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Theorem trnfsetN 30344
Description: The mapping from fiducial atom to set of translations. (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
trnfsetN  |-  ( K  e.  C  ->  T  =  ( d  e.  A  |->  { 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:    A, d    f, d, q, r, K   
f, L    W, q,
r
Allowed substitution hints:    A( f, r, q)    C( f, r, q, d)    .+ ( f, r, q, d)    S( f, r, q, d)    T( f, r, q, d)    L( r, q, d)    M( f, r, q, d)    ._|_ ( f, r, q, d)    W( f, d)

Proof of Theorem trnfsetN
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2796 . 2  |-  ( K  e.  C  ->  K  e.  _V )
2 trnset.t . . 3  |-  T  =  ( Trn `  K
)
3 fveq2 5525 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 trnset.a . . . . . 6  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2333 . . . . 5  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
6 fveq2 5525 . . . . . . . 8  |-  ( k  =  K  ->  ( Dil `  k )  =  ( Dil `  K
) )
7 trnset.l . . . . . . . 8  |-  L  =  ( Dil `  K
)
86, 7syl6eqr 2333 . . . . . . 7  |-  ( k  =  K  ->  ( Dil `  k )  =  L )
98fveq1d 5527 . . . . . 6  |-  ( k  =  K  ->  (
( Dil `  k
) `  d )  =  ( L `  d ) )
10 fveq2 5525 . . . . . . . . 9  |-  ( k  =  K  ->  ( WAtoms `
 k )  =  ( WAtoms `  K )
)
11 trnset.w . . . . . . . . 9  |-  W  =  ( WAtoms `  K )
1210, 11syl6eqr 2333 . . . . . . . 8  |-  ( k  =  K  ->  ( WAtoms `
 k )  =  W )
1312fveq1d 5527 . . . . . . 7  |-  ( k  =  K  ->  (
( WAtoms `  k ) `  d )  =  ( W `  d ) )
14 fveq2 5525 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( + P `  k )  =  ( + P `  K ) )
15 trnset.p . . . . . . . . . . . 12  |-  .+  =  ( + P `  K
)
1614, 15syl6eqr 2333 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( + P `  k )  =  .+  )
1716oveqd 5875 . . . . . . . . . 10  |-  ( k  =  K  ->  (
q ( + P `  k ) ( f `
 q ) )  =  ( q  .+  ( f `  q
) ) )
18 fveq2 5525 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( _|_ P `  k )  =  ( _|_ P `  K ) )
19 trnset.o . . . . . . . . . . . 12  |-  ._|_  =  ( _|_ P `  K
)
2018, 19syl6eqr 2333 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( _|_ P `  k )  =  ._|_  )
2120fveq1d 5527 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( _|_ P `  k ) `  {
d } )  =  (  ._|_  `  { d } ) )
2217, 21ineq12d 3371 . . . . . . . . 9  |-  ( k  =  K  ->  (
( q ( + P `  k ) ( f `  q
) )  i^i  (
( _|_ P `  k ) `  {
d } ) )  =  ( ( q 
.+  ( f `  q ) )  i^i  (  ._|_  `  { d } ) ) )
2316oveqd 5875 . . . . . . . . . 10  |-  ( k  =  K  ->  (
r ( + P `  k ) ( f `
 r ) )  =  ( r  .+  ( f `  r
) ) )
2423, 21ineq12d 3371 . . . . . . . . 9  |-  ( k  =  K  ->  (
( r ( + P `  k ) ( f `  r
) )  i^i  (
( _|_ P `  k ) `  {
d } ) )  =  ( ( r 
.+  ( f `  r ) )  i^i  (  ._|_  `  { d } ) ) )
2522, 24eqeq12d 2297 . . . . . . . 8  |-  ( k  =  K  ->  (
( ( q ( + P `  k
) ( f `  q ) )  i^i  ( ( _|_ P `  k ) `  {
d } ) )  =  ( ( r ( + P `  k ) ( f `
 r ) )  i^i  ( ( _|_
