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Theorem isltrn2N 33769
Description: The predicate "is a lattice translation". Version of isltrn 33768 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 33768 . 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 985 . . . . . 6  |-  ( ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  p  =/=  q )  -> 
( -.  p  .<_  W  /\  -.  q  .<_  W ) )
109imim1i 58 . . . . 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 969 . . . . . . . . 9  |-  ( ( p  =/=  q  /\  -.  p  .<_  W  /\  -.  q  .<_  W )  <-> 
( p  =/=  q  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) ) )
12 3anrot 970 . . . . . . . . 9  |-  ( ( p  =/=  q  /\  -.  p  .<_  W  /\  -.  q  .<_  W )  <-> 
( -.  p  .<_  W  /\  -.  q  .<_  W  /\  p  =/=  q
) )
13 df-ne 2613 . . . . . . . . . 10  |-  ( p  =/=  q  <->  -.  p  =  q )
1413anbi1i 695 . . . . . . . . 9  |-  ( ( p  =/=  q  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) )  <->  ( -.  p  =  q  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) ) )
1511, 12, 143bitr3i 275 . . . . . . . 8  |-  ( ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  p  =/=  q )  <->  ( -.  p  =  q  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) ) )
1615imbi1i 325 . . . . . . 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 446 . . . . . . 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 249 . . . . . 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 22 . . . . . . . . . 10  |-  ( p  =  q  ->  p  =  q )
20 fveq2 5696 . . . . . . . . . 10  |-  ( p  =  q  ->  ( F `  p )  =  ( F `  q ) )
2119, 20oveq12d 6114 . . . . . . . . 9  |-  ( p  =  q  ->  (
p  .\/  ( F `  p ) )  =  ( q  .\/  ( F `  q )
) )
2221oveq1d 6111 . . . . . . . 8  |-  ( p  =  q  ->  (
( p  .\/  ( F `  p )
)  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W ) )
2322a1d 25 . . . . . . 7  |-  ( p  =  q  ->  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  (
( p  .\/  ( F `  p )
)  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W ) ) )
24 pm2.61 171 . . . . . . 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 5 . . . . . 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 195 . . . . 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 188 . . . 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 2746 . . 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 694 . 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 261 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   lecple 14250   joincjn 15119   meetcmee 15120   Atomscatm 32913   LHypclh 33633   LDilcldil 33749   LTrncltrn 33750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-ltrn 33754
This theorem is referenced by: (None)
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