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Theorem trnfsetN 36277
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 3115 . 2  |-  ( K  e.  C  ->  K  e.  _V )
2 trnset.t . . 3  |-  T  =  ( Trn `  K
)
3 fveq2 5848 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 trnset.a . . . . . 6  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2513 . . . . 5  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
6 fveq2 5848 . . . . . . . 8  |-  ( k  =  K  ->  ( Dil `  k )  =  ( Dil `  K
) )
7 trnset.l . . . . . . . 8  |-  L  =  ( Dil `  K
)
86, 7syl6eqr 2513 . . . . . . 7  |-  ( k  =  K  ->  ( Dil `  k )  =  L )
98fveq1d 5850 . . . . . 6  |-  ( k  =  K  ->  (
( Dil `  k
) `  d )  =  ( L `  d ) )
10 fveq2 5848 . . . . . . . . 9  |-  ( k  =  K  ->  ( WAtoms `
 k )  =  ( WAtoms `  K )
)
11 trnset.w . . . . . . . . 9  |-  W  =  ( WAtoms `  K )
1210, 11syl6eqr 2513 . . . . . . . 8  |-  ( k  =  K  ->  ( WAtoms `
 k )  =  W )
1312fveq1d 5850 . . . . . . 7  |-  ( k  =  K  ->  (
( WAtoms `  k ) `  d )  =  ( W `  d ) )
14 fveq2 5848 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( +P `  k )  =  ( +P `  K ) )
15 trnset.p . . . . . . . . . . . 12  |-  .+  =  ( +P `  K
)
1614, 15syl6eqr 2513 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( +P `  k )  =  .+  )
1716oveqd 6287 . . . . . . . . . 10  |-  ( k  =  K  ->  (
q ( +P `  k ) ( f `
 q ) )  =  ( q  .+  ( f `  q
) ) )
18 fveq2 5848 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( _|_P `  k )  =  ( _|_P `  K ) )
19 trnset.o . . . . . . . . . . . 12  |-  ._|_  =  ( _|_P `  K
)
2018, 19syl6eqr 2513 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( _|_P `  k )  =  ._|_  )
2120fveq1d 5850 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( _|_P `  k ) `  {
d } )  =  (  ._|_  `  { d } ) )
2217, 21ineq12d 3687 . . . . . . . . 9  |-  ( k  =  K  ->  (
( q ( +P `  k ) ( f `  q
) )  i^i  (
( _|_P `  k ) `  {
d } ) )  =  ( ( q 
.+  ( f `  q ) )  i^i  (  ._|_  `  { d } ) ) )
2316oveqd 6287 . . . . . . . . . 10  |-  ( k  =  K  ->  (
r ( +P `  k ) ( f `
 r ) )  =  ( r  .+  ( f `  r
) ) )
2423, 21ineq12d 3687 . . . . . . . . 9  |-  ( k  =  K  ->  (
( r ( +P `  k ) ( f `  r
) )  i^i  (
( _|_P `  k ) `  {
d } ) )  =  ( ( r 
.+  ( f `  r ) )  i^i  (  ._|_  `  { d } ) ) )
2522, 24eqeq12d 2476 . . . . . . . 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 3065 . . . . . . 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 3065 . . . . . 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 3101 . . . . 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 4517 . . . 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 36228 . . . 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 5858 . . . . . 6  |-  ( Atoms `  K )  e.  _V
324, 31eqeltri 2538 . . . . 5  |-  A  e. 
_V
3332mptex 6118 . . . 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 5931 . . 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 2507 . 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 16 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 setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808   _Vcvv 3106    i^i cin 3460   {csn 4016    |-> cmpt 4497   ` cfv 5570  (class class class)co 6270   Atomscatm 35385   PSubSpcpsubsp 35617   +Pcpadd 35916   _|_PcpolN 36023   WAtomscwpointsN 36107   PAutcpautN 36108   DilcdilN 36223   TrnctrnN 36224
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-trnN 36228
This theorem is referenced by:  trnsetN  36278
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