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Theorem pmapglb 29227
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
Dummy variable  y is distinct from all other variables.
Allowed substitution hints:    G( x)    M( x)

Proof of Theorem pmapglb
StepHypRef Expression
1 df-rex 2551 . . . . . . 7  |-  ( E. x  e.  S  y  =  x  <->  E. x
( x  e.  S  /\  y  =  x
) )
2 equcom 1648 . . . . . . . . . . 11  |-  ( y  =  x  <->  x  =  y )
32anbi2i 677 . . . . . . . . . 10  |-  ( ( x  e.  S  /\  y  =  x )  <->  ( x  e.  S  /\  x  =  y )
)
4 ancom 439 . . . . . . . . . 10  |-  ( ( x  e.  S  /\  x  =  y )  <->  ( x  =  y  /\  x  e.  S )
)
53, 4bitri 242 . . . . . . . . 9  |-  ( ( x  e.  S  /\  y  =  x )  <->  ( x  =  y  /\  x  e.  S )
)
65exbii 1570 . . . . . . . 8  |-  ( E. x ( x  e.  S  /\  y  =  x )  <->  E. x
( x  =  y  /\  x  e.  S
) )
7 vex 2793 . . . . . . . . 9  |-  y  e. 
_V
8 eleq1 2345 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  e.  S  <->  y  e.  S ) )
97, 8ceqsexv 2825 . . . . . . . 8  |-  ( E. x ( x  =  y  /\  x  e.  S )  <->  y  e.  S )
106, 9bitri 242 . . . . . . 7  |-  ( E. x ( x  e.  S  /\  y  =  x )  <->  y  e.  S )
111, 10bitri 242 . . . . . 6  |-  ( E. x  e.  S  y  =  x  <->  y  e.  S )
1211abbii 2397 . . . . 5  |-  { y  |  E. x  e.  S  y  =  x }  =  { y  |  y  e.  S }
13 abid2 2402 . . . . 5  |-  { y  |  y  e.  S }  =  S
1412, 13eqtr2i 2306 . . . 4  |-  S  =  { y  |  E. x  e.  S  y  =  x }
1514fveq2i 5489 . . 3  |-  ( G `
 S )  =  ( G `  {
y  |  E. x  e.  S  y  =  x } )
1615fveq2i 5489 . 2  |-  ( M `
 ( G `  S ) )  =  ( M `  ( G `  { y  |  E. x  e.  S  y  =  x }
) )
17 dfss3 3172 . . 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 29226 . . 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 2329 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 6    /\ wa 360    /\ w3a 936   E.wex 1529    = wceq 1624    e. wcel 1685   {cab 2271    =/= wne 2448   A.wral 2545   E.wrex 2546    C_ wss 3154   (/)c0 3457   |^|_ciin 3908   ` cfv 5222   Basecbs 13143   glbcglb 14072   HLchlt 28808   pmapcpmap 28954
This theorem is referenced by:  pmapglb2N  29228  pmapmeet  29230
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-poset 14075  df-glb 14104  df-join 14105  df-meet 14106  df-lat 14147  df-clat 14209  df-ats 28725  df-hlat 28809  df-pmap 28961
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