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Theorem ltrnu 29440
Description: Uniqueness property of a lattice translation value for atoms not under the fiducial co-atom  W. Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
ltrnu.l  |-  .<_  =  ( le `  K )
ltrnu.j  |-  .\/  =  ( join `  K )
ltrnu.m  |-  ./\  =  ( meet `  K )
ltrnu.a  |-  A  =  ( Atoms `  K )
ltrnu.h  |-  H  =  ( LHyp `  K
)
ltrnu.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrnu  |-  ( ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( P  .\/  ( F `  P ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  Q ) )  ./\  W ) )

Proof of Theorem ltrnu
StepHypRef Expression
1 an4 800 . . 3  |-  ( ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  <->  ( ( P  e.  A  /\  Q  e.  A )  /\  ( -.  P  .<_  W  /\  -.  Q  .<_  W ) ) )
2 simpr 449 . . . . 5  |-  ( ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T )  /\  ( P  e.  A  /\  Q  e.  A )
)  ->  ( P  e.  A  /\  Q  e.  A ) )
3 simplr 734 . . . . . 6  |-  ( ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T )  /\  ( P  e.  A  /\  Q  e.  A )
)  ->  F  e.  T )
4 ltrnu.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
5 ltrnu.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
6 ltrnu.m . . . . . . . . 9  |-  ./\  =  ( meet `  K )
7 ltrnu.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
8 ltrnu.h . . . . . . . . 9  |-  H  =  ( LHyp `  K
)
9 eqid 2256 . . . . . . . . 9  |-  ( (
LDil `  K ) `  W )  =  ( ( LDil `  K
) `  W )
10 ltrnu.t . . . . . . . . 9  |-  T  =  ( ( LTrn `  K
) `  W )
114, 5, 6, 7, 8, 9, 10isltrn 29438 . . . . . . . 8  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( F  e.  T  <->  ( F  e.  ( (
LDil `  K ) `  W )  /\  A. p  e.  A  A. q  e.  A  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  (
( p  .\/  ( F `  p )
)  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W ) ) ) ) )
1211ad2antrr 709 . . . . . . 7  |-  ( ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T )  /\  ( P  e.  A  /\  Q  e.  A )
)  ->  ( F  e.  T  <->  ( F  e.  ( ( LDil `  K
) `  W )  /\  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  (
( p  .\/  ( F `  p )
)  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W ) ) ) ) )
13 simpr 449 . . . . . . 7  |-  ( ( F  e.  ( (
LDil `  K ) `  W )  /\  A. p  e.  A  A. q  e.  A  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  (
( p  .\/  ( F `  p )
)  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W ) ) )  ->  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  (
( p  .\/  ( F `  p )
)  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W ) ) )
1412, 13syl6bi 221 . . . . . 6  |-  ( ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T )  /\  ( P  e.  A  /\  Q  e.  A )
)  ->  ( F  e.  T  ->  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  ( ( p 
.\/  ( F `  p ) )  ./\  W )  =  ( ( q  .\/  ( F `
 q ) ) 
./\  W ) ) ) )
153, 14mpd 16 . . . . 5  |-  ( ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T )  /\  ( P  e.  A  /\  Q  e.  A )
)  ->  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  ( ( p 
.\/  ( F `  p ) )  ./\  W )  =  ( ( q  .\/  ( F `
 q ) ) 
./\  W ) ) )
16 breq1 3966 . . . . . . . . 9  |-  ( p  =  P  ->  (
p  .<_  W  <->  P  .<_  W ) )
1716notbid 287 . . . . . . . 8  |-  ( p  =  P  ->  ( -.  p  .<_  W  <->  -.  P  .<_  W ) )
1817anbi1d 688 . . . . . . 7  |-  ( p  =  P  ->  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  <->  ( -.  P  .<_  W  /\  -.  q  .<_  W ) ) )
19 id 21 . . . . . . . . . 10  |-  ( p  =  P  ->  p  =  P )
20 fveq2 5423 . . . . . . . . . 10  |-  ( p  =  P  ->  ( F `  p )  =  ( F `  P ) )
