Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isltrn2N Structured version   Unicode version

Theorem isltrn2N 30919
Description: The predicate "is a lattice translation". Version of isltrn 30918 that considers only different  p and  q. TODO: Can this eliminate some separate proofs for the 
p  =  q case? (Contributed by NM, 22-Apr-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
ltrnset.l  |-  .<_  =  ( le `  K )
ltrnset.j  |-  .\/  =  ( join `  K )
ltrnset.m  |-  ./\  =  ( meet `  K )
ltrnset.a  |-  A  =  ( Atoms `  K )
ltrnset.h  |-  H  =  ( LHyp `  K
)
ltrnset.d  |-  D  =  ( ( LDil `  K
) `  W )
ltrnset.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
isltrn2N  |-  ( ( K  e.  B  /\  W  e.  H )  ->  ( F  e.  T  <->  ( F  e.  D  /\  A. p  e.  A  A. q  e.  A  (
( -.  p  .<_  W  /\  -.  q  .<_  W  /\  p  =/=  q
)  ->  ( (
p  .\/  ( F `  p ) )  ./\  W )  =  ( ( q  .\/  ( F `
 q ) ) 
./\  W ) ) ) ) )
Distinct variable groups:    q, p, A    K, p, q    W, p, q    F, p, q
Allowed substitution hints:    B( q, p)    D( q, p)    T( q, p)    H( q, p)    .\/ ( q, p)   
.<_ ( q, p)    ./\ ( q, p)

Proof of Theorem isltrn2N
StepHypRef Expression
1 ltrnset.l . . 3  |-  .<_  =  ( le `  K )
2 ltrnset.j . . 3  |-  .\/  =  ( join `  K )
3 ltrnset.m . . 3  |-  ./\  =  ( meet `  K )
4 ltrnset.a . . 3  |-  A  =  ( Atoms `  K )
5 ltrnset.h . . 3  |-  H  =  ( LHyp `  K
)
6 ltrnset.d . . 3  |-  D  =  ( ( LDil `  K
) `  W )
7 ltrnset.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
81, 2, 3, 4, 5, 6, 7isltrn 30918 . 2  |-  ( ( K  e.  B  /\  W  e.  H )  ->  ( F  e.  T  <->  ( F  e.  D  /\  A. p  e.  A  A. q  e.  A  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  (
( p  .\/  ( F `  p )
)  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W ) ) ) ) )
9 3simpa 955 . . . . . 6  |-  ( ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  p  =/=  q )  -> 
( -.  p  .<_  W  /\  -.  q  .<_  W ) )
109imim1i 57 . . . . 5  |-  ( ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  (
( p  .\/  ( F `  p )
)  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W ) )  ->  (
( -.  p  .<_  W  /\  -.  q  .<_  W  /\  p  =/=  q
)  ->  ( (
p  .\/  ( F `  p ) )  ./\  W )  =  ( ( q  .\/  ( F `
 q ) ) 
./\  W ) ) )
11 3anass 941 . . . . . . . . 9  |-  ( ( p  =/=  q  /\  -.  p  .<_  W  /\  -.  q  .<_  W )  <-> 
( p  =/=  q  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) ) )
12 3anrot 942 . . . . . . . . 9  |-  ( ( p  =/=  q  /\  -.  p  .<_  W  /\  -.  q  .<_  W )  <-> 
( -.  p  .<_  W  /\  -.  q  .<_  W  /\  p  =/=  q
) )
13 df-ne 2603 . . . . . . . . . 10  |-  ( p  =/=  q  <->  -.  p  =  q )
1413anbi1i 678 . . . . . . . . 9  |-  ( ( p  =/=  q  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) )  <->  ( -.  p  =  q  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) ) )
1511, 12, 143bitr3i 268 . . . . . . . 8  |-  ( ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  p  =/=  q )  <->  ( -.  p  =  q  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) ) )
1615imbi1i 317 . . . . . . 7  |-  ( ( ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  p  =/=  q
)  ->  ( (
p  .\/  ( F `  p ) )  ./\  W )  =  ( ( q  .\/  ( F `
 q ) ) 
./\  W ) )  <-> 
( ( -.  p  =  q  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) )  ->  ( (
p  .\/  ( F `  p ) )  ./\  W )  =  ( ( q  .\/  ( F `
 q ) ) 
./\  W ) ) )
