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Theorem pol0N 30768
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 3658 . . 3  |-  (/)  C_  A
2 eqid 2438 . . . 4  |-  ( oc
`  K )  =  ( oc `  K
)
3 polssat.a . . . 4  |-  A  =  ( Atoms `  K )
4 eqid 2438 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
5 polssat.p . . . 4  |-  ._|_  =  ( _|_ P `  K
)
62, 3, 4, 5polvalN 30764 . . 3  |-  ( ( K  e.  B  /\  (/)  C_  A )  ->  (  ._|_  `  (/) )  =  ( A  i^i  |^|_ p  e.  (/)  ( ( pmap `  K ) `  (
( oc `  K
) `  p )
) ) )
71, 6mpan2 654 . 2  |-  ( K  e.  B  ->  (  ._|_  `  (/) )  =  ( A  i^i  |^|_ p  e.  (/)  ( ( pmap `  K ) `  (
( oc `  K
) `  p )
) ) )
8 0iin 4151 . . . 4  |-  |^|_ p  e.  (/)  ( ( pmap `  K ) `  (
( oc `  K
) `  p )
)  =  _V
98ineq2i 3541 . . 3  |-  ( A  i^i  |^|_ p  e.  (/)  ( ( pmap `  K
) `  ( ( oc `  K ) `  p ) ) )  =  ( A  i^i  _V )
10 inv1 3656 . . 3  |-  ( A  i^i  _V )  =  A
119, 10eqtri 2458 . 2  |-  ( A  i^i  |^|_ p  e.  (/)  ( ( pmap `  K
) `  ( ( oc `  K ) `  p ) ) )  =  A
127, 11syl6eq 2486 1  |-  ( K  e.  B  ->  (  ._|_  `  (/) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2958    i^i cin 3321    C_ wss 3322   (/)c0 3630   |^|_ciin 4096   ` cfv 5456   occoc 13539   Atomscatm 30123   pmapcpmap 30356   _|_ PcpolN 30761
This theorem is referenced by:  2pol0N  30770  1psubclN  30803  osumcllem9N  30823  pexmidN  30828  pexmidlem6N  30834  pexmidALTN  30837
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-polarityN 30762
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