Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dicval Unicode version

Theorem dicval 31988
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 31987 . . . 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 5543 . 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 4042 . . . . . 6  |-  ( r  =  Q  ->  (
r  .<_  W  <->  Q  .<_  W ) )
1312notbid 285 . . . . 5  |-  ( r  =  Q  ->  ( -.  r  .<_  W  <->  -.  Q  .<_  W ) )
1413elrab 2936 . . . 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 2305 . . . . . . . . 9  |-  ( q  =  Q  ->  (
( g `  P
)  =  q  <->  ( g `  P )  =  Q ) )
1716riotabidv 6322 . . . . . . . 8  |-  ( q  =  Q  ->  ( iota_ g  e.  T ( g `  P )  =  q )  =  ( iota_ g  e.  T
( g `  P
)  =  Q ) )
1817fveq2d 5545 . . . . . . 7  |-  ( q  =  Q  ->  (
s `  ( iota_ g  e.  T ( g `  P )  =  q ) )  =  ( s `  ( iota_ g  e.  T ( g `
 P )  =  Q ) ) )
1918eqeq2d 2307 . . . . . 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 4098 . . . 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 2296 . . . 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 5555 . . . . . . . . . . 11  |-  ( (
TEndo `  K ) `  W )  e.  _V
246, 23eqeltri 2366 . . . . . . . . . 10  |-  E  e. 
_V
2524uniex 4532 . . . . . . . . 9  |-  U. E  e.  _V
2625rnex 4958 . . . . . . . 8  |-  ran  U. E  e.  _V
2726uniex 4532 . . . . . . 7  |-  U. ran  U. E  e.  _V
2827pwex 4209 . . . . . 6  |-  ~P U. ran  U. E  e.  _V
2928, 24xpex 4817 . . . . 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 5567 . . . . . . . . . . 11  |-  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  C_  U. ran  s
32 elssuni 3871 . . . . . . . . . . . . 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 4923 . . . . . . . . . . . 12  |-  ( s 
C_  U. E  ->  ran  s  C_  ran  U. E
)
35 uniss 3864 . . . . . . . . . . . 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 3203 . . . . . . . . . 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 4191 . . . . . . . . . 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 2370 . . . . . . . 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 4308 . . . . . 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 4711 . . . . . 6  |-  ( ~P
U. ran  U. E  X.  E )  =  { <. f ,  s >.  |  ( f  e. 
~P U. ran  U. E  /\  s  e.  E
) }
4543, 44sseqtr4i 3224 . . . . 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 4175 . . . 4  |-  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E ) }  e.  _V
4721, 22, 46fvmpt 5618 . . 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 2328 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 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   class class class wbr 4039   {copab 4092    e. cmpt 4093    X. cxp 4703   ran crn 4706   ` cfv 5271   iota_crio 6313   lecple 13231   occoc 13232   Atomscatm 30075   LHypclh 30795   LTrncltrn 30912   TEndoctendo 31563   DIsoCcdic 31984
This theorem is referenced by:  dicopelval  31989  dicelvalN  31990  dicval2  31991  dicfnN  31995  dicvalrelN  31997  dicssdvh  31998  dicelval1sta  31999  dihpN  32148
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-13 1698  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-pow 4204  ax-pr 4230  ax-un 4528
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-pw 3640  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-riota 6320  df-dic 31985
  Copyright terms: Public domain W3C validator