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Theorem istrnN 30346
Description: The predicate "is a translation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
trnset.a  |-  A  =  ( Atoms `  K )
trnset.s  |-  S  =  ( PSubSp `  K )
trnset.p  |-  .+  =  ( + P `  K
)
trnset.o  |-  ._|_  =  ( _|_ P `  K
)
trnset.w  |-  W  =  ( WAtoms `  K )
trnset.m  |-  M  =  ( PAut `  K
)
trnset.l  |-  L  =  ( Dil `  K
)
trnset.t  |-  T  =  ( Trn `  K
)
Assertion
Ref Expression
istrnN  |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( F  e.  ( T `  D )  <-> 
( F  e.  ( L `  D )  /\  A. q  e.  ( W `  D
) A. r  e.  ( W `  D
) ( ( q 
.+  ( F `  q ) )  i^i  (  ._|_  `  { D } ) )  =  ( ( r  .+  ( F `  r ) )  i^i  (  ._|_  `  { D } ) ) ) ) )
Distinct variable groups:    r, q, K    W, q, r    D, q, r    F, q, r
Allowed substitution hints:    A( r, q)    B( r, q)    .+ ( r, q)    S( r, q)    T( r, q)    L( r, q)    M( r, q)    ._|_ ( r, q)

Proof of Theorem istrnN
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 trnset.a . . . 4  |-  A  =  ( Atoms `  K )
2 trnset.s . . . 4  |-  S  =  ( PSubSp `  K )
3 trnset.p . . . 4  |-  .+  =  ( + P `  K
)
4 trnset.o . . . 4  |-  ._|_  =  ( _|_ P `  K
)
5 trnset.w . . . 4  |-  W  =  ( WAtoms `  K )
6 trnset.m . . . 4  |-  M  =  ( PAut `  K
)
7 trnset.l . . . 4  |-  L  =  ( Dil `  K
)
8 trnset.t . . . 4  |-  T  =  ( Trn `  K
)
91, 2, 3, 4, 5, 6, 7, 8trnsetN 30345 . . 3  |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( T `  D
)  =  { f  e.  ( L `  D )  |  A. q  e.  ( W `  D ) A. r  e.  ( W `  D
) ( ( q 
.+  ( f `  q ) )  i^i  (  ._|_  `  { D } ) )  =  ( ( r  .+  ( f `  r
) )  i^i  (  ._|_  `  { D }
) ) } )
109eleq2d 2350 . 2  |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( F  e.  ( T `  D )  <-> 
F  e.  { f  e.  ( L `  D )  |  A. q  e.  ( W `  D ) A. r  e.  ( W `  D
) ( ( q 
.+  ( f `  q ) )  i^i  (  ._|_  `  { D } ) )  =  ( ( r  .+  ( f `  r
) )  i^i  (  ._|_  `  { D }
) ) } ) )
11 fveq1 5524 . . . . . . 7  |-  ( f  =  F  ->  (
f `  q )  =  ( F `  q ) )
1211oveq2d 5874 . . . . . 6  |-  ( f  =  F  ->  (
q  .+  ( f `  q ) )  =  ( q  .+  ( F `  q )
) )
1312ineq1d 3369 . . . . 5  |-  ( f  =  F  ->  (
( q  .+  (
f `  q )
)  i^i  (  ._|_  `  { D } ) )  =  ( ( q  .+  ( F `
 q ) )  i^i  (  ._|_  `  { D } ) ) )
14 fveq1 5524 . . . . . . 7  |-  ( f  =  F  ->  (
f `  r )  =  ( F `  r ) )
1514oveq2d 5874 . . . . . 6  |-  ( f  =  F  ->  (
r  .+  ( f `  r ) )  =  ( r  .+  ( F `  r )
) )
1615ineq1d 3369 . . . . 5  |-  ( f  =  F  ->  (
( r  .+  (
f `  r )
)  i^i  (  ._|_  `  { D } ) )  =  ( ( r  .+  ( F `
 r ) )  i^i  (  ._|_  `  { D } ) ) )
1713, 16eqeq12d 2297 . . . 4  |-  ( f  =  F  ->  (
( ( q  .+  ( f `  q
) )  i^i  (  ._|_  `  { D }
) )  =  ( ( r  .+  (
f `  r )
)  i^i  (  ._|_  `  { D } ) )  <->  ( ( q 
.+  ( F `  q ) )  i^i  (  ._|_  `  { D } ) )  =  ( ( r  .+  ( F `  r ) )  i^i  (  ._|_  `  { D } ) ) ) )
18172ralbidv 2585 . . 3  |-  ( f  =  F  ->  ( A. q  e.  ( W `  D ) A. r  e.  ( W `  D )
( ( q  .+  ( f `  q
) )  i^i  (  ._|_  `  { D }
) )  =  ( ( r  .+  (
f `  r )
)  i^i  (  ._|_  `  { D } ) )  <->  A. q  e.  ( W `  D ) A. r  e.  ( W `  D ) ( ( q  .+  ( F `  q ) )  i^i  (  ._|_  `  { D } ) )  =  ( ( r  .+  ( F `
 r ) )  i^i  (  ._|_  `  { D } ) ) ) )
1918elrab 2923 . 2  |-  ( F  e.  { f  e.  ( L `  D
)  |  A. q  e.  ( W `  D
) A. r  e.  ( W `  D
) ( ( q 
.+  ( f `  q ) )  i^i  (  ._|_  `  { D } ) )  =  ( ( r  .+  ( f `  r
) )  i^i  (  ._|_  `  { D }
) ) }  <->  ( F  e.  ( L `  D
)  /\  A. q  e.  ( W `  D
) A. r  e.  ( W `  D
) ( ( q 
.+  ( F `  q ) )  i^i  (  ._|_  `  { D } ) )  =  ( ( r  .+  ( F `  r ) )  i^i  (  ._|_  `  { D } ) ) ) )
2010, 19syl6bb 252 1  |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( F  e.  ( T `  D )  <-> 
( F  e.  ( L `  D )  /\  A. q  e.  ( W `  D
) A. r  e.  ( W `  D
) ( ( q 
.+  ( F `  q ) )  i^i  (  ._|_  `  { D } ) )  =  ( ( r  .+  ( F `  r ) )  i^i  (  ._|_  `  { D } ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547    i^i cin 3151   {csn 3640   ` cfv 5255  (class class class)co 5858   Atomscatm 29453   PSubSpcpsubsp 29685   + Pcpadd 29984   _|_ PcpolN 30091   WAtomscwpointsN 30175   PAutcpautN 30176   DilcdilN 30291   TrnctrnN 30292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-trnN 30296
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