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Theorem isltrn2N 30602
Description: The predicate "is a lattice translation". Version of isltrn 30601 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 30601 . 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 954 . . . . . 6  |-  ( ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  p  =/=  q )  -> 
( -.  p  .<_  W  /\  -.  q  .<_  W ) )
109imim1i 56 . . . . 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 940 . . . . . . . . 9  |-  ( ( p  =/=  q  /\  -.  p  .<_  W  /\  -.  q  .<_  W )  <-> 
( p  =/=  q  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) ) )
12 3anrot 941 . . . . . . . . 9  |-  ( ( p  =/=  q  /\  -.  p  .<_  W  /\  -.  q  .<_  W )  <-> 
( -.  p  .<_  W  /\  -.  q  .<_  W  /\  p  =/=  q
) )
13 df-ne 2569 . . . . . . . . . 10  |-  ( p  =/=  q  <->  -.  p  =  q )
1413anbi1i 677 . . . . . . . . 9  |-  ( ( p  =/=  q  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) )  <->  ( -.  p  =  q  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) ) )
1511, 12, 143bitr3i 267 . . . . . . . 8  |-  ( ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  p  =/=  q )  <->  ( -.  p  =  q  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) ) )
1615imbi1i 316 . . . . . . 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 434 . . . . . . 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 241 . . . . . 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 20 . . . . . . . . . 10  |-  ( p  =  q  ->  p  =  q )
20 fveq2 5687 . . . . . . . . . 10  |-  ( p  =  q  ->  ( F `  p )  =  ( F `  q ) )
2119, 20oveq12d 6058 . . . . . . . . 9  |-  ( p  =  q  ->  (
p  .\/  ( F `  p ) )  =  ( q  .\/  ( F `  q )
) )
2221oveq1d 6055 . . . . . . . 8  |-  ( p  =  q  ->  (
( p  .\/  ( F `  p )
)  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W ) )
2322a1d 23 . . . . . . 7  |-  ( p  =  q  ->  (
( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  (
( p  .\/  ( F `  p )
)  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W ) ) )
24 pm2.61 165 . . . . . . 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 188 . . . . 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 181 . . . 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 2692 . . 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 676 . 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 253 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   LHypclh 30466   LDilcldil 30582   LTrncltrn 30583
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-ltrn 30587
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