Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2polpmapN Unicode version

Theorem 2polpmapN 30724
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 2296 . . . 4  |-  ( oc
`  K )  =  ( oc `  K
)
3 2polpmap.m . . . 4  |-  M  =  ( pmap `  K
)
4 2polpmap.p . . . 4  |-  ._|_  =  ( _|_ P `  K
)
51, 2, 3, 4polpmapN 30723 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  (  ._|_  `  ( M `
 X ) )  =  ( M `  ( ( oc `  K ) `  X
) ) )
65fveq2d 5545 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  (  ._|_  `  (  ._|_  `  ( M `  X
) ) )  =  (  ._|_  `  ( M `
 ( ( oc
`  K ) `  X ) ) ) )
7 hlop 30174 . . . 4  |-  ( K  e.  HL  ->  K  e.  OP )
81, 2opoccl 30006 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
97, 8sylan 457 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
101, 2, 3, 4polpmapN 30723 . . 3  |-  ( ( K  e.  HL  /\  ( ( oc `  K ) `  X
)  e.  B )  ->  (  ._|_  `  ( M `  ( ( oc `  K ) `  X ) ) )  =  ( M `  ( ( oc `  K ) `  (
( oc `  K
) `  X )
) ) )
119, 10syldan 456 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  (  ._|_  `  ( M `
 ( ( oc
`  K ) `  X ) ) )  =  ( M `  ( ( oc `  K ) `  (
( oc `  K
) `  X )
) ) )
121, 2opococ 30007 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  (
( oc `  K
) `  X )
)  =  X )
137, 12sylan 457 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( oc `  K ) `  (
( oc `  K
) `  X )
)  =  X )
1413fveq2d 5545 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  (
( oc `  K
) `  ( ( oc `  K ) `  X ) ) )  =  ( M `  X ) )
156, 11, 143eqtrd 2332 1  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  (  ._|_  `  (  ._|_  `  ( M `  X
) ) )  =  ( M `  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   ` cfv 5271   Basecbs 13164   occoc 13232   OPcops 29984   HLchlt 30162   pmapcpmap 30308   _|_ PcpolN 30713
This theorem is referenced by:  pmapsubclN  30757  ispsubcl2N  30758
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-rmo 2564  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-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-pmap 30315  df-polarityN 30714
  Copyright terms: Public domain W3C validator