2119, 20oveq12d 5775 . . . . . . . . 9  |-  ( p  =  P  ->  (
p  .\/  ( F `  p ) )  =  ( P  .\/  ( F `  P )
) )
2221oveq1d 5772 . . . . . . . 8  |-  ( p  =  P  ->  (
( p  .\/  ( F `  p )
)  ./\  W )  =  ( ( P 
.\/  ( F `  P ) )  ./\  W ) )
2322eqeq1d 2264 . . . . . . 7  |-  ( p  =  P  ->  (
( ( p  .\/  ( F `  p ) )  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W )  <->  ( ( P 
.\/  ( F `  P ) )  ./\  W )  =  ( ( q  .\/  ( F `
 q ) ) 
./\  W ) ) )
2418, 23imbi12d 313 . . . . . 6  |-  ( p  =  P  ->  (
( ( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  (
( p  .\/  ( F `  p )
)  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W ) )  <->  ( ( -.  P  .<_  W  /\  -.  q  .<_  W )  ->  ( ( P 
.\/  ( F `  P ) )  ./\  W )  =  ( ( q  .\/  ( F `
 q ) ) 
./\  W ) ) ) )
25 breq1 3966 . . . . . . . . 9  |-  ( q  =  Q  ->  (
q  .<_  W  <->  Q  .<_  W ) )
2625notbid 287 . . . . . . . 8  |-  ( q  =  Q  ->  ( -.  q  .<_  W  <->  -.  Q  .<_  W ) )
2726anbi2d 687 . . . . . . 7  |-  ( q  =  Q  ->  (
( -.  P  .<_  W  /\  -.  q  .<_  W )  <->  ( -.  P  .<_  W  /\  -.  Q  .<_  W ) ) )
28 id 21 . . . . . . . . . 10  |-  ( q  =  Q  ->  q  =  Q )
29 fveq2 5423 . . . . . . . . . 10  |-  ( q  =  Q  ->  ( F `  q )  =  ( F `  Q ) )
3028, 29oveq12d 5775 . . . . . . . . 9  |-  ( q  =  Q  ->  (
q  .\/  ( F `  q ) )  =  ( Q  .\/  ( F `  Q )
) )
3130oveq1d 5772 . . . . . . . 8  |-  ( q  =  Q  ->  (
( q  .\/  ( F `  q )
)  ./\  W )  =  ( ( Q 
.\/  ( F `  Q ) )  ./\  W ) )
3231eqeq2d 2267 . . . . . . 7  |-  ( q  =  Q  ->  (
( ( P  .\/  ( F `  P ) )  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W )  <->  ( ( P 
.\/  ( F `  P ) )  ./\  W )  =  ( ( Q  .\/  ( F `
 Q ) ) 
./\  W ) ) )
3327, 32imbi12d 313 . . . . . 6  |-  ( q  =  Q  ->  (
( ( -.  P  .<_  W  /\  -.  q  .<_  W )  ->  (
( P  .\/  ( F `  P )
)  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W ) )  <->  ( ( -.  P  .<_  W  /\  -.  Q  .<_  W )  ->  ( ( P 
.\/  ( F `  P ) )  ./\  W )  =  ( ( Q  .\/  ( F `
 Q ) ) 
./\  W ) ) ) )
3424, 33rcla42v 2841 . . . . 5  |-  ( ( P  e.  A  /\  Q  e.  A )  ->  ( A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  ( ( p 
.\/  ( F `  p ) )  ./\  W )  =  ( ( q  .\/  ( F `
 q ) ) 
./\  W ) )  ->  ( ( -.  P  .<_  W  /\  -.  Q  .<_  W )  ->  ( ( P 
.\/  ( F `  P ) )  ./\  W )  =  ( ( Q  .\/  ( F `
 Q ) ) 
./\  W ) ) ) )
352, 15, 34sylc 58 . . . 4  |-  ( ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T )  /\  ( P  e.  A  /\  Q  e.  A )
)  ->  ( ( -.  P  .<_  W  /\  -.  Q  .<_  W )  ->  ( ( P 
.\/  ( F `  P ) )  ./\  W )  =  ( ( Q  .\/  ( F `
 Q ) ) 
./\  W ) ) )
3635impr 605 . . 3  |-  ( ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T )  /\  (
( P  e.  A  /\  Q  e.  A
)  /\  ( -.  P  .<_  W  /\  -.  Q  .<_  W ) ) )  ->  ( ( P  .\/  ( F `  P ) )  ./\  W )  =  ( ( Q  .\/  ( F `
 Q ) ) 
./\  W ) )
371, 36sylan2b 463 . 2  |-  ( ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( ( P 
.\/  ( F `  P ) )  ./\  W )  =  ( ( Q  .\/  ( F `
 Q ) ) 
./\  W ) )
38373impb 1152 1  |-  ( ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( P  .\/  ( F `  P ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  Q ) )  ./\  W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   A.wral 2516   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   lecple 13142   joincjn 14005   meetcmee 14006   Atomscatm 28583   LHypclh 29303   LDilcldil 29419   LTrncltrn 29420
This theorem is referenced by:  ltrncnv  29465  trlval2  29482  cdlemg14f  29972  cdlemg14g  29973
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-ltrn 29424
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