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Theorem trnfsetN 31014
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 2966 . 2  |-  ( K  e.  C  ->  K  e.  _V )
2 trnset.t . . 3  |-  T  =  ( Trn `  K
)
3 fveq2 5730 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 trnset.a . . . . . 6  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2488 . . . . 5  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
6 fveq2 5730 . . . . . . . 8  |-  ( k  =  K  ->  ( Dil `  k )  =  ( Dil `  K
) )
7 trnset.l . . . . . . . 8  |-  L  =  ( Dil `  K
)
86, 7syl6eqr 2488 . . . . . . 7  |-  ( k  =  K  ->  ( Dil `  k )  =  L )
98fveq1d 5732 . . . . . 6  |-  ( k  =  K  ->  (
( Dil `  k
) `  d )  =  ( L `  d ) )
10 fveq2 5730 . . . . . . . . 9  |-  ( k  =  K  ->  ( WAtoms `
 k )  =  ( WAtoms `  K )
)
11 trnset.w . . . . . . . . 9  |-  W  =  ( WAtoms `  K )
1210, 11syl6eqr 2488 . . . . . . . 8  |-  ( k  =  K  ->  ( WAtoms `
 k )  =  W )
1312fveq1d 5732 . . . . . . 7  |-  ( k  =  K  ->  (
( WAtoms `  k ) `  d )  =  ( W `  d ) )
14 fveq2 5730 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( + P `  k )  =  ( + P `  K ) )
15 trnset.p . . . . . . . . . . . 12  |-  .+  =  ( + P `  K
)
1614, 15syl6eqr 2488 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( + P `  k )  =  .+  )
1716oveqd 6100 . . . . . . . . . 10  |-  ( k  =  K  ->  (
q ( + P `  k ) ( f `
 q ) )  =  ( q  .+  ( f `  q
) ) )
18 fveq2 5730 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( _|_ P `  k )  =  ( _|_ P `  K ) )
19 trnset.o . . . . . . . . . . . 12  |-  ._|_  =  ( _|_ P `  K
)
2018, 19syl6eqr 2488 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( _|_ P `  k )  =  ._|_  )
2120fveq1d 5732 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( _|_ P `  k ) `  {
d } )  =  (  ._|_  `  { d } ) )
2217, 21ineq12d 3545 . . . . . . . . 9  |-  ( k  =  K  ->  (
( q ( + P `  k ) ( f `  q
) )  i^i  (
( _|_ P `  k ) `  {
d } ) )  =  ( ( q 
.+  ( f `  q ) )  i^i  (  ._|_  `  { d } ) ) )
2316oveqd 6100 . . . . . . . . . 10  |-  ( k  =  K  ->  (
r ( + P `  k ) ( f `
 r ) )  =  ( r  .+  ( f `  r
) ) )
2423, 21ineq12d 3545 . . . . . . . . 9  |-  ( k  =  K  ->  (
( r ( + P `  k ) ( f `  r
) )  i^i  (
( _|_ P `  k ) `  {
d } ) )  =  ( ( r 
.+  ( f `  r ) )  i^i  (  ._|_  `  { d } ) ) )
2522, 24eqeq12d 2452 . . . . . . . 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 2918 . . . . . . 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 2918 . . . . . 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 2953 . . . . 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 4289 . . . 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 30966 . . . 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 5744 . . . . . 6  |-  ( Atoms `  K )  e.  _V
324, 31eqeltri 2508 . . . . 5  |-  A  e. 
_V
3332mptex 5968 . . . 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 5808 . . 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 2482 . 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 set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711   _Vcvv 2958    i^i cin 3321   {csn 3816    e. cmpt 4268   ` cfv 5456  (class class class)co 6083   Atomscatm 30123   PSubSpcpsubsp 30355   + Pcpadd 30654   _|_ PcpolN 30761   WAtomscwpointsN 30845   PAutcpautN 30846   DilcdilN 30961   TrnctrnN 30962
This theorem is referenced by:  trnsetN  31015
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 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pr 4405
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-trnN 30966
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