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Theorem dicffval 31182
Description: The partial isomorphism C for a lattice  K. (Contributed by NM, 15-Dec-2013.)
Hypotheses
Ref Expression
dicval.l  |-  .<_  =  ( le `  K )
dicval.a  |-  A  =  ( Atoms `  K )
dicval.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
dicffval  |-  ( K  e.  V  ->  ( DIsoC `  K )  =  ( w  e.  H  |->  ( q  e.  {
r  e.  A  |  -.  r  .<_  w }  |->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  w )
( g `  (
( oc `  K
) `  w )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  w )
) } ) ) )
Distinct variable groups:    A, r    w, H    f, g, q, r, s, w, K
Allowed substitution hints:    A( w, f, g, s, q)    H( f, g, s, r, q)    .<_ ( w, f, g, s, r, q)    V( w, f, g, s, r, q)

Proof of Theorem dicffval
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2830 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5563 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 dicval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2366 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5563 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
6 dicval.a . . . . . . 7  |-  A  =  ( Atoms `  K )
75, 6syl6eqr 2366 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
8 fveq2 5563 . . . . . . . . 9  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
9 dicval.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
108, 9syl6eqr 2366 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1110breqd 4071 . . . . . . 7  |-  ( k  =  K  ->  (
r ( le `  k ) w  <->  r  .<_  w ) )
1211notbid 285 . . . . . 6  |-  ( k  =  K  ->  ( -.  r ( le `  k ) w  <->  -.  r  .<_  w ) )
137, 12rabeqbidv 2817 . . . . 5  |-  ( k  =  K  ->  { r  e.  ( Atoms `  k
)  |  -.  r
( le `  k
) w }  =  { r  e.  A  |  -.  r  .<_  w }
)
14 fveq2 5563 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
1514fveq1d 5565 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
16 fveq2 5563 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( oc `  k )  =  ( oc `  K
) )
1716fveq1d 5565 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
( oc `  k
) `  w )  =  ( ( oc
`  K ) `  w ) )
1817fveq2d 5567 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
g `  ( ( oc `  k ) `  w ) )  =  ( g `  (
( oc `  K
) `  w )
) )
1918eqeq1d 2324 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( g `  (
( oc `  k
) `  w )
)  =  q  <->  ( g `  ( ( oc `  K ) `  w
) )  =  q ) )
2015, 19riotaeqbidv 6349 . . . . . . . . 9  |-  ( k  =  K  ->  ( iota_ g  e.  ( (
LTrn `  k ) `  w ) ( g `
 ( ( oc
`  k ) `  w ) )  =  q )  =  (
iota_ g  e.  (
( LTrn `  K ) `  w ) ( g `
 ( ( oc
`  K ) `  w ) )  =  q ) )
2120fveq2d 5567 . . . . . . . 8  |-  ( k  =  K  ->  (
s `  ( iota_ g  e.  ( ( LTrn `  k
) `  w )
( g `  (
( oc `  k
) `  w )
)  =  q ) )  =  ( s `
 ( iota_ g  e.  ( ( LTrn `  K
) `  w )
( g `  (
( oc `  K
) `  w )
)  =  q ) ) )
2221eqeq2d 2327 . . . . . . 7  |-  ( k  =  K  ->  (
f  =  ( s `
 ( iota_ g  e.  ( ( LTrn `  k
) `  w )
( g `  (
( oc `  k
) `  w )
)  =  q ) )  <->  f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K ) `  w
) ( g `  ( ( oc `  K ) `  w
) )  =  q ) ) ) )
23 fveq2 5563 . . . . . . . . 9  |-  ( k  =  K  ->  ( TEndo `  k )  =  ( TEndo `  K )
)
2423fveq1d 5565 . . . . . . . 8  |-  ( k  =  K  ->  (
( TEndo `  k ) `  w )  =  ( ( TEndo `  K ) `  w ) )
2524eleq2d 2383 . . . . . . 7  |-  ( k  =  K  ->  (
s  e.  ( (
TEndo `  k ) `  w )  <->  s  e.  ( ( TEndo `  K
