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Theorem polval2N 30095
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 30094 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( P `  X
)  =  ( A  i^i  |^|_ p  e.  X  ( M `  (  ._|_  `  p ) ) ) )
6 hlop 29552 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
76ad2antrr 706 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  K  e.  OP )
8 ssel2 3175 . . . . . . 7  |-  ( ( X  C_  A  /\  p  e.  X )  ->  p  e.  A )
98adantll 694 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  p  e.  A )
10 eqid 2283 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1110, 2atbase 29479 . . . . . 6  |-  ( p  e.  A  ->  p  e.  ( Base `  K
) )
129, 11syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  p  e.  ( Base `  K )
)
1310, 1opoccl 29384 . . . . 5  |-  ( ( K  e.  OP  /\  p  e.  ( Base `  K ) )  -> 
(  ._|_  `  p )  e.  ( Base `  K
) )
147, 12, 13syl2anc 642 . . . 4  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  (  ._|_  `  p )  e.  (
Base `  K )
)
1514ralrimiva 2626 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  A. p  e.  X  (  ._|_  `  p )  e.  ( Base `  K
) )
16 eqid 2283 . . . 4  |-  ( glb `  K )  =  ( glb `  K )
1710, 16, 2, 3pmapglb2xN 29961 . . 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 456 . 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 29567 . . . . 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 456 . . . 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 29385 . . . . . . . . . . 11  |-  ( ( K  e.  OP  /\  p  e.  ( Base `  K ) )  -> 
(  ._|_  `  (  ._|_  `  p ) )  =  p )
237, 12, 22syl2anc 642 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  (  ._|_  `  (  ._|_  `  p ) )  =  p )
2423eqeq2d 2294 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  ( x  =  (  ._|_  `  (  ._|_  `  p ) )  <-> 
x  =  p ) )
2524rexbidva 2560 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( E. p  e.  X  x  =  ( 
._|_  `  (  ._|_  `  p
) )  <->  E. p  e.  X  x  =  p ) )
2625abbidv 2397 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  { x  |  E. p  e.  X  x  =  (  ._|_  `  (  ._|_  `  p ) ) }  =  { x  |  E. p  e.  X  x  =  p }
)
27 df-rex 2549 . . . . . . . . . 10  |-  ( E. p  e.  X  x  =  p  <->  E. p
( p  e.  X  /\  x  =  p
) )
28 equcom 1647 . . . . . . . . . . . . 13  |-  ( x  =  p  <->  p  =  x )
2928anbi2i 675 . . . . . . . . . . . 12  |-  ( ( p  e.  X  /\  x  =  p )  <->  ( p  e.  X  /\  p  =  x )
)
30 ancom 437 . . . . . . . . . . . 12  |-  ( ( p  e.  X  /\  p  =  x )  <->  ( p  =  x  /\  p  e.  X )
)
3129, 30bitri 240 . . . . . . . . . . 11  |-  ( ( p  e.  X  /\  x  =  p )  <->  ( p  =  x  /\  p  e.  X )
)
3231exbii 1569 . . . . . . . . . 10  |-  ( E. p ( p  e.  X  /\  x  =  p )  <->  E. p
( p  =  x  /\  p  e.  X
) )
33 vex 2791 . . . . . . . . . . 11  |-  x  e. 
_V
34 eleq1 2343 . . . . . . . . . . 11  |-  ( p  =  x  ->  (
p  e.  X  <->  x  e.  X ) )
3533, 34ceqsexv 2823 . . . . . . . . . 10  |-  ( E. p ( p  =  x  /\  p  e.  X )  <->  x  e.  X )
3627, 32, 353bitri 262 . . . . . . . . 9  |-  ( E. p  e.  X  x  =  p  <->  x  e.  X )
3736abbii 2395 . . . . . . . 8  |-  { x  |  E. p  e.  X  x  =  p }  =  { x  |  x  e.  X }
38 abid2 2400 . . . . . . . 8  |-  { x  |  x  e.  X }  =  X
3937, 38eqtri 2303 . . . . . . 7  |-  { x  |  E. p  e.  X  x  =  p }  =  X
4026, 39syl6eq 2331 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  { x  |  E. p  e.  X  x  =  (  ._|_  `  (  ._|_  `  p ) ) }  =  X )
4140fveq2d 5529 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( U `  {
x  |  E. p  e.  X  x  =  (  ._|_  `  (  ._|_  `  p ) ) } )  =  ( U `
 X ) )
4241fveq2d 5529 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  ( U `  { x  |  E. p  e.  X  x  =  (  ._|_  `  (  ._|_  `  p ) ) } ) )  =  (  ._|_  `  ( U `
 X ) ) )
4321, 42eqtrd 2315 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( glb `  K
) `  { x  |  E. p  e.  X  x  =  (  ._|_  `  p ) } )  =  (  ._|_  `  ( U `  X )
) )
4443fveq2d 5529 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( M `  (
( glb `  K
) `  { x  |  E. p  e.  X  x  =  (  ._|_  `  p ) } ) )  =  ( M `
 (  ._|_  `  ( U `  X )
) ) )
455, 18, 443eqtr2d 2321 1  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( P `  X
)  =  ( M `
 (  ._|_  `  ( U `  X )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   |^|_ciin 3906   ` cfv 5255   Basecbs 13148   occoc 13216   lubclub 14076   glbcglb 14077   OPcops 29362   Atomscatm 29453   HLchlt 29540   pmapcpmap 29686   _|_ PcpolN 30091
This theorem is referenced by:  polsubN  30096  pol1N  30099  polpmapN  30101  2polvalN  30103  3polN  30105  poldmj1N  30117  pnonsingN  30122  ispsubcl2N  30136  polsubclN  30141  poml4N  30142
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-ats 29457  df-hlat 29541  df-pmap 29693  df-polarityN 30092
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