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Theorem pmapmeet 29962
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 2283 . . . 4  |-  ( glb `  K )  =  ( glb `  K )
3 pmapmeet.m . . . 4  |-  ./\  =  ( meet `  K )
41, 2, 3meetval 14129 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( ( glb `  K ) `
 { X ,  Y } ) )
54fveq2d 5529 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( P `  ( X  ./\  Y ) )  =  ( P `  ( ( glb `  K
) `  { X ,  Y } ) ) )
6 simp1 955 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  HL )
7 prssi 3771 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  { X ,  Y }  C_  B )
873adant1 973 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  { X ,  Y }  C_  B )
9 prnzg 3746 . . . 4  |-  ( X  e.  B  ->  { X ,  Y }  =/=  (/) )
1093ad2ant2 977 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  { X ,  Y }  =/=  (/) )
11 pmapmeet.p . . . 4  |-  P  =  ( pmap `  K
)
121, 2, 11pmapglb 29959 . . 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 1182 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( P `  (
( glb `  K
) `  { X ,  Y } ) )  =  |^|_ x  e.  { X ,  Y } 
( P `  x
) )
14 fveq2 5525 . . . 4  |-  ( x  =  X  ->  ( P `  x )  =  ( P `  X ) )
15 fveq2 5525 . . . 4  |-  ( x  =  Y  ->  ( P `  x )  =  ( P `  Y ) )
1614, 15iinxprg 3979 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
|^|_ x  e.  { X ,  Y }  ( P `
 x )  =  ( ( P `  X )  i^i  ( P `  Y )
) )
17163adant1 973 . 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 2319 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 934    = wceq 1623    e. wcel 1684    =/= wne 2446    i^i cin 3151    C_ wss 3152   (/)c0 3455   {cpr 3641   |^|_ciin 3906   ` cfv 5255  (class class class)co 5858   Basecbs 13148   glbcglb 14077   meetcmee 14079   Atomscatm 29453   HLchlt 29540   pmapcpmap 29686
This theorem is referenced by:  hlmod1i  30045  poldmj1N  30117  pmapj2N  30118  pnonsingN  30122  psubclinN  30137  poml4N  30142  pl42lem1N  30168  pl42lem2N  30169
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-glb 14109  df-join 14110  df-meet 14111  df-lat 14152  df-clat 14214  df-ats 29457  df-hlat 29541  df-pmap 29693
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