17 impexp 435 . . . . . . 7  |-  ( ( ( -.  p  =  q  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) )  ->  ( (
p  .\/  ( F `  p ) )  ./\  W )  =  ( ( q  .\/  ( F `
 q ) ) 
./\  W ) )  <-> 
( -.  p  =  q  ->  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  ( ( p 
.\/  ( F `  p ) )  ./\  W )  =  ( ( q  .\/  ( F `
 q ) ) 
./\  W ) ) ) )
1816, 17bitri 242 . . . . . 6  |-  ( ( ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  p  =/=  q
)  ->  ( (
p  .\/  ( F `  p ) )  ./\  W )  =  ( ( q  .\/  ( F `
 q ) ) 
./\  W ) )  <-> 
( -.  p  =  q  ->  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  ( ( p 
.\/  ( F `  p ) )  ./\  W )  =  ( ( q  .\/  ( F `
 q ) ) 
./\  W ) ) ) )
19 id 21 . . . . . . . . . 10  |-  ( p  =  q  ->  p  =  q )
20 fveq2 5730 . . . . . . . . . 10  |-  ( p  =  q  ->  ( F `  p )  =  ( F `  q ) )
2119, 20oveq12d 6101 . . . . . . . . 9  |-  ( p  =  q  ->  (
p  .\/  ( F `  p ) )  =  ( q  .\/  ( F `  q )
) )
2221oveq1d 6098 . . . . . . . 8  |-  ( p  =  q  ->  (
( p  .\/  ( F `  p )
)  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W ) )
2322a1d 24 . . . . . . 7  |-  ( p  =  q  ->  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  (
( p  .\/  ( F `  p )
)  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W ) ) )
24 pm2.61 166 . . . . . . 7  |-  ( ( p  =  q  -> 
( ( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  (
( p  .\/  ( F `  p )
)  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W ) ) )  -> 
( ( -.  p  =  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 ) ) ) )
2523, 24ax-mp 8 . . . . . 6  |-  ( ( -.  p  =  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 ) ) )
2618, 25sylbi 189 . . . . 5  |-  ( ( ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  p  =/=  q
)  ->  ( (
p  .\/  ( F `  p ) )  ./\  W )  =  ( ( q  .\/  ( F `
 q ) ) 
./\  W ) )  ->  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  ( ( p 
.\/  ( F `  p ) )  ./\  W )  =  ( ( q  .\/  ( F `
 q ) ) 
./\  W ) ) )
2710, 26impbii 182 . . . 4  |-  ( ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  (
( p  .\/  ( F `  p )
)  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W ) )  <->  ( ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  p  =/=  q )  -> 
( ( p  .\/  ( F `  p ) )  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W ) ) )
28272ralbii 2733 . . 3  |-  ( 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  =/=  q )  -> 
( ( p  .\/  ( F `  p ) )  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W ) ) )
2928anbi2i 677 . 2  |-  ( ( F  e.  D  /\  A. p  e.  A  A. q  e.  A  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  (
( p  .\/  ( F `  p )
)  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W ) ) )  <->  ( F  e.  D  /\  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  p  =/=  q )  -> 
( ( p  .\/  ( F `  p ) )  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W ) ) ) )
308, 29syl6bb 254 1  |-  ( ( K  e.  B  /\  W  e.  H )  ->  ( F  e.  T  <->  ( F  e.  D  /\  A. p  e.  A  A. q  e.  A  (
( -.  p  .<_  W  /\  -.  q  .<_  W  /\  p  =/=  q
)  ->  ( (
p  .\/  ( F `  p ) )  ./\  W )  =  ( ( q  .\/  ( F `
 q ) ) 
./\  W ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   lecple 13538   joincjn 14403   meetcmee 14404   Atomscatm 30063   LHypclh 30783   LDilcldil 30899   LTrncltrn 30900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-ltrn 30904
  Copyright terms: Public domain W3C validator