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Theorem dicval 31975
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 31974 . . . 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 453 . . 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 5731 . 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 449 . . . 4  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
12 breq1 4216 . . . . . 6  |-  ( r  =  Q  ->  (
r  .<_  W  <->  Q  .<_  W ) )
1312notbid 287 . . . . 5  |-  ( r  =  Q  ->  ( -.  r  .<_  W  <->  -.  Q  .<_  W ) )
1413elrab 3093 . . . 4  |-  ( Q  e.  { r  e.  A  |  -.  r  .<_  W }  <->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
1511, 14sylibr 205 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  Q  e.  { r  e.  A  |  -.  r  .<_  W } )
16 eqeq2 2446 . . . . . . . . 9  |-  ( q  =  Q  ->  (
( g `  P
)  =  q  <->  ( g `  P )  =  Q ) )
1716riotabidv 6552 . . . . . . . 8  |-  ( q  =  Q  ->  ( iota_ g  e.  T ( g `  P )  =  q )  =  ( iota_ g  e.  T
( g `  P
)  =  Q ) )
1817fveq2d 5733 . . . . . . 7  |-  ( q  =  Q  ->  (
s `  ( iota_ g  e.  T ( g `  P )  =  q ) )  =  ( s `  ( iota_ g  e.  T ( g `
 P )  =  Q ) ) )
1918eqeq2d 2448 . . . . . 6  |-  ( q  =  Q  ->  (
f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  q ) )  <->  f  =  ( s `  ( iota_ g  e.  T ( g `  P )  =  Q ) ) ) )
2019anbi1d 687 . . . . 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 4272 . . . 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 2437 . . . 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 5743 . . . . . . . . . . 11  |-  ( (
TEndo `  K ) `  W )  e.  _V
246, 23eqeltri 2507 . . . . . . . . . 10  |-  E  e. 
_V
2524uniex 4706 . . . . . . . . 9  |-  U. E  e.  _V
2625rnex 5134 . . . . . . . 8  |-  ran  U. E  e.  _V
2726uniex 4706 . . . . . . 7  |-  U. ran  U. E  e.  _V
2827pwex 4383 . . . . . 6  |-  ~P U. ran  U. E  e.  _V
2928, 24xpex 4991 . . . . 5  |-  ( ~P
U. ran  U. E  X.  E )  e.  _V
30 simpl 445 . . . . . . . . 9  |-  ( ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E )  ->  f  =  ( s `  ( iota_ g  e.  T
( g `  P
)  =  Q ) ) )
31 fvssunirn 5755 . . . . . . . . . . 11  |-  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  C_  U. ran  s
32 elssuni 4044 . . . . . . . . . . . . 13  |-  ( s  e.  E  ->  s  C_ 
U. E )
3332adantl 454 . . . . . . . . . . . 12  |-  ( ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E )  ->  s  C_ 
U. E )
34 rnss 5099 . . . . . . . . . . . 12  |-  ( s 
C_  U. E  ->  ran  s  C_  ran  U. E
)
35 uniss 4037 . . . . . . . . . . . 12  |-  ( ran  s  C_  ran  U. E  ->  U. ran  s  C_  U.
ran  U. E )
3633, 34, 353syl 19 . . . . . . . . . . 11  |-  ( ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E )  ->  U. ran  s  C_  U. ran  U. E )
3731, 36syl5ss 3360 . . . . . . . . . 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 4365 . . . . . . . . . 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 205 . . . . . . . . 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 2511 . . . . . . . 8  |-  ( ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E )  ->  f  e.  ~P U. ran  U. E )
41 simpr 449 . . . . . . . 8  |-  ( ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E )  ->  s  e.  E )
4240, 41jca 520 . . . . . . 7  |-  ( ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E )  ->  (
f  e.  ~P U. ran  U. E  /\  s  e.  E ) )
4342ssopab2i 4483 . . . . . 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 4885 . . . . . 6  |-  ( ~P
U. ran  U. E  X.  E )  =  { <. f ,  s >.  |  ( f  e. 
~P U. ran  U. E  /\  s  e.  E
) }
4543, 44sseqtr4i 3382 . . . . 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 4349 . . . 4  |-  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E ) }  e.  _V
4721, 22, 46fvmpt 5807 . . 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 16 . 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 2469 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 360    = wceq 1653    e. wcel 1726   {crab 2710   _Vcvv 2957    C_ wss 3321   ~Pcpw 3800   U.cuni 4016   class class class wbr 4213   {copab 4266    e. cmpt 4267    X. cxp 4877   ran crn 4880   ` cfv 5455   iota_crio 6543   lecple 13537   occoc 13538   Atomscatm 30062   LHypclh 30782   LTrncltrn 30899   TEndoctendo 31550   DIsoCcdic 31971
This theorem is referenced by:  dicopelval  31976  dicelvalN  31977  dicval2  31978  dicfnN  31982  dicvalrelN  31984  dicssdvh  31985  dicelval1sta  31986  dihpN  32135
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-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-riota 6550  df-dic 31972
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