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Theorem pmapmeet 34786
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 eqid 2467 . . . 4  |-  ( glb `  K )  =  ( glb `  K )
2 pmapmeet.m . . . 4  |-  ./\  =  ( meet `  K )
3 simp1 996 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  HL )
4 simp2 997 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
5 simp3 998 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
61, 2, 3, 4, 5meetval 15509 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( ( glb `  K ) `
 { X ,  Y } ) )
76fveq2d 5870 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( P `  ( X  ./\  Y ) )  =  ( P `  ( ( glb `  K
) `  { X ,  Y } ) ) )
8 prssi 4183 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  { X ,  Y }  C_  B )
983adant1 1014 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  { X ,  Y }  C_  B )
10 prnzg 4147 . . . 4  |-  ( X  e.  B  ->  { X ,  Y }  =/=  (/) )
11103ad2ant2 1018 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  { X ,  Y }  =/=  (/) )
12 pmapmeet.b . . . 4  |-  B  =  ( Base `  K
)
13 pmapmeet.p . . . 4  |-  P  =  ( pmap `  K
)
1412, 1, 13pmapglb 34783 . . 3  |-  ( ( K  e.  HL  /\  { X ,  Y }  C_  B  /\  { X ,  Y }  =/=  (/) )  -> 
( P `  (
( glb `  K
) `  { X ,  Y } ) )  =  |^|_ x  e.  { X ,  Y } 
( P `  x
) )
153, 9, 11, 14syl3anc 1228 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( P `  (
( glb `  K
) `  { X ,  Y } ) )  =  |^|_ x  e.  { X ,  Y } 
( P `  x
) )
16 fveq2 5866 . . . 4  |-  ( x  =  X  ->  ( P `  x )  =  ( P `  X ) )
17 fveq2 5866 . . . 4  |-  ( x  =  Y  ->  ( P `  x )  =  ( P `  Y ) )
1816, 17iinxprg 4403 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
|^|_ x  e.  { X ,  Y }  ( P `
 x )  =  ( ( P `  X )  i^i  ( P `  Y )
) )
19183adant1 1014 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  -> 
|^|_ x  e.  { X ,  Y }  ( P `
 x )  =  ( ( P `  X )  i^i  ( P `  Y )
) )
207, 15, 193eqtrd 2512 1  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( P `  ( X  ./\  Y ) )  =  ( ( P `
 X )  i^i  ( P `  Y
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662    i^i cin 3475    C_ wss 3476   (/)c0 3785   {cpr 4029   |^|_ciin 4326   ` cfv 5588  (class class class)co 6285   Basecbs 14493   glbcglb 15433   meetcmee 15435   Atomscatm 34277   HLchlt 34364   pmapcpmap 34510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-poset 15436  df-lub 15464  df-glb 15465  df-join 15466  df-meet 15467  df-lat 15536  df-clat 15598  df-ats 34281  df-hlat 34365  df-pmap 34517
This theorem is referenced by:  hlmod1i  34869  poldmj1N  34941  pmapj2N  34942  pnonsingN  34946  psubclinN  34961  poml4N  34966  pl42lem1N  34992  pl42lem2N  34993
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