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Theorem isltrn2N 36241
Description: The predicate "is a lattice translation". Version of isltrn 36240 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 36240 . 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 991 . . . . . 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 975 . . . . . . . . 9  |-  ( ( p  =/=  q  /\  -.  p  .<_  W  /\  -.  q  .<_  W )  <-> 
( p  =/=  q  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W ) ) )
12 3anrot 976 . . . . . . . . 9  |-  ( ( p  =/=  q  /\  -.  p  .<_  W  /\  -.  q  .<_  W )  <-> 
( -.  p  .<_  W  /\  -.  q  .<_  W  /\  p  =/=  q
) )
13 df-ne 2651 . . . . . . . . . 10  |-  ( p  =/=  q  <->  -.  p  =  q )
1413anbi1i 693 . . . . . . . . 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 323 . . . . . . 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 444 . . . . . . 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 5848 . . . . . . . . . 10  |-  ( p  =  q  ->  ( F `  p )  =  ( F `  q ) )
2119, 20oveq12d 6288 . . . . . . . . 9  |-  ( p  =  q  ->  (
p  .\/  ( F `  p ) )  =  ( q  .\/  ( F `  q )
) )
2221oveq1d 6285 . . . . . . . 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 2886 . . 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 692 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   lecple 14791   joincjn 15772   meetcmee 15773   Atomscatm 35385   LHypclh 36105   LDilcldil 36221   LTrncltrn 36222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-ltrn 36226
This theorem is referenced by: (None)
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