Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemg8a Unicode version

Theorem cdlemg8a 30741
Description: TODO: FIX COMMENT (Contributed by NM, 29-Apr-2013.)
Hypotheses
Ref Expression
cdlemg8.l  |-  .<_  =  ( le `  K )
cdlemg8.j  |-  .\/  =  ( join `  K )
cdlemg8.m  |-  ./\  =  ( meet `  K )
cdlemg8.a  |-  A  =  ( Atoms `  K )
cdlemg8.h  |-  H  =  ( LHyp `  K
)
cdlemg8.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
cdlemg8a  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) )

Proof of Theorem cdlemg8a
StepHypRef Expression
1 simp1 957 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 simp2r 984 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
3 cdlemg8.l . . . 4  |-  .<_  =  ( le `  K )
4 cdlemg8.m . . . 4  |-  ./\  =  ( meet `  K )
5 eqid 2387 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
6 cdlemg8.a . . . 4  |-  A  =  ( Atoms `  K )
7 cdlemg8.h . . . 4  |-  H  =  ( LHyp `  K
)
83, 4, 5, 6, 7lhpmat 30144 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Q  ./\  W
)  =  ( 0.
`  K ) )
91, 2, 8syl2anc 643 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) )  -> 
( Q  ./\  W
)  =  ( 0.
`  K ) )
10 cdlemg8.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
113, 6, 7, 10cdlemg6 30737 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) )  -> 
( F `  ( G `  Q )
)  =  Q )
1211oveq2d 6036 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) )  -> 
( Q  .\/  ( F `  ( G `  Q ) ) )  =  ( Q  .\/  Q ) )
13 simp1l 981 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) )  ->  K  e.  HL )
14 simp2rl 1026 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) )  ->  Q  e.  A )
15 cdlemg8.j . . . . . 6  |-  .\/  =  ( join `  K )
1615, 6hlatjidm 29483 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( Q  .\/  Q
)  =  Q )
1713, 14, 16syl2anc 643 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) )  -> 
( Q  .\/  Q
)  =  Q )
1812, 17eqtrd 2419 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) )  -> 
( Q  .\/  ( F `  ( G `  Q ) ) )  =  Q )
1918oveq1d 6035 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) )  -> 
( ( Q  .\/  ( F `  ( G `
 Q ) ) )  ./\  W )  =  ( Q  ./\  W ) )
20 simp33 995 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) )  -> 
( F `  ( G `  P )
)  =  P )
2120oveq2d 6036 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) )  -> 
( P  .\/  ( F `  ( G `  P ) ) )  =  ( P  .\/  P ) )
22 simp2ll 1024 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) )  ->  P  e.  A )
2315, 6hlatjidm 29483 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( P  .\/  P
)  =  P )
2413, 22, 23syl2anc 643 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) )  -> 
( P  .\/  P
)  =  P )
2521, 24eqtrd 2419 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) )  -> 
( P  .\/  ( F `  ( G `  P ) ) )  =  P )
2625oveq1d 6035 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( P  ./\  W ) )
27 simp2l 983 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
283, 4, 5, 6, 7lhpmat 30144 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P  ./\  W
)  =  ( 0.
`  K ) )
291, 27, 28syl2anc 643 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) )  -> 
( P  ./\  W
)  =  ( 0.
`  K ) )
3026, 29eqtrd 2419 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( 0. `  K ) )
319, 19, 303eqtr4rd 2430 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   lecple 13463   joincjn 14328   meetcmee 14329   0.cp0 14393   Atomscatm 29378   HLchlt 29465   LHypclh 30098   LTrncltrn 30215
This theorem is referenced by:  cdlemg8  30745
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-map 6956  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-p1 14396  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-llines 29612  df-lplanes 29613  df-lvols 29614  df-lines 29615  df-psubsp 29617  df-pmap 29618  df-padd 29910  df-lhyp 30102  df-laut 30103  df-ldil 30218  df-ltrn 30219  df-trl 30273
  Copyright terms: Public domain W3C validator