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Theorem pol0N 34580
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 3807 . . 3  |-  (/)  C_  A
2 eqid 2460 . . . 4  |-  ( oc
`  K )  =  ( oc `  K
)
3 polssat.a . . . 4  |-  A  =  ( Atoms `  K )
4 eqid 2460 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
5 polssat.p . . . 4  |-  ._|_  =  ( _|_P `  K
)
62, 3, 4, 5polvalN 34576 . . 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 4376 . . . 4  |-  |^|_ p  e.  (/)  ( ( pmap `  K ) `  (
( oc `  K
) `  p )
)  =  _V
98ineq2i 3690 . . 3  |-  ( A  i^i  |^|_ p  e.  (/)  ( ( pmap `  K
) `  ( ( oc `  K ) `  p ) ) )  =  ( A  i^i  _V )
10 inv1 3805 . . 3  |-  ( A  i^i  _V )  =  A
119, 10eqtri 2489 . 2  |-  ( A  i^i  |^|_ p  e.  (/)  ( ( pmap `  K
) `  ( ( oc `  K ) `  p ) ) )  =  A
127, 11syl6eq 2517 1  |-  ( K  e.  B  ->  (  ._|_  `  (/) )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   _Vcvv 3106    i^i cin 3468    C_ wss 3469   (/)c0 3778   |^|_ciin 4319   ` cfv 5579   occoc 14552   Atomscatm 33935   pmapcpmap 34168   _|_PcpolN 34573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-polarityN 34574
This theorem is referenced by:  2pol0N  34582  1psubclN  34615  osumcllem9N  34635  pexmidN  34640  pexmidlem6N  34646  pexmidALTN  34649
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