) `  w )
) )
2622, 25anbi12d 691 . . . . . 6  |-  ( k  =  K  ->  (
( f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  k ) `  w
) ( g `  ( ( oc `  k ) `  w
) )  =  q ) )  /\  s  e.  ( ( TEndo `  k
) `  w )
)  <->  ( f  =  ( s `  ( iota_ g  e.  ( (
LTrn `  K ) `  w ) ( g `
 ( ( oc
`  K ) `  w ) )  =  q ) )  /\  s  e.  ( ( TEndo `  K ) `  w ) ) ) )
2726opabbidv 4119 . . . . 5  |-  ( k  =  K  ->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  ( ( LTrn `  k
) `  w )
( g `  (
( oc `  k
) `  w )
)  =  q ) )  /\  s  e.  ( ( TEndo `  k
) `  w )
) }  =  { <. f ,  s >.  |  ( f  =  ( s `  ( iota_ g  e.  ( (
LTrn `  K ) `  w ) ( g `
 ( ( oc
`  K ) `  w ) )  =  q ) )  /\  s  e.  ( ( TEndo `  K ) `  w ) ) } )
2813, 27mpteq12dv 4135 . . . 4  |-  ( k  =  K  ->  (
q  e.  { r  e.  ( Atoms `  k
)  |  -.  r
( le `  k
) w }  |->  {
<. f ,  s >.  |  ( f  =  ( s `  ( iota_ g  e.  ( (
LTrn `  k ) `  w ) ( g `
 ( ( oc
`  k ) `  w ) )  =  q ) )  /\  s  e.  ( ( TEndo `  k ) `  w ) ) } )  =  ( q  e.  { r  e.  A  |  -.  r  .<_  w }  |->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  ( ( LTrn `  K
) `  w )
( g `  (
( oc `  K
) `  w )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  w )
) } ) )
294, 28mpteq12dv 4135 . . 3  |-  ( k  =  K  ->  (
w  e.  ( LHyp `  k )  |->  ( q  e.  { r  e.  ( Atoms `  k )  |  -.  r ( le
`  k ) w }  |->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  ( ( LTrn `  k
) `  w )
( g `  (
( oc `  k
) `  w )
)  =  q ) )  /\  s  e.  ( ( TEndo `  k
) `  w )
) } ) )  =  ( w  e.  H  |->  ( q  e. 
{ r  e.  A  |  -.  r  .<_  w }  |->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  w )
( g `  (
( oc `  K
) `  w )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  w )
) } ) ) )
30 df-dic 31181 . . 3  |-  DIsoC  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( q  e.  { r  e.  ( Atoms `  k )  |  -.  r ( le
`  k ) w }  |->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  ( ( LTrn `  k
) `  w )
( g `  (
( oc `  k
) `  w )
)  =  q ) )  /\  s  e.  ( ( TEndo `  k
) `  w )
) } ) ) )
31 fvex 5577 . . . . 5  |-  ( LHyp `  K )  e.  _V
323, 31eqeltri 2386 . . . 4  |-  H  e. 
_V
3332mptex 5787 . . 3  |-  ( w  e.  H  |->  ( q  e.  { r  e.  A  |  -.  r  .<_  w }  |->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  ( ( LTrn `  K
) `  w )
( g `  (
( oc `  K
) `  w )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  w )
) } ) )  e.  _V
3429, 30, 33fvmpt 5640 . 2  |-  ( K  e.  _V  ->  ( DIsoC `  K )  =  ( w  e.  H  |->  ( q  e.  {
r  e.  A  |  -.  r  .<_  w }  |->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  w )
( g `  (
( oc `  K
) `  w )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  w )
) } ) ) )
351, 34syl 15 1  |-  ( K  e.  V  ->  ( DIsoC `  K )  =  ( w  e.  H  |->  ( q  e.  {
r  e.  A  |  -.  r  .<_  w }  |->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  w )
( g `  (
( oc `  K
) `  w )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  w )
) } ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   {crab 2581   _Vcvv 2822   class class class wbr 4060   {copab 4113    e. cmpt 4114   ` cfv 5292   iota_crio 6339   lecple 13262   occoc 13263   Atomscatm 29271   LHypclh 29991   LTrncltrn 30108   TEndoctendo 30759   DIsoCcdic 31180
This theorem is referenced by:  dicfval  31183
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-riota 6346  df-dic 31181
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