_ s  /  t ]_ D
)  .\/  ( x  ./\ 
W ) ) ) ) ,  x ) )
12 fveq1 5668 . . . . 5  |-  ( F  =  G  ->  ( F `  P )  =  ( G `  P ) )
13 fveq1 5668 . . . . 5  |-  ( F  =  G  ->  ( F `  Q )  =  ( G `  Q ) )
1412, 13oveq12d 6039 . . . 4  |-  ( F  =  G  ->  (
( F `  P
)  .\/  ( F `  Q ) )  =  ( ( G `  P )  .\/  ( G `  Q )
) )
1512oveq1d 6036 . . . 4  |-  ( F  =  G  ->  (
( F `  P
)  .\/  V )  =  ( ( G `
 P )  .\/  V ) )
1614, 15eqeq12d 2402 . . 3  |-  ( F  =  G  ->  (
( ( F `  P )  .\/  ( F `  Q )
)  =  ( ( F `  P ) 
.\/  V )  <->  ( ( G `  P )  .\/  ( G `  Q
) )  =  ( ( G `  P
)  .\/  V )
) )
17 vex 2903 . . . . 5  |-  s  e. 
_V
18 eqid 2388 . . . . . 6  |-  ( ( s  .\/  U ) 
./\  ( q  .\/  ( ( p  .\/  s )  ./\  W
) ) )  =  ( ( s  .\/  U )  ./\  ( q  .\/  ( ( p  .\/  s )  ./\  W
) ) )
199, 18cdleme31sc 30499 . . . . 5  |-  ( s  e.  _V  ->  [_ s  /  t ]_ D  =  ( ( s 
.\/  U )  ./\  ( q  .\/  (
( p  .\/  s
)  ./\  W )
) ) )
2017, 19ax-mp 8 . . . 4  |-  [_ s  /  t ]_ D  =  ( ( s 
.\/  U )  ./\  ( q  .\/  (
( p  .\/  s
)  ./\  W )
) )
21 eqid 2388 . . . 4  |-  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( p 
.\/  q ) )  ->  y  =  E ) )  =  (
iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( p  .\/  q
) )  ->  y  =  E ) )
22 eqid 2388 . . . 4  |-  if ( s  .<_  ( p  .\/  q ) ,  (
iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( p  .\/  q
) )  ->  y  =  E ) ) , 
[_ s  /  t ]_ D )  =  if ( s  .<_  ( p 
.\/  q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( p  .\/  q
) )  ->  y  =  E ) ) , 
[_ s  /  t ]_ D )
23 eqid 2388 . . . 4  |-  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  (
x  ./\  W )
)  =  x )  ->  z  =  ( if ( s  .<_  ( p  .\/  q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( p 
.\/  q ) )  ->  y  =  E ) ) ,  [_ s  /  t ]_ D
)  .\/  ( x  ./\ 
W ) ) ) )  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  (
x  ./\  W )
)  =  x )  ->  z  =  ( if ( s  .<_  ( p  .\/  q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( p 
.\/  q ) )  ->  y  =  E ) ) ,  [_ s  /  t ]_ D
)  .\/  ( x  ./\ 
W ) ) ) )
24 cdlemg2klem.v . . . 4  |-  V  =  ( ( P  .\/  Q )  ./\  W )
251, 2, 3, 4, 5, 6, 8, 20, 9, 10, 21, 22, 23, 11, 24cdleme42keg 30601 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  A  /\  -.  p  .<_  W )  /\  ( q  e.  A  /\  -.  q  .<_  W ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  (
( G `  P
)  .\/  ( G `  Q ) )  =  ( ( G `  P )  .\/  V
) )
261, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 25cdlemg2ce 30707 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  (
( F `  P
)  .\/  ( F `  Q ) )  =  ( ( F `  P )  .\/  V
) )
27263com23 1159 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  ->  (
( F `  P
)  .\/  ( F `  Q ) )  =  ( ( F `  P )  .\/  V
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551   A.wral 2650   _Vcvv 2900   [_csb 3195   ifcif 3683   class class class wbr 4154    e. cmpt 4208   ` cfv 5395  (class class class)co 6021   iota_crio 6479   Basecbs 13397   lecple 13464   joincjn 14329   meetcmee 14330   Atomscatm 29379   HLchlt 29466   LHypclh 30099   LTrncltrn 30216
This theorem is referenced by:  cdlemg2k  30716
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-undef 6480  df-riota 6486  df-map 6957  df-poset 14331  df-plt 14343  df-lub 14359  df-glb 14360  df-join 14361  df-meet 14362  df-p0 14396  df-p1 14397  df-lat 14403  df-clat 14465  df-oposet 29292  df-ol 29294  df-oml 29295  df-covers 29382  df-ats 29383  df-atl 29414  df-cvlat 29438  df-hlat 29467  df-llines 29613  df-lplanes 29614  df-lvols 29615  df-lines 29616  df-psubsp 29618  df-pmap 29619  df-padd 29911  df-lhyp 30103  df-laut 30104  df-ldil 30219  df-ltrn 30220  df-trl 30274
  Copyright terms: Public domain W3C validator