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Theorem pol0N 34213
 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 3924 . . 3 ∅ ⊆ 𝐴
2 eqid 2610 . . . 4 (oc‘𝐾) = (oc‘𝐾)
3 polssat.a . . . 4 𝐴 = (Atoms‘𝐾)
4 eqid 2610 . . . 4 (pmap‘𝐾) = (pmap‘𝐾)
5 polssat.p . . . 4 = (⊥𝑃𝐾)
62, 3, 4, 5polvalN 34209 . . 3 ((𝐾𝐵 ∧ ∅ ⊆ 𝐴) → ( ‘∅) = (𝐴 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))))
71, 6mpan2 703 . 2 (𝐾𝐵 → ( ‘∅) = (𝐴 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))))
8 0iin 4514 . . . 4 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)) = V
98ineq2i 3773 . . 3 (𝐴 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))) = (𝐴 ∩ V)
10 inv1 3922 . . 3 (𝐴 ∩ V) = 𝐴
119, 10eqtri 2632 . 2 (𝐴 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))) = 𝐴
127, 11syl6eq 2660 1 (𝐾𝐵 → ( ‘∅) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ∩ ciin 4456  ‘cfv 5804  occoc 15776  Atomscatm 33568  pmapcpmap 33801  ⊥𝑃cpolN 34206 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-polarityN 34207 This theorem is referenced by:  2pol0N  34215  1psubclN  34248  osumcllem9N  34268  pexmidN  34273  pexmidlem6N  34279  pexmidALTN  34282
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