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Theorem pol0N 33393
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 3791 . . 3  |-  (/)  C_  A
2 eqid 2422 . . . 4  |-  ( oc
`  K )  =  ( oc `  K
)
3 polssat.a . . . 4  |-  A  =  ( Atoms `  K )
4 eqid 2422 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
5 polssat.p . . . 4  |-  ._|_  =  ( _|_P `  K
)
62, 3, 4, 5polvalN 33389 . . 3  |-  ( ( K  e.  B  /\  (/)  C_  A )  ->  (  ._|_  `  (/) )  =  ( A  i^i  |^|_ p  e.  (/)  ( ( pmap `  K ) `  (
( oc `  K
) `  p )
) ) )
71, 6mpan2 675 . 2  |-  ( K  e.  B  ->  (  ._|_  `  (/) )  =  ( A  i^i  |^|_ p  e.  (/)  ( ( pmap `  K ) `  (
( oc `  K
) `  p )
) ) )
8 0iin 4354 . . . 4  |-  |^|_ p  e.  (/)  ( ( pmap `  K ) `  (
( oc `  K
) `  p )
)  =  _V
98ineq2i 3661 . . 3  |-  ( A  i^i  |^|_ p  e.  (/)  ( ( pmap `  K
) `  ( ( oc `  K ) `  p ) ) )  =  ( A  i^i  _V )
10 inv1 3789 . . 3  |-  ( A  i^i  _V )  =  A
119, 10eqtri 2451 . 2  |-  ( A  i^i  |^|_ p  e.  (/)  ( ( pmap `  K
) `  ( ( oc `  K ) `  p ) ) )  =  A
127, 11syl6eq 2479 1  |-  ( K  e.  B  ->  (  ._|_  `  (/) )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1868   _Vcvv 3081    i^i cin 3435    C_ wss 3436   (/)c0 3761   |^|_ciin 4297   ` cfv 5598   occoc 15186   Atomscatm 32748   pmapcpmap 32981   _|_PcpolN 33386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4765  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-polarityN 33387
This theorem is referenced by:  2pol0N  33395  1psubclN  33428  osumcllem9N  33448  pexmidN  33453  pexmidlem6N  33459  pexmidALTN  33462
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