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Theorem pmapglb 28648
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 2514 . . . . . . 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 2730 . . . . . . . . 9  |-  y  e. 
_V
8 eleq1 2313 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  e.  S  <->  y  e.  S ) )
97, 8ceqsexv 2761 . . . . . . . 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 2361 . . . . 5  |-  { y  |  E. x  e.  S  y  =  x }  =  { y  |  y  e.  S }
13 abid2 2366 . . . . 5  |-  { y  |  y  e.  S }  =  S
1412, 13eqtr2i 2274 . . . 4  |-  S  =  { y  |  E. x  e.  S  y  =  x }
1514fveq2i 5380 . . 3  |-  ( G `
 S )  =  ( G `  {
y  |  E. x  e.  S  y  =  x } )
1615fveq2i 5380 . 2  |-  ( M `
 ( G `  S ) )  =  ( M `  ( G `  { y  |  E. x  e.  S  y  =  x }
) )
17 dfss3 3093 . . 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 28647 . . 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 2297 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 2239    =/= wne 2412   A.wral 2509   E.wrex 2510    C_ wss 3078   (/)c0 3362   |^|_ciin 3804   ` cfv 4592   Basecbs 13022   glbcglb 13921   HLchlt 28229   pmapcpmap 28375
This theorem is referenced by:  pmapglb2N  28649  pmapmeet  28651
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-glb 13953  df-join 13954  df-meet 13955  df-lat 13996  df-clat 14058  df-ats 28146  df-hlat 28230  df-pmap 28382
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