P `  k ) `  { d } ) )  <->  ( ( q 
.+  ( f `  q ) )  i^i  (  ._|_  `  { d } ) )  =  ( ( r  .+  ( f `  r
) )  i^i  (  ._|_  `  { d } ) ) ) )
2613, 25raleqbidv 2748 . . . . . . 7  |-  ( k  =  K  ->  ( A. r  e.  (
( WAtoms `  k ) `  d ) ( ( q ( + P `  k ) ( f `
 q ) )  i^i  ( ( _|_
P `  k ) `  { d } ) )  =  ( ( r ( + P `  k ) ( f `
 r ) )  i^i  ( ( _|_
P `  k ) `  { d } ) )  <->  A. r  e.  ( W `  d ) ( ( q  .+  ( f `  q
) )  i^i  (  ._|_  `  { d } ) )  =  ( ( r  .+  (
f `  r )
)  i^i  (  ._|_  `  { d } ) ) ) )
2713, 26raleqbidv 2748 . . . . . 6  |-  ( k  =  K  ->  ( A. q  e.  (
( WAtoms `  k ) `  d ) A. r  e.  ( ( WAtoms `  k
) `  d )
( ( q ( + P `  k
) ( f `  q ) )  i^i  ( ( _|_ P `  k ) `  {
d } ) )  =  ( ( r ( + P `  k ) ( f `
 r ) )  i^i  ( ( _|_
P `  k ) `  { d } ) )  <->  A. q  e.  ( W `  d ) A. r  e.  ( W `  d ) ( ( q  .+  ( f `  q
) )  i^i  (  ._|_  `  { d } ) )  =  ( ( r  .+  (
f `  r )
)  i^i  (  ._|_  `  { d } ) ) ) )
289, 27rabeqbidv 2783 . . . . 5  |-  ( k  =  K  ->  { f  e.  ( ( Dil `  k ) `  d
)  |  A. q  e.  ( ( WAtoms `  k
) `  d ) A. r  e.  (
( WAtoms `  k ) `  d ) ( ( q ( + P `  k ) ( f `
 q ) )  i^i  ( ( _|_
P `  k ) `  { d } ) )  =  ( ( r ( + P `  k ) ( f `
 r ) )  i^i  ( ( _|_
P `  k ) `  { 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 } ) ) } )
295, 28mpteq12dv 4098 . . . 4  |-  ( k  =  K  ->  (
d  e.  ( Atoms `  k )  |->  { f  e.  ( ( Dil `  k ) `  d
)  |  A. q  e.  ( ( WAtoms `  k
) `  d ) A. r  e.  (
( WAtoms `  k ) `  d ) ( ( q ( + P `  k ) ( f `
 q ) )  i^i  ( ( _|_
P `  k ) `  { d } ) )  =  ( ( r ( + P `  k ) ( f `
 r ) )  i^i  ( ( _|_
P `  k ) `  { d } ) ) } )  =  ( d  e.  A  |->  { 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 } ) ) } ) )
30 df-trnN 30296 . . . 4  |-  Trn  =  ( k  e.  _V  |->  ( d  e.  (
Atoms `  k )  |->  { f  e.  ( ( Dil `  k ) `
 d )  | 
A. q  e.  ( ( WAtoms `  k ) `  d ) A. r  e.  ( ( WAtoms `  k
) `  d )
( ( q ( + P `  k
) ( f `  q ) )  i^i  ( ( _|_ P `  k ) `  {
d } ) )  =  ( ( r ( + P `  k ) ( f `
 r ) )  i^i  ( ( _|_
P `  k ) `  { d } ) ) } ) )
31 fvex 5539 . . . . . 6  |-  ( Atoms `  K )  e.  _V
324, 31eqeltri 2353 . . . . 5  |-  A  e. 
_V
3332mptex 5746 . . . 4  |-  ( d  e.  A  |->  { 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 } ) ) } )  e.  _V
3429, 30, 33fvmpt 5602 . . 3  |-  ( K  e.  _V  ->  ( Trn `  K )  =  ( d  e.  A  |->  { 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 } ) ) } ) )
352, 34syl5eq 2327 . 2  |-  ( K  e.  _V  ->  T  =  ( d  e.  A  |->  { 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 } ) ) } ) )
361, 35syl 15 1  |-  ( K  e.  C  ->  T  =  ( d  e.  A  |->  { 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    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    i^i cin 3151   {csn 3640    e. cmpt 4077   ` 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 is referenced by:  trnsetN  30345
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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