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Theorem pol0N 35513
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 = (⊥𝑃𝐾)
Assertion
Ref Expression
pol0N (𝐾𝐵 → ( ‘∅) = 𝐴)

Proof of Theorem pol0N
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 0ss 4005 . . 3 ∅ ⊆ 𝐴
2 eqid 2651 . . . 4 (oc‘𝐾) = (oc‘𝐾)
3 polssat.a . . . 4 𝐴 = (Atoms‘𝐾)
4 eqid 2651 . . . 4 (pmap‘𝐾) = (pmap‘𝐾)
5 polssat.p . . . 4 = (⊥𝑃𝐾)
62, 3, 4, 5polvalN 35509 . . 3 ((𝐾𝐵 ∧ ∅ ⊆ 𝐴) → ( ‘∅) = (𝐴 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))))
71, 6mpan2 707 . 2 (𝐾𝐵 → ( ‘∅) = (𝐴 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))))
8 0iin 4610 . . . 4 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)) = V
98ineq2i 3844 . . 3 (𝐴 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))) = (𝐴 ∩ V)
10 inv1 4003 . . 3 (𝐴 ∩ V) = 𝐴
119, 10eqtri 2673 . 2 (𝐴 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))) = 𝐴
127, 11syl6eq 2701 1 (𝐾𝐵 → ( ‘∅) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  cin 3606  wss 3607  c0 3948   ciin 4553  cfv 5926  occoc 15996  Atomscatm 34868  pmapcpmap 35101  𝑃cpolN 35506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-polarityN 35507
This theorem is referenced by:  2pol0N  35515  1psubclN  35548  osumcllem9N  35568  pexmidN  35573  pexmidlem6N  35579  pexmidALTN  35582
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