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Theorem pmapmeet 30584
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 2296 . . . 4  |-  ( glb `  K )  =  ( glb `  K )
3 pmapmeet.m . . . 4  |-  ./\  =  ( meet `  K )
41, 2, 3meetval 14145 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( ( glb `  K ) `
 { X ,  Y } ) )
54fveq2d 5545 . 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 3787 . . . 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 3759 . . . 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 30581 . . 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 5541 . . . 4  |-  ( x  =  X  ->  ( P `  x )  =  ( P `  X ) )
15 fveq2 5541 . . . 4  |-  ( x  =  Y  ->  ( P `  x )  =  ( P `  Y ) )
1614, 15iinxprg 3995 . . 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 2332 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 1632    e. wcel 1696    =/= wne 2459    i^i cin 3164    C_ wss 3165   (/)c0 3468   {cpr 3654   |^|_ciin 3922   ` cfv 5271  (class class class)co 5874   Basecbs 13164   glbcglb 14093   meetcmee 14095   Atomscatm 30075   HLchlt 30162   pmapcpmap 30308
This theorem is referenced by:  hlmod1i  30667  poldmj1N  30739  pmapj2N  30740  pnonsingN  30744  psubclinN  30759  poml4N  30764  pl42lem1N  30790  pl42lem2N  30791
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-glb 14125  df-join 14126  df-meet 14127  df-lat 14168  df-clat 14230  df-ats 30079  df-hlat 30163  df-pmap 30315
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