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Theorem pmapglb 29110
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
StepHypRef Expression
1 df-rex 2522 . . . . . . 7  |-  ( E. x  e.  S  y  =  x  <->  E. x
( x  e.  S  /\  y  =  x
) )
2 equcom 1824 . . . . . . . . . . 11  |-  ( y  =  x  <->  x  =  y )
32anbi2i 678 . . . . . . . . . 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 1580 . . . . . . . 8  |-  ( E. x ( x  e.  S  /\  y  =  x )  <->  E. x
( x  =  y  /\  x  e.  S
) )
7 vex 2760 . . . . . . . . 9  |-  y  e. 
_V
8 eleq1 2316 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  e.  S  <->  y  e.  S ) )
97, 8ceqsexv 2791 . . . . . . . 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 2368 . . . . 5  |-  { y  |  E. x  e.  S  y  =  x }  =  { y  |  y  e.  S }
13 abid2 2373 . . . . 5  |-  { y  |  y  e.  S }  =  S
1412, 13eqtr2i 2277 . . . 4  |-  S  =  { y  |  E. x  e.  S  y  =  x }
1514fveq2i 5447 . . 3  |-  ( G `
 S )  =  ( G `  {
y  |  E. x  e.  S  y  =  x } )
1615fveq2i 5447 . 2  |-  ( M `
 ( G `  S ) )  =  ( M `  ( G `  { y  |  E. x  e.  S  y  =  x }
) )
17 dfss3 3131 . . 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 29109 . . 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 1224 . 2  |-  ( ( K  e.  HL  /\  S  C_  B  /\  S  =/=  (/) )  ->  ( M `  ( G `  { y  |  E. x  e.  S  y  =  x } ) )  =  |^|_ x  e.  S  ( M `  x ) )
2316, 22syl5eq 2300 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 939   E.wex 1537    = wceq 1619    e. wcel 1621   {cab 2242    =/= wne 2419   A.wral 2516   E.wrex 2517    C_ wss 3113   (/)c0 3416   |^|_ciin 3866   ` cfv 4659   Basecbs 13096   glbcglb 14025   HLchlt 28691   pmapcpmap 28837
This theorem is referenced by:  pmapglb2N  29111  pmapmeet  29113
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-undef 6250  df-riota 6258  df-poset 14028  df-glb 14057  df-join 14058  df-meet 14059  df-lat 14100  df-clat 14162  df-ats 28608  df-hlat 28692  df-pmap 28844
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