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Theorem trnfsetN 30966
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 2809 . 2  |-  ( K  e.  C  ->  K  e.  _V )
2 trnset.t . . 3  |-  T  =  ( Trn `  K
)
3 fveq2 5541 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 trnset.a . . . . . 6  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2346 . . . . 5  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
6 fveq2 5541 . . . . . . . 8  |-  ( k  =  K  ->  ( Dil `  k )  =  ( Dil `  K
) )
7 trnset.l . . . . . . . 8  |-  L  =  ( Dil `  K
)
86, 7syl6eqr 2346 . . . . . . 7  |-  ( k  =  K  ->  ( Dil `  k )  =  L )
98fveq1d 5543 . . . . . 6  |-  ( k  =  K  ->  (
( Dil `  k
) `  d )  =  ( L `  d ) )
10 fveq2 5541 . . . . . . . . 9  |-  ( k  =  K  ->  ( WAtoms `
 k )  =  ( WAtoms `  K )
)
11 trnset.w . . . . . . . . 9  |-  W  =  ( WAtoms `  K )
1210, 11syl6eqr 2346 . . . . . . . 8  |-  ( k  =  K  ->  ( WAtoms `
 k )  =  W )
1312fveq1d 5543 . . . . . . 7  |-  ( k  =  K  ->  (
( WAtoms `  k ) `  d )  =  ( W `  d ) )
14 fveq2 5541 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( + P `  k )  =  ( + P `  K ) )
15 trnset.p . . . . . . . . . . . 12  |-  .+  =  ( + P `  K
)
1614, 15syl6eqr 2346 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( + P `  k )  =  .+  )
1716oveqd 5891 . . . . . . . . . 10  |-  ( k  =  K  ->  (
q ( + P `  k ) ( f `
 q ) )  =  ( q  .+  ( f `  q
) ) )
18 fveq2 5541 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( _|_ P `  k )  =  ( _|_ P `  K ) )
19 trnset.o . . . . . . . . . . . 12  |-  ._|_  =  ( _|_ P `  K
)
2018, 19syl6eqr 2346 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( _|_ P `  k )  =  ._|_  )
2120fveq1d 5543 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( _|_ P `  k ) `  {
d } )  =  (  ._|_  `  { d } ) )
2217, 21ineq12d 3384 . . . . . . . . 9  |-  ( k  =  K  ->  (
( q ( + P `  k ) ( f `  q
) )  i^i  (
( _|_ P `  k ) `  {
d } ) )  =  ( ( q 
.+  ( f `  q ) )  i^i  (  ._|_  `  { d } ) ) )
2316oveqd 5891 . . . . . . . . . 10  |-  ( k  =  K  ->  (
r ( + P `  k ) ( f `
 r ) )  =  ( r  .+  ( f `  r
) ) )
2423, 21ineq12d 3384 . . . . . . . . 9  |-  ( k  =  K  ->  (
( r ( + P `  k ) ( f `  r
) )  i^i  (
( _|_ P `  k ) `  {
d } ) )  =  ( ( r 
.+  ( f `  r ) )  i^i  (  ._|_  `  { d } ) ) )
2522, 24eqeq12d 2310 . . . . . . . 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 2761 . . . . . . 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 2761 . . . . . 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 2796 . . . . 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 4114 . . . 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 30918 . . . 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 5555 . . . . . 6  |-  ( Atoms `  K )  e.  _V
324, 31eqeltri 2366 . . . . 5  |-  A  e. 
_V
3332mptex 5762 . . . 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 5618 . . 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 2340 . 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 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801    i^i cin 3164   {csn 3653    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   Atomscatm 30075   PSubSpcpsubsp 30307   + Pcpadd 30606   _|_ PcpolN 30713   WAtomscwpointsN 30797   PAutcpautN 30798   DilcdilN 30913   TrnctrnN 30914
This theorem is referenced by:  trnsetN  30967
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-trnN 30918
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