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Theorem polval2N 34995
Description: Alternate expression for value of the projective subspace polarity function. Equation for polarity in [Holland95] p. 223. (Contributed by NM, 22-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
polval2.u  |-  U  =  ( lub `  K
)
polval2.o  |-  ._|_  =  ( oc `  K )
polval2.a  |-  A  =  ( Atoms `  K )
polval2.m  |-  M  =  ( pmap `  K
)
polval2.p  |-  P  =  ( _|_P `  K )
Assertion
Ref Expression
polval2N  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( P `  X
)  =  ( M `
 (  ._|_  `  ( U `  X )
) ) )

Proof of Theorem polval2N
Dummy variables  x  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 polval2.o . . 3  |-  ._|_  =  ( oc `  K )
2 polval2.a . . 3  |-  A  =  ( Atoms `  K )
3 polval2.m . . 3  |-  M  =  ( pmap `  K
)
4 polval2.p . . 3  |-  P  =  ( _|_P `  K )
51, 2, 3, 4polvalN 34994 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( P `  X
)  =  ( A  i^i  |^|_ p  e.  X  ( M `  (  ._|_  `  p ) ) ) )
6 hlop 34452 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
76ad2antrr 725 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  K  e.  OP )
8 ssel2 3504 . . . . . . 7  |-  ( ( X  C_  A  /\  p  e.  X )  ->  p  e.  A )
98adantll 713 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  p  e.  A )
10 eqid 2467 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1110, 2atbase 34379 . . . . . 6  |-  ( p  e.  A  ->  p  e.  ( Base `  K
) )
129, 11syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  p  e.  ( Base `  K )
)
1310, 1opoccl 34284 . . . . 5  |-  ( ( K  e.  OP  /\  p  e.  ( Base `  K ) )  -> 
(  ._|_  `  p )  e.  ( Base `  K
) )
147, 12, 13syl2anc 661 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  (  ._|_  `  p )  e.  (
Base `  K )
)
1514ralrimiva 2881 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  A. p  e.  X  (  ._|_  `  p )  e.  ( Base `  K
) )
16 eqid 2467 . . . 4  |-  ( glb `  K )  =  ( glb `  K )
1710, 16, 2, 3pmapglb2xN 34861 . . 3  |-  ( ( K  e.  HL  /\  A. p  e.  X  ( 
._|_  `  p )  e.  ( Base `  K
) )  ->  ( M `  ( ( glb `  K ) `  { x  |  E. p  e.  X  x  =  (  ._|_  `  p
) } ) )  =  ( A  i^i  |^|_
p  e.  X  ( M `  (  ._|_  `  p ) ) ) )
1815, 17syldan 470 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( M `  (
( glb `  K
) `  { x  |  E. p  e.  X  x  =  (  ._|_  `  p ) } ) )  =  ( A  i^i  |^|_ p  e.  X  ( M `  (  ._|_  `  p ) ) ) )
19 polval2.u . . . . . 6  |-  U  =  ( lub `  K
)
2010, 19, 16, 1glbconxN 34467 . . . . 5  |-  ( ( K  e.  HL  /\  A. p  e.  X  ( 
._|_  `  p )  e.  ( Base `  K
) )  ->  (
( glb `  K
) `  { x  |  E. p  e.  X  x  =  (  ._|_  `  p ) } )  =  (  ._|_  `  ( U `  { x  |  E. p  e.  X  x  =  (  ._|_  `  (  ._|_  `  p ) ) } ) ) )
2115, 20syldan 470 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( glb `  K
) `  { x  |  E. p  e.  X  x  =  (  ._|_  `  p ) } )  =  (  ._|_  `  ( U `  { x  |  E. p  e.  X  x  =  (  ._|_  `  (  ._|_  `  p ) ) } ) ) )
2210, 1opococ 34285 . . . . . . . . . . 11  |-  ( ( K  e.  OP  /\  p  e.  ( Base `  K ) )  -> 
(  ._|_  `  (  ._|_  `  p ) )  =  p )
237, 12, 22syl2anc 661 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  (  ._|_  `  (  ._|_  `  p ) )  =  p )
2423eqeq2d 2481 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  ( x  =  (  ._|_  `  (  ._|_  `  p ) )  <-> 
x  =  p ) )
2524rexbidva 2975 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( E. p  e.  X  x  =  ( 
