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Theorem ltrnu 30932
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
Dummy variables  q  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 an4 797 . . 3  |-  ( ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  <->  ( ( P  e.  A  /\  Q  e.  A )  /\  ( -.  P  .<_  W  /\  -.  Q  .<_  W ) ) )
2 simpr 447 . . . . 5  |-  ( ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T )  /\  ( P  e.  A  /\  Q  e.  A )
)  ->  ( P  e.  A  /\  Q  e.  A ) )
3 simplr 731 . . . . . 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 2296 . . . . . . . . 9  |-  ( (
LDil `  K ) `  W )  =  ( ( LDil `  K
) `  W )
10 ltrnu.t . . . . . . . . 9  |-  T  =  ( ( LTrn `  K
) `  W )
114, 5, 6, 7, 8, 9, 10isltrn 30930 . . . . . . . 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 706 . . . . . . 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 447 . . . . . . 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 219 . . . . . 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 14 . . . . 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 4042 . . . . . . . . 9  |-  ( p  =  P  ->  (
p  .<_  W  <->  P  .<_  W ) )
1716notbid 285 . . . . . . . 8  |-  ( p  =  P  ->  ( -.  p  .<_  W  <->  -.  P  .<_  W ) )
1817anbi1d 685 . . . . . . 7  |-  ( p  =  P  ->  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  <->  ( -.  P  .<_  W  /\  -.  q  .<_  W ) ) )
19 id 19 . . . . . . . . . 10  |-  ( p  =  P  ->  p  =  P )
20 fveq2 5541 . . . . . . . . . 10  |-  ( p  =  P  ->  ( F `  p )  =  ( F `  P ) )
2119, 20oveq12d 5892 . . . . . . . . 9  |-  ( p  =  P  ->  (
p  .\/  ( F `  p ) )  =  ( P  .\/  ( F `  P )
) )
2221oveq1d 5889 . . . . . . . 8  |-  ( p  =  P  ->  (
( p  .\/  ( F `  p )
)  ./\  W )  =  ( ( P 
.\/  ( F `  P ) )  ./\  W ) )
2322eqeq1d 2304 . . . . . . 7  |-  ( p  =  P  ->  (
( ( p  .\/  ( F `  p ) )  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W )  <->  ( ( P 
.\/  ( F `  P ) )  ./\  W )  =  ( ( q  .\/  ( F `
 q ) ) 
./\  W ) ) )
2418, 23imbi12d 311 . . . . . 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 4042 . . . . . . . . 9  |-  ( q  =  Q  ->  (
q  .<_  W  <->  Q  .<_  W ) )
2625notbid 285 . . . . . . . 8  |-  ( q  =  Q  ->  ( -.  q  .<_  W  <->  -.  Q  .<_  W ) )
2726anbi2d 684 . . . . . . 7  |-  ( q  =  Q  ->  (
( -.  P  .<_  W  /\  -.  q  .<_  W )  <->  ( -.  P  .<_  W  /\  -.  Q  .<_  W ) ) )
28 id 19 . . . . . . . . . 10  |-  ( q  =  Q  ->  q  =  Q )
29 fveq2 5541 . . . . . . . . . 10  |-  ( q  =  Q  ->  ( F `  q )  =  ( F `  Q ) )
3028, 29oveq12d 5892 . . . . . . . . 9  |-  ( q  =  Q  ->  (
q  .\/  ( F `  q ) )  =  ( Q  .\/  ( F `  Q )
) )
3130oveq1d 5889 . . . . . . . 8  |-  ( q  =  Q  ->  (
( q  .\/  ( F `  q )
)  ./\  W )  =  ( ( Q 
.\/  ( F `  Q ) )  ./\  W ) )
3231eqeq2d 2307 . . . . . . 7  |-  ( q  =  Q  ->  (
( ( P  .\/  ( F `  P ) )  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W )  <->  ( ( P 
.\/  ( F `  P ) )  ./\  W )  =  ( ( Q  .\/  ( F `
 Q ) ) 
./\  W ) ) )
3327, 32imbi12d 311 . . . . . 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 2903 . . . . 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 56 . . . 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 602 . . 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 461 . 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 1147 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 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   lecple 13231   joincjn 14094   meetcmee 14095   Atomscatm 30075   LHypclh 30795   LDilcldil 30911   LTrncltrn 30912
This theorem is referenced by:  ltrncnv  30957  trlval2  30974  cdlemg14f  31464  cdlemg14g  31465
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-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-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2561  df-rex 2562  df-reu 2563  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-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  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-ltrn 30916
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