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

Theorem istrnN 30955
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 30954 . . 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 2504 . 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 5728 . . . . . . 7  |-  ( f  =  F  ->  (
f `  q )  =  ( F `  q ) )
1211oveq2d 6098 . . . . . 6  |-  ( f  =  F  ->  (
q  .+  ( f `  q ) )  =  ( q  .+  ( F `  q )
) )
1312ineq1d 3542 . . . . 5  |-  ( f  =  F  ->  (
( q  .+  (
f `  q )
)  i^i  (  ._|_  `  { D } ) )  =  ( ( q  .+  ( F `
 q ) )  i^i  (  ._|_  `  { D } ) ) )
14 fveq1 5728 . . . . . . 7  |-  ( f  =  F  ->  (
f `  r )  =  ( F `  r ) )
1514oveq2d 6098 . . . . . 6  |-  ( f  =  F  ->  (
r  .+  ( f `  r ) )  =  ( r  .+  ( F `  r )
) )
1615ineq1d 3542 . . . . 5  |-  ( f  =  F  ->  (
( r  .+  (
f `  r )
)  i^i  (  ._|_  `  { D } ) )  =  ( ( r  .+  ( F `
 r ) )  i^i  (  ._|_  `  { D } ) ) )
1713, 16eqeq12d 2451 . . . 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 2748 . . 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 3093 . 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 254 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2706   {crab 2710    i^i cin 3320   {csn 3815   ` cfv 5455  (class class class)co 6082   Atomscatm 30062   PSubSpcpsubsp 30294   + Pcpadd 30593   _|_ PcpolN 30700   WAtomscwpointsN 30784   PAutcpautN 30785   DilcdilN 30900   TrnctrnN 30901
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 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pr 4404
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-trnN 30905
  Copyright terms: Public domain W3C validator