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Theorem ltrnu 29577
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 2284 . . . . . . . . 9  |-  ( (
LDil `  K ) `  W )  =  ( ( LDil `  K
) `  W )
10 ltrnu.t . . . . . . . . 9  |-  T  =  ( ( LTrn `  K
) `  W )
114, 5, 6, 7, 8, 9, 10isltrn 29575 . . . . . . . 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 4027 . . . . . . . . 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 5485 . . . . . . . . . 10  |-  ( p  =  P  ->  ( F `  p )  =  ( F `  P ) )
2119, 20oveq12d 5837 . . . . . . . . 9  |-  ( p  =  P  ->  (
p  .\/  ( F `  p ) )  =  ( P  .\/  ( F `  P )
) )
2221oveq1d 5834 . . . . . . . 8  |-  ( p  =  P  ->  (
( p  .\/  ( F `  p )
)  ./\  W )  =  ( ( P 
.\/  ( F `  P ) )  ./\  W ) )
2322eqeq1d 2292 . . . . . . 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 4027 . . . . . . . . 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 5485 . . . . . . . . . 10  |-  ( q  =  Q  ->  ( F `  q )  =  ( F `  Q ) )
3028, 29oveq12d 5837 . . . . . . . . 9  |-  ( q  =  Q  ->  (
q  .\/  ( F `  q ) )  =  ( Q  .\/  ( F `  Q )
) )
3130oveq1d 5834 . . . . . . . 8  |-  ( q  =  Q  ->  (
( q  .\/  ( F `  q )
)  ./\  W )  =  ( ( Q 
.\/  ( F `  Q ) )  ./\  W ) )
3231eqeq2d 2295 . . . . . . 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, 33rspc2v 2891 . . . . 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 1628    e. wcel 1688   A.wral 2544   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   lecple 13209   joincjn 14072   meetcmee 14073   Atomscatm 28720   LHypclh 29440   LDilcldil 29556   LTrncltrn 29557
This theorem is referenced by:  ltrncnv  29602  trlval2  29619  cdlemg14f  30109  cdlemg14g  30110
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-ltrn 29561
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