Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  polfvalN Unicode version

Theorem polfvalN 30715
Description: The projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
polfval.o  |-  ._|_  =  ( oc `  K )
polfval.a  |-  A  =  ( Atoms `  K )
polfval.m  |-  M  =  ( pmap `  K
)
polfval.p  |-  P  =  ( _|_ P `  K )
Assertion
Ref Expression
polfvalN  |-  ( K  e.  B  ->  P  =  ( m  e. 
~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p ) ) ) ) )
Distinct variable groups:    A, m    m, p, K
Allowed substitution hints:    A( p)    B( m, p)    P( m, p)    M( m, p)    ._|_ ( m, p)

Proof of Theorem polfvalN
Dummy variable  h is distinct from all other variables.
StepHypRef Expression
1 elex 2809 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 polfval.p . . 3  |-  P  =  ( _|_ P `  K )
3 fveq2 5541 . . . . . . 7  |-  ( h  =  K  ->  ( Atoms `  h )  =  ( Atoms `  K )
)
4 polfval.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2346 . . . . . 6  |-  ( h  =  K  ->  ( Atoms `  h )  =  A )
65pweqd 3643 . . . . 5  |-  ( h  =  K  ->  ~P ( Atoms `  h )  =  ~P A )
7 fveq2 5541 . . . . . . . . . 10  |-  ( h  =  K  ->  ( pmap `  h )  =  ( pmap `  K
) )
8 polfval.m . . . . . . . . . 10  |-  M  =  ( pmap `  K
)
97, 8syl6eqr 2346 . . . . . . . . 9  |-  ( h  =  K  ->  ( pmap `  h )  =  M )
10 fveq2 5541 . . . . . . . . . . 11  |-  ( h  =  K  ->  ( oc `  h )  =  ( oc `  K
) )
11 polfval.o . . . . . . . . . . 11  |-  ._|_  =  ( oc `  K )
1210, 11syl6eqr 2346 . . . . . . . . . 10  |-  ( h  =  K  ->  ( oc `  h )  = 
._|_  )
1312fveq1d 5543 . . . . . . . . 9  |-  ( h  =  K  ->  (
( oc `  h
) `  p )  =  (  ._|_  `  p
) )
149, 13fveq12d 5547 . . . . . . . 8  |-  ( h  =  K  ->  (
( pmap `  h ) `  ( ( oc `  h ) `  p
) )  =  ( M `  (  ._|_  `  p ) ) )
1514adantr 451 . . . . . . 7  |-  ( ( h  =  K  /\  p  e.  m )  ->  ( ( pmap `  h
) `  ( ( oc `  h ) `  p ) )  =  ( M `  (  ._|_  `  p ) ) )
1615iineq2dv 3943 . . . . . 6  |-  ( h  =  K  ->  |^|_ p  e.  m  ( ( pmap `  h ) `  ( ( oc `  h ) `  p
) )  =  |^|_ p  e.  m  ( M `
 (  ._|_  `  p
) ) )
175, 16ineq12d 3384 . . . . 5  |-  ( h  =  K  ->  (
( Atoms `  h )  i^i  |^|_ p  e.  m  ( ( pmap `  h
) `  ( ( oc `  h ) `  p ) ) )  =  ( A  i^i  |^|_
p  e.  m  ( M `  (  ._|_  `  p ) ) ) )
186, 17mpteq12dv 4114 . . . 4  |-  ( h  =  K  ->  (
m  e.  ~P ( Atoms `  h )  |->  ( ( Atoms `  h )  i^i  |^|_ p  e.  m  ( ( pmap `  h
) `  ( ( oc `  h ) `  p ) ) ) )  =  ( m  e.  ~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p
) ) ) ) )
19 df-polarityN 30714 . . . 4  |-  _|_ P  =  ( h  e. 
_V  |->  ( m  e. 
~P ( Atoms `  h
)  |->  ( ( Atoms `  h )  i^i  |^|_ p  e.  m  ( (
pmap `  h ) `  ( ( oc `  h ) `  p
) ) ) ) )
20 fvex 5555 . . . . . . 7  |-  ( Atoms `  K )  e.  _V
214, 20eqeltri 2366 . . . . . 6  |-  A  e. 
_V
2221pwex 4209 . . . . 5  |-  ~P A  e.  _V
2322mptex 5762 . . . 4  |-  ( m  e.  ~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p
) ) ) )  e.  _V
2418, 19, 23fvmpt 5618 . . 3  |-  ( K  e.  _V  ->  ( _|_ P `  K )  =  ( m  e. 
~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p ) ) ) ) )
252, 24syl5eq 2340 . 2  |-  ( K  e.  _V  ->  P  =  ( m  e. 
~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p ) ) ) ) )
261, 25syl 15 1  |-  ( K  e.  B  ->  P  =  ( m  e. 
~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164   ~Pcpw 3638   |^|_ciin 3922    e. cmpt 4093   ` cfv 5271   occoc 13232   Atomscatm 30075   pmapcpmap 30308   _|_ PcpolN 30713
This theorem is referenced by:  polvalN  30716
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-polarityN 30714
  Copyright terms: Public domain W3C validator