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Theorem pmapglb 29935
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 2648 . . . . . . 7  |-  ( E. x  e.  S  y  =  x  <->  E. x
( x  e.  S  /\  y  =  x
) )
2 equcom 1687 . . . . . . . . . . 11  |-  ( y  =  x  <->  x  =  y )
32anbi2i 676 . . . . . . . . . 10  |-  ( ( x  e.  S  /\  y  =  x )  <->  ( x  e.  S  /\  x  =  y )
)
4 ancom 438 . . . . . . . . . 10  |-  ( ( x  e.  S  /\  x  =  y )  <->  ( x  =  y  /\  x  e.  S )
)
53, 4bitri 241 . . . . . . . . 9  |-  ( ( x  e.  S  /\  y  =  x )  <->  ( x  =  y  /\  x  e.  S )
)
65exbii 1589 . . . . . . . 8  |-  ( E. x ( x  e.  S  /\  y  =  x )  <->  E. x
( x  =  y  /\  x  e.  S
) )
7 vex 2895 . . . . . . . . 9  |-  y  e. 
_V
8 eleq1 2440 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  e.  S  <->  y  e.  S ) )
97, 8ceqsexv 2927 . . . . . . . 8  |-  ( E. x ( x  =  y  /\  x  e.  S )  <->  y  e.  S )
106, 9bitri 241 . . . . . . 7  |-  ( E. x ( x  e.  S  /\  y  =  x )  <->  y  e.  S )
111, 10bitri 241 . . . . . 6  |-  ( E. x  e.  S  y  =  x  <->  y  e.  S )
1211abbii 2492 . . . . 5  |-  { y  |  E. x  e.  S  y  =  x }  =  { y  |  y  e.  S }
13 abid2 2497 . . . . 5  |-  { y  |  y  e.  S }  =  S
1412, 13eqtr2i 2401 . . . 4  |-  S  =  { y  |  E. x  e.  S  y  =  x }
1514fveq2i 5664 . . 3  |-  ( G `
 S )  =  ( G `  {
y  |  E. x  e.  S  y  =  x } )
1615fveq2i 5664 . 2  |-  ( M `
 ( G `  S ) )  =  ( M `  ( G `  { y  |  E. x  e.  S  y  =  x }
) )
17 dfss3 3274 . . 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 29934 . . 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 1221 . 2  |-  ( ( K  e.  HL  /\  S  C_  B  /\  S  =/=  (/) )  ->  ( M `  ( G `  { y  |  E. x  e.  S  y  =  x } ) )  =  |^|_ x  e.  S  ( M `  x ) )
2316, 22syl5eq 2424 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 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2366    =/= wne 2543   A.wral 2642   E.wrex 2643    C_ wss 3256   (/)c0 3564   |^|_ciin 4029   ` cfv 5387   Basecbs 13389   glbcglb 14320   HLchlt 29516   pmapcpmap 29662
This theorem is referenced by:  pmapglb2N  29936  pmapmeet  29938
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-undef 6472  df-riota 6478  df-poset 14323  df-glb 14352  df-join 14353  df-meet 14354  df-lat 14395  df-clat 14457  df-ats 29433  df-hlat 29517  df-pmap 29669
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