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Theorem dicval 31366
Description: The partial isomorphism C for a lattice  K. (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
dicval.l  |-  .<_  =  ( le `  K )
dicval.a  |-  A  =  ( Atoms `  K )
dicval.h  |-  H  =  ( LHyp `  K
)
dicval.p  |-  P  =  ( ( oc `  K ) `  W
)
dicval.t  |-  T  =  ( ( LTrn `  K
) `  W )
dicval.e  |-  E  =  ( ( TEndo `  K
) `  W )
dicval.i  |-  I  =  ( ( DIsoC `  K
) `  W )
Assertion
Ref Expression
dicval  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E ) } )
Distinct variable groups:    f, g,
s, K    T, g    f, W, g, s    f, E, s    P, f    Q, f, g, s    T, f
Allowed substitution hints:    A( f, g, s)    P( g, s)    T( s)    E( g)    H( f, g, s)    I( f, g, s)    .<_ ( f, g, s)    V( f, g, s)

Proof of Theorem dicval
Dummy variables  r 
q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dicval.l . . . . 5  |-  .<_  =  ( le `  K )
2 dicval.a . . . . 5  |-  A  =  ( Atoms `  K )
3 dicval.h . . . . 5  |-  H  =  ( LHyp `  K
)
4 dicval.p . . . . 5  |-  P  =  ( ( oc `  K ) `  W
)
5 dicval.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
6 dicval.e . . . . 5  |-  E  =  ( ( TEndo `  K
) `  W )
7 dicval.i . . . . 5  |-  I  =  ( ( DIsoC `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dicfval 31365 . . . 4  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( q  e.  { r  e.  A  |  -.  r  .<_  W }  |->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  q ) )  /\  s  e.  E ) } ) )
98adantr 451 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  I  =  ( q  e.  { r  e.  A  |  -.  r  .<_  W }  |->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  T
( g `  P
)  =  q ) )  /\  s  e.  E ) } ) )
109fveq1d 5527 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =  ( ( q  e.  { r  e.  A  |  -.  r  .<_  W }  |->  {
<. f ,  s >.  |  ( f  =  ( s `  ( iota_ g  e.  T ( g `  P )  =  q ) )  /\  s  e.  E
) } ) `  Q ) )
11 simpr 447 . . . 4  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
12 breq1 4026 . . . . . 6  |-  ( r  =  Q  ->  (
r  .<_  W  <->  Q  .<_  W ) )
1312notbid 285 . . . . 5  |-  ( r  =  Q  ->  ( -.  r  .<_  W  <->  -.  Q  .<_  W ) )
1413elrab 2923 . . . 4  |-  ( Q  e.  { r  e.  A  |  -.  r  .<_  W }  <->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
1511, 14sylibr 203 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  Q  e.  { r  e.  A  |  -.  r  .<_  W } )
16 eqeq2 2292 . . . . . . . . 9  |-  ( q  =  Q  ->  (
( g `  P
)  =  q  <->  ( g `  P )  =  Q ) )
1716riotabidv 6306 . . . . . . . 8  |-  ( q  =  Q  ->  ( iota_ g  e.  T ( g `  P )  =  q )  =  ( iota_ g  e.  T
( g `  P
)  =  Q ) )
1817fveq2d 5529 . . . . . . 7  |-  ( q  =  Q  ->  (
s `  ( iota_ g  e.  T ( g `  P )  =  q ) )  =  ( s `  ( iota_ g  e.  T ( g `
 P )  =  Q ) ) )
1918eqeq2d 2294 . . . . . 6  |-  ( q  =  Q  ->  (
f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  q ) )  <->  f  =  ( s `  ( iota_ g  e.  T ( g `  P )  =  Q ) ) ) )
2019anbi1d 685 . . . . 5  |-  ( q  =  Q  ->  (
( f  =  ( s `  ( iota_ g  e.  T ( g `
 P )  =  q ) )  /\  s  e.  E )  <->  ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E ) ) )
2120opabbidv 4082 . . . 4  |-  ( q  =  Q  ->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  q ) )  /\  s  e.  E ) }  =  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  T
( g `  P
)  =  Q ) )  /\  s  e.  E ) } )
22 eqid 2283 . . . 4  |-  ( q  e.  { r  e.  A  |  -.  r  .<_  W }  |->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  q ) )  /\  s  e.  E ) } )  =  ( q  e. 
{ r  e.  A  |  -.  r  .<_  W }  |->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  T
( g `  P
)  =  q ) )  /\  s  e.  E ) } )
23 fvex 5539 . . . . . . . . . . 11  |-  ( (
TEndo `  K ) `  W )  e.  _V
246, 23eqeltri 2353 . . . . . . . . . 10  |-  E  e. 