._|_  `  (  ._|_  `  p
) )  <->  E. p  e.  X  x  =  p ) )
2625abbidv 2603 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  { x  |  E. p  e.  X  x  =  (  ._|_  `  (  ._|_  `  p ) ) }  =  { x  |  E. p  e.  X  x  =  p }
)
27 df-rex 2823 . . . . . . . . . 10  |-  ( E. p  e.  X  x  =  p  <->  E. p
( p  e.  X  /\  x  =  p
) )
28 equcom 1743 . . . . . . . . . . . . 13  |-  ( x  =  p  <->  p  =  x )
2928anbi2i 694 . . . . . . . . . . . 12  |-  ( ( p  e.  X  /\  x  =  p )  <->  ( p  e.  X  /\  p  =  x )
)
30 ancom 450 . . . . . . . . . . . 12  |-  ( ( p  e.  X  /\  p  =  x )  <->  ( p  =  x  /\  p  e.  X )
)
3129, 30bitri 249 . . . . . . . . . . 11  |-  ( ( p  e.  X  /\  x  =  p )  <->  ( p  =  x  /\  p  e.  X )
)
3231exbii 1644 . . . . . . . . . 10  |-  ( E. p ( p  e.  X  /\  x  =  p )  <->  E. p
( p  =  x  /\  p  e.  X
) )
33 vex 3121 . . . . . . . . . . 11  |-  x  e. 
_V
34 eleq1 2539 . . . . . . . . . . 11  |-  ( p  =  x  ->  (
p  e.  X  <->  x  e.  X ) )
3533, 34ceqsexv 3155 . . . . . . . . . 10  |-  ( E. p ( p  =  x  /\  p  e.  X )  <->  x  e.  X )
3627, 32, 353bitri 271 . . . . . . . . 9  |-  ( E. p  e.  X  x  =  p  <->  x  e.  X )
3736abbii 2601 . . . . . . . 8  |-  { x  |  E. p  e.  X  x  =  p }  =  { x  |  x  e.  X }
38 abid2 2607 . . . . . . . 8  |-  { x  |  x  e.  X }  =  X
3937, 38eqtri 2496 . . . . . . 7  |-  { x  |  E. p  e.  X  x  =  p }  =  X
4026, 39syl6eq 2524 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  { x  |  E. p  e.  X  x  =  (  ._|_  `  (  ._|_  `  p ) ) }  =  X )
4140fveq2d 5875 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( U `  {
x  |  E. p  e.  X  x  =  (  ._|_  `  (  ._|_  `  p ) ) } )  =  ( U `
 X ) )
4241fveq2d 5875 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  ( U `  { x  |  E. p  e.  X  x  =  (  ._|_  `  (  ._|_  `  p ) ) } ) )  =  (  ._|_  `  ( U `
 X ) ) )
4321, 42eqtrd 2508 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( glb `  K
) `  { x  |  E. p  e.  X  x  =  (  ._|_  `  p ) } )  =  (  ._|_  `  ( U `  X )
) )
4443fveq2d 5875 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( M `  (
( glb `  K
) `  { x  |  E. p  e.  X  x  =  (  ._|_  `  p ) } ) )  =  ( M `
 (  ._|_  `  ( U `  X )
) ) )
455, 18, 443eqtr2d 2514 1  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( P `  X
)  =  ( M `
 (  ._|_  `  ( U `  X )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452   A.wral 2817   E.wrex 2818    i^i cin 3480    C_ wss 3481   |^|_ciin 4331   ` cfv 5593   Basecbs 14502   occoc 14575   lubclub 15441   glbcglb 15442   OPcops 34262   Atomscatm 34353   HLchlt 34440   pmapcpmap 34586   _|_PcpolN 34991
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 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-riotaBAD 34049
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-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-iin 4333  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-undef 7012  df-poset 15445  df-lub 15473  df-glb 15474  df-join 15475  df-meet 15476  df-p1 15539  df-lat 15545  df-clat 15607  df-oposet 34266  df-ol 34268  df-oml 34269  df-ats 34357  df-hlat 34441  df-pmap 34593  df-polarityN 34992
This theorem is referenced by:  polsubN  34996  pol1N  34999  polpmapN  35001  2polvalN  35003  3polN  35005  poldmj1N  35017  pnonsingN  35022  ispsubcl2N  35036  polsubclN  35041  poml4N  35042
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