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Theorem polval2N 30022
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 30021 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( P `  X
)  =  ( A  i^i  |^|_ p  e.  X  ( M `  (  ._|_  `  p ) ) ) )
6 hlop 29479 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
76ad2antrr 707 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  K  e.  OP )
8 ssel2 3288 . . . . . . 7  |-  ( ( X  C_  A  /\  p  e.  X )  ->  p  e.  A )
98adantll 695 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  p  e.  A )
10 eqid 2389 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1110, 2atbase 29406 . . . . . 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 29311 . . . . 5  |-  ( ( K  e.  OP  /\  p  e.  ( Base `  K ) )  -> 
(  ._|_  `  p )  e.  ( Base `  K
) )
147, 12, 13syl2anc 643 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  (  ._|_  `  p )  e.  (
Base `  K )
)
1514ralrimiva 2734 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  A. p  e.  X  (  ._|_  `  p )  e.  ( Base `  K
) )
16 eqid 2389 . . . 4  |-  ( glb `  K )  =  ( glb `  K )
1710, 16, 2, 3pmapglb2xN 29888 . . 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 457 . 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 29494 . . . . 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 457 . . . 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 29312 . . . . . . . . . . 11  |-  ( ( K  e.  OP  /\  p  e.  ( Base `  K ) )  -> 
(  ._|_  `  (  ._|_  `  p ) )  =  p )
237, 12, 22syl2anc 643 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  (  ._|_  `  (  ._|_  `  p ) )  =  p )
2423eqeq2d 2400 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  ( x  =  (  ._|_  `  (  ._|_  `  p ) )  <-> 
x  =  p ) )
2524rexbidva 2668 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( E. p  e.  X  x  =  ( 
._|_  `  (  ._|_  `  p
) )  <->  E. p  e.  X  x  =  p ) )
2625abbidv 2503 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  { x  |  E. p  e.  X  x  =  (  ._|_  `  (  ._|_  `  p ) ) }  =  { x  |  E. p  e.  X  x  =  p }
)
27 df-rex 2657 . . . . . . . . . 10  |-  ( E. p  e.  X  x  =  p  <->  E. p
( p  e.  X  /\  x  =  p
) )
28 equcom 1687 . . . . . . . . . . . . 13  |-  ( x  =  p  <->  p  =  x )
2928anbi2i 676 . . . . . . . . . . . 12  |-  ( ( p  e.  X  /\  x  =  p )  <->  ( p  e.  X  /\  p  =  x )
)
30 ancom 438 . . . . . . . . . . . 12  |-  ( ( p  e.  X  /\  p  =  x )  <->  ( p  =  x  /\  p  e.  X )
)
3129, 30bitri 241 . . . . . . . . . . 11  |-  ( ( p  e.  X  /\  x  =  p )  <->  ( p  =  x  /\  p  e.  X )
)
3231exbii 1589 . . . . . . . . . 10  |-  ( E. p ( p  e.  X  /\  x  =  p )  <->  E. p
( p  =  x  /\  p  e.  X
) )
33 vex 2904 . . . . . . . . . . 11  |-  x  e. 
_V
34 eleq1 2449 . . . . . . . . . . 11  |-  ( p  =  x  ->  (
p  e.  X  <->  x  e.  X ) )
3533, 34ceqsexv 2936 . . . . . . . . . 10  |-  ( E. p ( p  =  x  /\  p  e.  X )  <->  x  e.  X )
3627, 32, 353bitri 263 . . . . . . . . 9  |-  ( E. p  e.  X  x  =  p  <->  x  e.  X )
3736abbii 2501 . . . . . . . 8  |-  { x  |  E. p  e.  X  x  =  p }  =  { x  |  x  e.  X }
38 abid2 2506 . . . . . . . 8  |-  { x  |  x  e.  X }  =  X
3937, 38eqtri 2409 . . . . . . 7  |-  { x  |  E. p  e.  X  x  =  p }  =  X
4026, 39syl6eq 2437 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  { x  |  E. p  e.  X  x  =  (  ._|_  `  (  ._|_  `  p ) ) }  =  X )
4140fveq2d 5674 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( U `  {
x  |  E. p  e.  X  x  =  (  ._|_  `  (  ._|_  `  p ) ) } )  =  ( U `
 X ) )
4241fveq2d 5674 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  ( U `  { x  |  E. p  e.  X  x  =  (  ._|_  `  (  ._|_  `  p ) ) } ) )  =  (  ._|_  `  ( U `
 X ) ) )
4321, 42eqtrd 2421 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( glb `  K
) `  { x  |  E. p  e.  X  x  =  (  ._|_  `  p ) } )  =  (  ._|_  `  ( U `  X )
) )
4443fveq2d 5674 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( M `  (
( glb `  K
) `  { x  |  E. p  e.  X  x  =  (  ._|_  `  p ) } ) )  =  ( M `
 (  ._|_  `  ( U `  X )
) ) )
455, 18, 443eqtr2d 2427 1  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( P `  X
)  =  ( M `
 (  ._|_  `  ( U `  X )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2375   A.wral 2651   E.wrex 2652    i^i cin 3264    C_ wss 3265   |^|_ciin 4038   ` cfv 5396   Basecbs 13398   occoc 13466   lubclub 14328   glbcglb 14329   OPcops 29289   Atomscatm 29380   HLchlt 29467   pmapcpmap 29613   _|_ PcpolN 30018
This theorem is referenced by:  polsubN  30023  pol1N  30026  polpmapN  30028  2polvalN  30030  3polN  30032  poldmj1N  30044  pnonsingN  30049  ispsubcl2N  30063  polsubclN  30068  poml4N  30069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-poset 14332  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-p1 14398  df-lat 14404  df-clat 14466  df-oposet 29293  df-ol 29295  df-oml 29296  df-ats 29384  df-hlat 29468  df-pmap 29620  df-polarityN 30019
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