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Theorem pmapglb 30581
Description: The projective map of the GLB of a set of lattice elements  S. Variant of Theorem 15.5.2 of [MaedaMaeda] p. 62. (Contributed by NM, 5-Dec-2011.)
Hypotheses
Ref Expression
pmapglb.b  |-  B  =  ( Base `  K
)
pmapglb.g  |-  G  =  ( glb `  K
)
pmapglb.m  |-  M  =  ( pmap `  K
)
Assertion
Ref Expression
pmapglb  |-  ( ( K  e.  HL  /\  S  C_  B  /\  S  =/=  (/) )  ->  ( M `  ( G `  S ) )  = 
|^|_ x  e.  S  ( M `  x ) )
Distinct variable groups:    x, B    x, K    x, S
Allowed substitution hints:    G( x)    M( x)

Proof of Theorem pmapglb
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-rex 2562 . . . . . . 7  |-  ( E. x  e.  S  y  =  x  <->  E. x
( x  e.  S  /\  y  =  x
) )
2 equcom 1665 . . . . . . . . . . 11  |-  ( y  =  x  <->  x  =  y )
32anbi2i 675 . . . . . . . . . 10  |-  ( ( x  e.  S  /\  y  =  x )  <->  ( x  e.  S  /\  x  =  y )
)
4 ancom 437 . . . . . . . . . 10  |-  ( ( x  e.  S  /\  x  =  y )  <->  ( x  =  y  /\  x  e.  S )
)
53, 4bitri 240 . . . . . . . . 9  |-  ( ( x  e.  S  /\  y  =  x )  <->  ( x  =  y  /\  x  e.  S )
)
65exbii 1572 . . . . . . . 8  |-  ( E. x ( x  e.  S  /\  y  =  x )  <->  E. x
( x  =  y  /\  x  e.  S
) )
7 vex 2804 . . . . . . . . 9  |-  y  e. 
_V
8 eleq1 2356 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  e.  S  <->  y  e.  S ) )
97, 8ceqsexv 2836 . . . . . . . 8  |-  ( E. x ( x  =  y  /\  x  e.  S )  <->  y  e.  S )
106, 9bitri 240 . . . . . . 7  |-  ( E. x ( x  e.  S  /\  y  =  x )  <->  y  e.  S )
111, 10bitri 240 . . . . . 6  |-  ( E. x  e.  S  y  =  x  <->  y  e.  S )
1211abbii 2408 . . . . 5  |-  { y  |  E. x  e.  S  y  =  x }  =  { y  |  y  e.  S }
13 abid2 2413 . . . . 5  |-  { y  |  y  e.  S }  =  S
1412, 13eqtr2i 2317 . . . 4  |-  S  =  { y  |  E. x  e.  S  y  =  x }
1514fveq2i 5544 . . 3  |-  ( G `
 S )  =  ( G `  {
y  |  E. x  e.  S  y  =  x } )
1615fveq2i 5544 . 2  |-  ( M `
 ( G `  S ) )  =  ( M `  ( G `  { y  |  E. x  e.  S  y  =  x }
) )
17 dfss3 3183 . . 3  |-  ( S 
C_  B  <->  A. x  e.  S  x  e.  B )
18 pmapglb.b . . . 4  |-  B  =  ( Base `  K
)
19 pmapglb.g . . . 4  |-  G  =  ( glb `  K
)
20 pmapglb.m . . . 4  |-  M  =  ( pmap `  K
)
2118, 19, 20pmapglbx 30580 . . 3  |-  ( ( K  e.  HL  /\  A. x  e.  S  x  e.  B  /\  S  =/=  (/) )  ->  ( M `  ( G `  { y  |  E. x  e.  S  y  =  x } ) )  =  |^|_ x  e.  S  ( M `  x ) )
2217, 21syl3an2b 1219 . 2  |-  ( ( K  e.  HL  /\  S  C_  B  /\  S  =/=  (/) )  ->  ( M `  ( G `  { y  |  E. x  e.  S  y  =  x } ) )  =  |^|_ x  e.  S  ( M `  x ) )
2316, 22syl5eq 2340 1  |-  ( ( K  e.  HL  /\  S  C_  B  /\  S  =/=  (/) )  ->  ( M `  ( G `  S ) )  = 
|^|_ x  e.  S  ( M `  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   E.wrex 2557    C_ wss 3165   (/)c0 3468   |^|_ciin 3922   ` cfv 5271   Basecbs 13164   glbcglb 14093   HLchlt 30162   pmapcpmap 30308
This theorem is referenced by:  pmapglb2N  30582  pmapmeet  30584
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-nel 2462  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-iin 3924  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-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-glb 14125  df-join 14126  df-meet 14127  df-lat 14168  df-clat 14230  df-ats 30079  df-hlat 30163  df-pmap 30315
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