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Theorem pmapmeet 30572
Description: The projective map of a meet. (Contributed by NM, 25-Jan-2012.)
Hypotheses
Ref Expression
pmapmeet.b  |-  B  =  ( Base `  K
)
pmapmeet.m  |-  ./\  =  ( meet `  K )
pmapmeet.a  |-  A  =  ( Atoms `  K )
pmapmeet.p  |-  P  =  ( pmap `  K
)
Assertion
Ref Expression
pmapmeet  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( P `  ( X  ./\  Y ) )  =  ( ( P `
 X )  i^i  ( P `  Y
) ) )

Proof of Theorem pmapmeet
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pmapmeet.b . . . 4  |-  B  =  ( Base `  K
)
2 eqid 2438 . . . 4  |-  ( glb `  K )  =  ( glb `  K )
3 pmapmeet.m . . . 4  |-  ./\  =  ( meet `  K )
41, 2, 3meetval 14454 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( ( glb `  K ) `
 { X ,  Y } ) )
54fveq2d 5734 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( P `  ( X  ./\  Y ) )  =  ( P `  ( ( glb `  K
) `  { X ,  Y } ) ) )
6 simp1 958 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  HL )
7 prssi 3956 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  { X ,  Y }  C_  B )
873adant1 976 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  { X ,  Y }  C_  B )
9 prnzg 3926 . . . 4  |-  ( X  e.  B  ->  { X ,  Y }  =/=  (/) )
1093ad2ant2 980 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  { X ,  Y }  =/=  (/) )
11 pmapmeet.p . . . 4  |-  P  =  ( pmap `  K
)
121, 2, 11pmapglb 30569 . . 3  |-  ( ( K  e.  HL  /\  { X ,  Y }  C_  B  /\  { X ,  Y }  =/=  (/) )  -> 
( P `  (
( glb `  K
) `  { X ,  Y } ) )  =  |^|_ x  e.  { X ,  Y } 
( P `  x
) )
136, 8, 10, 12syl3anc 1185 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( P `  (
( glb `  K
) `  { X ,  Y } ) )  =  |^|_ x  e.  { X ,  Y } 
( P `  x
) )
14 fveq2 5730 . . . 4  |-  ( x  =  X  ->  ( P `  x )  =  ( P `  X ) )
15 fveq2 5730 . . . 4  |-  ( x  =  Y  ->  ( P `  x )  =  ( P `  Y ) )
1614, 15iinxprg 4170 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
|^|_ x  e.  { X ,  Y }  ( P `
 x )  =  ( ( P `  X )  i^i  ( P `  Y )
) )
17163adant1 976 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  -> 
|^|_ x  e.  { X ,  Y }  ( P `
 x )  =  ( ( P `  X )  i^i  ( P `  Y )
) )
185, 13, 173eqtrd 2474 1  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( P `  ( X  ./\  Y ) )  =  ( ( P `
 X )  i^i  ( P `  Y
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601    i^i cin 3321    C_ wss 3322   (/)c0 3630   {cpr 3817   |^|_ciin 4096   ` cfv 5456  (class class class)co 6083   Basecbs 13471   glbcglb 14402   meetcmee 14404   Atomscatm 30063   HLchlt 30150   pmapcpmap 30296
This theorem is referenced by:  hlmod1i  30655  poldmj1N  30727  pmapj2N  30728  pnonsingN  30732  psubclinN  30747  poml4N  30752  pl42lem1N  30778  pl42lem2N  30779
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-glb 14434  df-join 14435  df-meet 14436  df-lat 14477  df-clat 14539  df-ats 30067  df-hlat 30151  df-pmap 30303
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