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Theorem 2polpmapN 30710
Description: Double polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
2polpmap.b  |-  B  =  ( Base `  K
)
2polpmap.m  |-  M  =  ( pmap `  K
)
2polpmap.p  |-  ._|_  =  ( _|_ P `  K
)
Assertion
Ref Expression
2polpmapN  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  (  ._|_  `  (  ._|_  `  ( M `  X
) ) )  =  ( M `  X
) )

Proof of Theorem 2polpmapN
StepHypRef Expression
1 2polpmap.b . . . 4  |-  B  =  ( Base `  K
)
2 eqid 2436 . . . 4  |-  ( oc
`  K )  =  ( oc `  K
)
3 2polpmap.m . . . 4  |-  M  =  ( pmap `  K
)
4 2polpmap.p . . . 4  |-  ._|_  =  ( _|_ P `  K
)
51, 2, 3, 4polpmapN 30709 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  (  ._|_  `  ( M `
 X ) )  =  ( M `  ( ( oc `  K ) `  X
) ) )
65fveq2d 5732 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  (  ._|_  `  (  ._|_  `  ( M `  X
) ) )  =  (  ._|_  `  ( M `
 ( ( oc
`  K ) `  X ) ) ) )
7 hlop 30160 . . . 4  |-  ( K  e.  HL  ->  K  e.  OP )
81, 2opoccl 29992 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
97, 8sylan 458 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
101, 2, 3, 4polpmapN 30709 . . 3  |-  ( ( K  e.  HL  /\  ( ( oc `  K ) `  X
)  e.  B )  ->  (  ._|_  `  ( M `  ( ( oc `  K ) `  X ) ) )  =  ( M `  ( ( oc `  K ) `  (
( oc `  K
) `  X )
) ) )
119, 10syldan 457 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  (  ._|_  `  ( M `
 ( ( oc
`  K ) `  X ) ) )  =  ( M `  ( ( oc `  K ) `  (
( oc `  K
) `  X )
) ) )
121, 2opococ 29993 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  (
( oc `  K
) `  X )
)  =  X )
137, 12sylan 458 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( oc `  K ) `  (
( oc `  K
) `  X )
)  =  X )
1413fveq2d 5732 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  (
( oc `  K
) `  ( ( oc `  K ) `  X ) ) )  =  ( M `  X ) )
156, 11, 143eqtrd 2472 1  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  (  ._|_  `  (  ._|_  `  ( M `  X
) ) )  =  ( M `  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ` cfv 5454   Basecbs 13469   occoc 13537   OPcops 29970   HLchlt 30148   pmapcpmap 30294   _|_ PcpolN 30699
This theorem is referenced by:  pmapsubclN  30743  ispsubcl2N  30744
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-pmap 30301  df-polarityN 30700
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