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Theorem pmapmeet 33091
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 2420 . . . 4  |-  ( glb `  K )  =  ( glb `  K )
2 pmapmeet.m . . . 4  |-  ./\  =  ( meet `  K )
3 simp1 1005 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  HL )
4 simp2 1006 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
5 simp3 1007 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
61, 2, 3, 4, 5meetval 16217 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( ( glb `  K ) `
 { X ,  Y } ) )
76fveq2d 5876 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( P `  ( X  ./\  Y ) )  =  ( P `  ( ( glb `  K
) `  { X ,  Y } ) ) )
8 prssi 4150 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  { X ,  Y }  C_  B )
983adant1 1023 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  { X ,  Y }  C_  B )
10 prnzg 4114 . . . 4  |-  ( X  e.  B  ->  { X ,  Y }  =/=  (/) )
11103ad2ant2 1027 . . 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 33088 . . 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 1264 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( P `  (
( glb `  K
) `  { X ,  Y } ) )  =  |^|_ x  e.  { X ,  Y } 
( P `  x
) )
16 fveq2 5872 . . . 4  |-  ( x  =  X  ->  ( P `  x )  =  ( P `  X ) )
17 fveq2 5872 . . . 4  |-  ( x  =  Y  ->  ( P `  x )  =  ( P `  Y ) )
1816, 17iinxprg 4374 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
|^|_ x  e.  { X ,  Y }  ( P `
 x )  =  ( ( P `  X )  i^i  ( P `  Y )
) )
19183adant1 1023 . 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 2465 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 982    = wceq 1437    e. wcel 1867    =/= wne 2616    i^i cin 3432    C_ wss 3433   (/)c0 3758   {cpr 3995   |^|_ciin 4294   ` cfv 5592  (class class class)co 6296   Basecbs 15081   glbcglb 16140   meetcmee 16142   Atomscatm 32582   HLchlt 32669   pmapcpmap 32815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-poset 16143  df-lub 16172  df-glb 16173  df-join 16174  df-meet 16175  df-lat 16244  df-clat 16306  df-ats 32586  df-hlat 32670  df-pmap 32822
This theorem is referenced by:  hlmod1i  33174  poldmj1N  33246  pmapj2N  33247  pnonsingN  33251  psubclinN  33266  poml4N  33271  pl42lem1N  33297  pl42lem2N  33298
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