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Theorem polval2N 29246
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
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 29245 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( P `  X
)  =  ( A  i^i  |^|_ p  e.  X  ( M `  (  ._|_  `  p ) ) ) )
6 hlop 28703 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
76ad2antrr 709 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  K  e.  OP )
8 ssel2 3136 . . . . . . 7  |-  ( ( X  C_  A  /\  p  e.  X )  ->  p  e.  A )
98adantll 697 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  p  e.  A )
10 eqid 2256 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1110, 2atbase 28630 . . . . . 6  |-  ( p  e.  A  ->  p  e.  ( Base `  K
) )
129, 11syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  p  e.  ( Base `  K )
)
1310, 1opoccl 28535 . . . . 5  |-  ( ( K  e.  OP  /\  p  e.  ( Base `  K ) )  -> 
(  ._|_  `  p )  e.  ( Base `  K
) )
147, 12, 13syl2anc 645 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  (  ._|_  `  p )  e.  (
Base `  K )
)
1514ralrimiva 2599 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  A. p  e.  X  (  ._|_  `  p )  e.  ( Base `  K
) )
16 eqid 2256 . . . 4  |-  ( glb `  K )  =  ( glb `  K )
1710, 16, 2, 3pmapglb2xN 29112 . . 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 458 . 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 28718 . . . . 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 458 . . . 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 28536 . . . . . . . . . . 11  |-  ( ( K  e.  OP  /\  p  e.  ( Base `  K ) )  -> 
(  ._|_  `  (  ._|_  `  p ) )  =  p )
237, 12, 22syl2anc 645 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  (  ._|_  `  (  ._|_  `  p ) )  =  p )
2423eqeq2d 2267 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  ( x  =  (  ._|_  `  (  ._|_  `  p ) )  <-> 
x  =  p ) )
2524rexbidva 2533 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( E. p  e.  X  x  =  ( 
._|_  `  (  ._|_  `  p
) )  <->  E. p  e.  X  x  =  p ) )
2625abbidv 2370 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  { x  |  E. p  e.  X  x  =  (  ._|_  `  (  ._|_  `  p ) ) }  =  { x  |  E. p  e.  X  x  =  p }
)
27 df-rex 2522 . . . . . . . . . 10  |-  ( E. p  e.  X  x  =  p  <->  E. p
( p  e.  X  /\  x  =  p
) )
28 equcom 1824 . . . . . . . . . . . . 13  |-  ( x  =  p  <->  p  =  x )
2928anbi2i 678 . . . . . . . . . . . 12  |-  ( ( p  e.  X  /\  x  =  p )  <->  ( p  e.  X  /\  p  =  x )
)
30 ancom 439 . . . . . . . . . . . 12  |-  ( ( p  e.  X  /\  p  =  x )  <->  ( p  =  x  /\  p  e.  X )
)
3129, 30bitri 242 . . . . . . . . . . 11  |-  ( ( p  e.  X  /\  x  =  p )  <->  ( p  =  x  /\  p  e.  X )
)
3231exbii 1580 . . . . . . . . . 10  |-  ( E. p ( p  e.  X  /\  x  =  p )  <->  E. p
( p  =  x  /\  p  e.  X
) )
33 vex 2760 . . . . . . . . . . 11  |-  x  e. 
_V
34 eleq1 2316 . . . . . . . . . . 11  |-  ( p  =  x  ->  (
p  e.  X  <->  x  e.  X ) )
3533, 34ceqsexv 2791 . . . . . . . . . 10  |-  ( E. p ( p  =  x  /\  p  e.  X )  <->  x  e.  X )
3627, 32, 353bitri 264 . . . . . . . . 9  |-  ( E. p  e.  X  x  =  p  <->  x  e.  X )
3736abbii 2368 . . . . . . . 8  |-  { x  |  E. p  e.  X  x  =  p }  =  { x  |  x  e.  X }
38 abid2 2373 . . . . . . . 8  |-  { x  |  x  e.  X }  =  X
3937, 38eqtri 2276 . . . . . . 7  |-  { x  |  E. p  e.  X  x  =  p }  =  X
4026, 39syl6eq 2304 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  { x  |  E. p  e.  X  x  =  (  ._|_  `  (  ._|_  `  p ) ) }  =  X )
4140fveq2d 5448 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( U `  {
x  |  E. p  e.  X  x  =  (  ._|_  `  (  ._|_  `  p ) ) } )  =  ( U `
 X ) )
4241fveq2d 5448 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  ( U `  { x  |  E. p  e.  X  x  =  (  ._|_  `  (  ._|_  `  p ) ) } ) )  =  (  ._|_  `  ( U `
 X ) ) )
4321, 42eqtrd 2288 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( glb `  K
) `  { x  |  E. p  e.  X  x  =  (  ._|_  `  p ) } )  =  (  ._|_  `  ( U `  X )
) )
4443fveq2d 5448 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( M `  (
( glb `  K
) `  { x  |  E. p  e.  X  x  =  (  ._|_  `  p ) } ) )  =  ( M `
 (  ._|_  `  ( U `  X )
) ) )
455, 18, 443eqtr2d 2294 1  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( P `  X
)  =  ( M `
 (  ._|_  `  ( U `  X )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621   {cab 2242   A.wral 2516   E.wrex 2517    i^i cin 3112    C_ wss 3113   |^|_ciin 3866   ` cfv 4659   Basecbs 13096   occoc 13164   lubclub 14024   glbcglb 14025   OPcops 28513   Atomscatm 28604   HLchlt 28691   pmapcpmap 28837   _|_ PcpolN 29242
This theorem is referenced by:  polsubN  29247  pol1N  29250  polpmapN  29252  2polvalN  29254  3polN  29256  poldmj1N  29268  pnonsingN  29273  ispsubcl2N  29287  polsubclN  29292  poml4N  29293
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-undef 6250  df-riota 6258  df-poset 14028  df-lub 14056  df-glb 14057  df-join 14058  df-meet 14059  df-p1 14094  df-lat 14100  df-clat 14162  df-oposet 28517  df-ol 28519  df-oml 28520  df-ats 28608  df-hlat 28692  df-pmap 28844  df-polarityN 29243
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