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Theorem pol0N 30098
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 3483 . . 3  |-  (/)  C_  A
2 eqid 2283 . . . 4  |-  ( oc
`  K )  =  ( oc `  K
)
3 polssat.a . . . 4  |-  A  =  ( Atoms `  K )
4 eqid 2283 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
5 polssat.p . . . 4  |-  ._|_  =  ( _|_ P `  K
)
62, 3, 4, 5polvalN 30094 . . 3  |-  ( ( K  e.  B  /\  (/)  C_  A )  ->  (  ._|_  `  (/) )  =  ( A  i^i  |^|_ p  e.  (/)  ( ( pmap `  K ) `  (
( oc `  K
) `  p )
) ) )
71, 6mpan2 652 . 2  |-  ( K  e.  B  ->  (  ._|_  `  (/) )  =  ( A  i^i  |^|_ p  e.  (/)  ( ( pmap `  K ) `  (
( oc `  K
) `  p )
) ) )
8 0iin 3960 . . . 4  |-  |^|_ p  e.  (/)  ( ( pmap `  K ) `  (
( oc `  K
) `  p )
)  =  _V
98ineq2i 3367 . . 3  |-  ( A  i^i  |^|_ p  e.  (/)  ( ( pmap `  K
) `  ( ( oc `  K ) `  p ) ) )  =  ( A  i^i  _V )
10 inv1 3481 . . 3  |-  ( A  i^i  _V )  =  A
119, 10eqtri 2303 . 2  |-  ( A  i^i  |^|_ p  e.  (/)  ( ( pmap `  K
) `  ( ( oc `  K ) `  p ) ) )  =  A
127, 11syl6eq 2331 1  |-  ( K  e.  B  ->  (  ._|_  `  (/) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152   (/)c0 3455   |^|_ciin 3906   ` cfv 5255   occoc 13216   Atomscatm 29453   pmapcpmap 29686   _|_ PcpolN 30091
This theorem is referenced by:  2pol0N  30100  1psubclN  30133  osumcllem9N  30153  pexmidN  30158  pexmidlem6N  30164  pexmidALTN  30167
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-polarityN 30092
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