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Theorem polval2N 28784
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 28783 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( P `  X
)  =  ( A  i^i  |^|_ p  e.  X  ( M `  (  ._|_  `  p ) ) ) )
6 hlop 28241 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
76ad2antrr 709 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  K  e.  OP )
8 ssel2 3098 . . . . . . 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 2253 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1110, 2atbase 28168 . . . . . 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 28073 . . . . 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 2588 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  A. p  e.  X  (  ._|_  `  p )  e.  ( Base `  K
) )
16 eqid 2253 . . . 4  |-  ( glb `  K )  =  ( glb `  K )
1710, 16, 2, 3pmapglb2xN 28650 . . 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 28256 . . . . 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 28074 . . . . . . . . . . 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 2264 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  X
)  ->  ( x  =  (  ._|_  `  (  ._|_  `  p ) )  <-> 
x  =  p ) )
2524rexbidva 2524 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( E. p  e.  X  x  =  ( 
._|_  `  (  ._|_  `  p
) )  <->  E. p  e.  X  x  =  p ) )
2625abbidv 2363 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  { x  |  E. p  e.  X  x  =  (  ._|_  `  (  ._|_  `  p ) ) }  =  { x  |  E. p  e.  X  x  =  p }
)
27 df-rex 2514 . . . . . . . . . 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 2730 . . . . . . . . . . 11  |-  x  e. 
_V
34 eleq1 2313 . . . . . . . . . . 11  |-  ( p  =  x  ->  (
p  e.  X  <->  x  e.  X ) )
3533, 34ceqsexv 2761 . . . . . . . . . 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 2361 . . . . . . . 8  |-  { x  |  E. p  e.  X  x  =  p }  =  { x  |  x  e.  X }
38 abid2 2366 . . . . . . . 8  |-  { x  |  x  e.  X }  =  X
3937, 38eqtri 2273 . . . . . . 7  |-  { x  |  E. p  e.  X  x  =  p }  =  X
4026, 39syl6eq 2301 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  { x  |  E. p  e.  X  x  =  (  ._|_  `  (  ._|_  `  p ) ) }  =  X )
4140fveq2d 5381 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( U `  {
x  |  E. p  e.  X  x  =  (  ._|_  `  (  ._|_  `  p ) ) } )  =  ( U `
 X ) )
4241fveq2d 5381 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  ( U `  { x  |  E. p  e.  X  x  =  (  ._|_  `  (  ._|_  `  p ) ) } ) )  =  (  ._|_  `  ( U `
 X ) ) )
4321, 42eqtrd 2285 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( glb `  K
) `  { x  |  E. p  e.  X  x  =  (  ._|_  `  p ) } )  =  (  ._|_  `  ( U `  X )
) )
4443fveq2d 5381 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( M `  (
( glb `  K
) `  { x  |  E. p  e.  X  x  =  (  ._|_  `  p ) } ) )  =  ( M `
 (  ._|_  `  ( U `  X )
) ) )
455, 18, 443eqtr2d 2291 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 2239   A.wral 2509   E.wrex 2510    i^i cin 3077    C_ wss 3078   |^|_ciin 3804   ` cfv 4592   Basecbs 13022   occoc 13090   lubclub 13920   glbcglb 13921   OPcops 28051   Atomscatm 28142   HLchlt 28229   pmapcpmap 28375   _|_ PcpolN 28780
This theorem is referenced by:  polsubN  28785  pol1N  28788  polpmapN  28790  2polvalN  28792  3polN  28794  poldmj1N  28806  pnonsingN  28811  ispsubcl2N  28825  polsubclN  28830  poml4N  28831
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-ats 28146  df-hlat 28230  df-pmap 28382  df-polarityN 28781
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