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Theorem pol0N 33892
Description: The polarity of the empty projective subspace is the whole space. (Contributed by NM, 29-Oct-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
polssat.a  |-  A  =  ( Atoms `  K )
polssat.p  |-  ._|_  =  ( _|_P `  K
)
Assertion
Ref Expression
pol0N  |-  ( K  e.  B  ->  (  ._|_  `  (/) )  =  A )

Proof of Theorem pol0N
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 0ss 3775 . . 3  |-  (/)  C_  A
2 eqid 2454 . . . 4  |-  ( oc
`  K )  =  ( oc `  K
)
3 polssat.a . . . 4  |-  A  =  ( Atoms `  K )
4 eqid 2454 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
5 polssat.p . . . 4  |-  ._|_  =  ( _|_P `  K
)
62, 3, 4, 5polvalN 33888 . . 3  |-  ( ( K  e.  B  /\  (/)  C_  A )  ->  (  ._|_  `  (/) )  =  ( A  i^i  |^|_ p  e.  (/)  ( ( pmap `  K ) `  (
( oc `  K
) `  p )
) ) )
71, 6mpan2 671 . 2  |-  ( K  e.  B  ->  (  ._|_  `  (/) )  =  ( A  i^i  |^|_ p  e.  (/)  ( ( pmap `  K ) `  (
( oc `  K
) `  p )
) ) )
8 0iin 4337 . . . 4  |-  |^|_ p  e.  (/)  ( ( pmap `  K ) `  (
( oc `  K
) `  p )
)  =  _V
98ineq2i 3658 . . 3  |-  ( A  i^i  |^|_ p  e.  (/)  ( ( pmap `  K
) `  ( ( oc `  K ) `  p ) ) )  =  ( A  i^i  _V )
10 inv1 3773 . . 3  |-  ( A  i^i  _V )  =  A
119, 10eqtri 2483 . 2  |-  ( A  i^i  |^|_ p  e.  (/)  ( ( pmap `  K
) `  ( ( oc `  K ) `  p ) ) )  =  A
127, 11syl6eq 2511 1  |-  ( K  e.  B  ->  (  ._|_  `  (/) )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3078    i^i cin 3436    C_ wss 3437   (/)c0 3746   |^|_ciin 4281   ` cfv 5527   occoc 14366   Atomscatm 33247   pmapcpmap 33480   _|_PcpolN 33885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-iin 4283  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-polarityN 33886
This theorem is referenced by:  2pol0N  33894  1psubclN  33927  osumcllem9N  33947  pexmidN  33952  pexmidlem6N  33958  pexmidALTN  33961
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