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Theorem ltrnu 35988
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 824 . . 3  |-  ( ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  <->  ( ( P  e.  A  /\  Q  e.  A )  /\  ( -.  P  .<_  W  /\  -.  Q  .<_  W ) ) )
2 simpr 461 . . . . 5  |-  ( ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T )  /\  ( P  e.  A  /\  Q  e.  A )
)  ->  ( P  e.  A  /\  Q  e.  A ) )
3 simplr 755 . . . . . 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 2457 . . . . . . . . 9  |-  ( (
LDil `  K ) `  W )  =  ( ( LDil `  K
) `  W )
10 ltrnu.t . . . . . . . . 9  |-  T  =  ( ( LTrn `  K
) `  W )
114, 5, 6, 7, 8, 9, 10isltrn 35986 . . . . . . . 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 725 . . . . . . 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 461 . . . . . . 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 228 . . . . . 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 15 . . . . 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 4459 . . . . . . . . 9  |-  ( p  =  P  ->  (
p  .<_  W  <->  P  .<_  W ) )
1716notbid 294 . . . . . . . 8  |-  ( p  =  P  ->  ( -.  p  .<_  W  <->  -.  P  .<_  W ) )
1817anbi1d 704 . . . . . . 7  |-  ( p  =  P  ->  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  <->  ( -.  P  .<_  W  /\  -.  q  .<_  W ) ) )
19 id 22 . . . . . . . . . 10  |-  ( p  =  P  ->  p  =  P )
20 fveq2 5872 . . . . . . . . . 10  |-  ( p  =  P  ->  ( F `  p )  =  ( F `  P ) )
2119, 20oveq12d 6314 . . . . . . . . 9  |-  ( p  =  P  ->  (
p  .\/  ( F `  p ) )  =  ( P  .\/  ( F `  P )
) )
2221oveq1d 6311 . . . . . . . 8  |-  ( p  =  P  ->  (
( p  .\/  ( F `  p )
)  ./\  W )  =  ( ( P 
.\/  ( F `  P ) )  ./\  W ) )
2322eqeq1d 2459 . . . . . . 7  |-  ( p  =  P  ->  (
( ( p  .\/  ( F `  p ) )  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W )  <->  ( ( P 
.\/  ( F `  P ) )  ./\  W )  =  ( ( q  .\/  ( F `
 q ) ) 
./\  W ) ) )
2418, 23imbi12d 320 . . . . . 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 4459 . . . . . . . . 9  |-  ( q  =  Q  ->  (
q  .<_  W  <->  Q  .<_  W ) )
2625notbid 294 . . . . . . . 8  |-  ( q  =  Q  ->  ( -.  q  .<_  W  <->  -.  Q  .<_  W ) )
2726anbi2d 703 . . . . . . 7  |-  ( q  =  Q  ->  (
( -.  P  .<_  W  /\  -.  q  .<_  W )  <->  ( -.  P  .<_  W  /\  -.  Q  .<_  W ) ) )
28 id 22 . . . . . . . . . 10  |-  ( q  =  Q  ->  q  =  Q )
29 fveq2 5872 . . . . . . . . . 10  |-  ( q  =  Q  ->  ( F `  q )  =  ( F `  Q ) )
3028, 29oveq12d 6314 . . . . . . . . 9  |-  ( q  =  Q  ->  (
q  .\/  ( F `  q ) )  =  ( Q  .\/  ( F `  Q )
) )
3130oveq1d 6311 . . . . . . . 8  |-  ( q  =  Q  ->  (
( q  .\/  ( F `  q )
)  ./\  W )  =  ( ( Q 
.\/  ( F `  Q ) )  ./\  W ) )
3231eqeq2d 2471 . . . . . . 7  |-  ( q  =  Q  ->  (
( ( P  .\/  ( F `  P ) )  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W )  <->  ( ( P 
.\/  ( F `  P ) )  ./\  W )  =  ( ( Q  .\/  ( F `
 Q ) ) 
./\  W ) ) )
3327, 32imbi12d 320 . . . . . 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 3219 . . . . 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 60 . . . 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 619 . . 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 475 . 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 1192 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   lecple 14719   joincjn 15700   meetcmee 15701   Atomscatm 35131   LHypclh 35851   LDilcldil 35967   LTrncltrn 35968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-ltrn 35972
This theorem is referenced by:  ltrncnv  36013  trlval2  36031  cdlemg14f  36522  cdlemg14g  36523
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