_V
2524uniex 4516 . . . . . . . . 9  |-  U. E  e.  _V
2625rnex 4942 . . . . . . . 8  |-  ran  U. E  e.  _V
2726uniex 4516 . . . . . . 7  |-  U. ran  U. E  e.  _V
2827pwex 4193 . . . . . 6  |-  ~P U. ran  U. E  e.  _V
2928, 24xpex 4801 . . . . 5  |-  ( ~P
U. ran  U. E  X.  E )  e.  _V
30 simpl 443 . . . . . . . . 9  |-  ( ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E )  ->  f  =  ( s `  ( iota_ g  e.  T
( g `  P
)  =  Q ) ) )
31 fvssunirn 5551 . . . . . . . . . . 11  |-  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  C_  U. ran  s
32 elssuni 3855 . . . . . . . . . . . . 13  |-  ( s  e.  E  ->  s  C_ 
U. E )
3332adantl 452 . . . . . . . . . . . 12  |-  ( ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E )  ->  s  C_ 
U. E )
34 rnss 4907 . . . . . . . . . . . 12  |-  ( s 
C_  U. E  ->  ran  s  C_  ran  U. E
)
35 uniss 3848 . . . . . . . . . . . 12  |-  ( ran  s  C_  ran  U. E  ->  U. ran  s  C_  U.
ran  U. E )
3633, 34, 353syl 18 . . . . . . . . . . 11  |-  ( ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E )  ->  U. ran  s  C_  U. ran  U. E )
3731, 36syl5ss 3190 . . . . . . . . . 10  |-  ( ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E )  ->  (
s `  ( iota_ g  e.  T ( g `  P )  =  Q ) )  C_  U. ran  U. E )
3827elpw2 4175 . . . . . . . . . 10  |-  ( ( s `  ( iota_ g  e.  T ( g `
 P )  =  Q ) )  e. 
~P U. ran  U. E  <->  ( s `  ( iota_ g  e.  T ( g `
 P )  =  Q ) )  C_  U.
ran  U. E )
3937, 38sylibr 203 . . . . . . . . 9  |-  ( ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E )  ->  (
s `  ( iota_ g  e.  T ( g `  P )  =  Q ) )  e.  ~P U.
ran  U. E )
4030, 39eqeltrd 2357 . . . . . . . 8  |-  ( ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E )  ->  f  e.  ~P U. ran  U. E )
41 simpr 447 . . . . . . . 8  |-  ( ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E )  ->  s  e.  E )
4240, 41jca 518 . . . . . . 7  |-  ( ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E )  ->  (
f  e.  ~P U. ran  U. E  /\  s  e.  E ) )
4342ssopab2i 4292 . . . . . 6  |-  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E ) }  C_  {
<. f ,  s >.  |  ( f  e. 
~P U. ran  U. E  /\  s  e.  E
) }
44 df-xp 4695 . . . . . 6  |-  ( ~P
U. ran  U. E  X.  E )  =  { <. f ,  s >.  |  ( f  e. 
~P U. ran  U. E  /\  s  e.  E
) }
4543, 44sseqtr4i 3211 . . . . 5  |-  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E ) }  C_  ( ~P U. ran  U. E  X.  E )
4629, 45ssexi 4159 . . . 4  |-  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E ) }  e.  _V
4721, 22, 46fvmpt 5602 . . 3  |-  ( Q  e.  { r  e.  A  |  -.  r  .<_  W }  ->  (
( q  e.  {
r  e.  A  |  -.  r  .<_  W }  |->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  T
( g `  P
)  =  q ) )  /\  s  e.  E ) } ) `
 Q )  =  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  T
( g `  P
)  =  Q ) )  /\  s  e.  E ) } )
4815, 47syl 15 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( q  e. 
{ r  e.  A  |  -.  r  .<_  W }  |->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  T
( g `  P
)  =  q ) )  /\  s  e.  E ) } ) `
 Q )  =  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  T
( g `  P
)  =  Q ) )  /\  s  e.  E ) } )
4910, 48eqtrd 2315 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E ) } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   class class class wbr 4023   {copab 4076    e. cmpt 4077    X. cxp 4687   ran crn 4690   ` cfv 5255   iota_crio 6297   lecple 13215   occoc 13216   Atomscatm 29453   LHypclh 30173   LTrncltrn 30290   TEndoctendo 30941   DIsoCcdic 31362
This theorem is referenced by:  dicopelval  31367  dicelvalN  31368  dicval2  31369  dicfnN  31373  dicvalrelN  31375  dicssdvh  31376  dicelval1sta  31377  dihpN  31526
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 6304  df-dic 31363
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