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Theorem trnfsetN 33895
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 3002 . 2  |-  ( K  e.  C  ->  K  e.  _V )
2 trnset.t . . 3  |-  T  =  ( Trn `  K
)
3 fveq2 5712 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 trnset.a . . . . . 6  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2493 . . . . 5  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
6 fveq2 5712 . . . . . . . 8  |-  ( k  =  K  ->  ( Dil `  k )  =  ( Dil `  K
) )
7 trnset.l . . . . . . . 8  |-  L  =  ( Dil `  K
)
86, 7syl6eqr 2493 . . . . . . 7  |-  ( k  =  K  ->  ( Dil `  k )  =  L )
98fveq1d 5714 . . . . . 6  |-  ( k  =  K  ->  (
( Dil `  k
) `  d )  =  ( L `  d ) )
10 fveq2 5712 . . . . . . . . 9  |-  ( k  =  K  ->  ( WAtoms `
 k )  =  ( WAtoms `  K )
)
11 trnset.w . . . . . . . . 9  |-  W  =  ( WAtoms `  K )
1210, 11syl6eqr 2493 . . . . . . . 8  |-  ( k  =  K  ->  ( WAtoms `
 k )  =  W )
1312fveq1d 5714 . . . . . . 7  |-  ( k  =  K  ->  (
( WAtoms `  k ) `  d )  =  ( W `  d ) )
14 fveq2 5712 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( +P `  k )  =  ( +P `  K ) )
15 trnset.p . . . . . . . . . . . 12  |-  .+  =  ( +P `  K
)
1614, 15syl6eqr 2493 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( +P `  k )  =  .+  )
1716oveqd 6129 . . . . . . . . . 10  |-  ( k  =  K  ->  (
q ( +P `  k ) ( f `
 q ) )  =  ( q  .+  ( f `  q
) ) )
18 fveq2 5712 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( _|_P `  k )  =  ( _|_P `  K ) )
19 trnset.o . . . . . . . . . . . 12  |-  ._|_  =  ( _|_P `  K
)
2018, 19syl6eqr 2493 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( _|_P `  k )  =  ._|_  )
2120fveq1d 5714 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( _|_P `  k ) `  {
d } )  =  (  ._|_  `  { d } ) )
2217, 21ineq12d 3574 . . . . . . . . 9  |-  ( k  =  K  ->  (
( q ( +P `  k ) ( f `  q
) )  i^i  (
( _|_P `  k ) `  {
d } ) )  =  ( ( q 
.+  ( f `  q ) )  i^i  (  ._|_  `  { d } ) ) )
2316oveqd 6129 . . . . . . . . . 10  |-  ( k  =  K  ->  (
r ( +P `  k ) ( f `
 r ) )  =  ( r  .+  ( f `  r
) ) )
2423, 21ineq12d 3574 . . . . . . . . 9  |-  ( k  =  K  ->  (
( r ( +P `  k ) ( f `  r
) )  i^i  (
( _|_P `  k ) `  {
d } ) )  =  ( ( r 
.+  ( f `  r ) )  i^i  (  ._|_  `  { d } ) ) )
2522, 24eqeq12d 2457 . . . . . . . 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 2952 . . . . . . 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 2952 . . . . . 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 2988 . . . . 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 4391 . . . 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 33847 . . . 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 5722 . . . . . 6  |-  ( Atoms `  K )  e.  _V
324, 31eqeltri 2513 . . . . 5  |-  A  e. 
_V
3332mptex 5969 . . . 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 5795 . . 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 2487 . 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 1369    e. wcel 1756   A.wral 2736   {crab 2740   _Vcvv 2993    i^i cin 3348   {csn 3898    e. cmpt 4371   ` cfv 5439  (class class class)co 6112   Atomscatm 33004   PSubSpcpsubsp 33236   +Pcpadd 33535   _|_PcpolN 33642   WAtomscwpointsN 33726   PAutcpautN 33727   DilcdilN 33842   TrnctrnN 33843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-trnN 33847
This theorem is referenced by:  trnsetN  33896
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