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Theorem pol0N 39382
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 4397 . . 3 ∅ ⊆ 𝐴
2 eqid 2728 . . . 4 (oc‘𝐾) = (oc‘𝐾)
3 polssat.a . . . 4 𝐴 = (Atoms‘𝐾)
4 eqid 2728 . . . 4 (pmap‘𝐾) = (pmap‘𝐾)
5 polssat.p . . . 4 = (⊥𝑃𝐾)
62, 3, 4, 5polvalN 39378 . . 3 ((𝐾𝐵 ∧ ∅ ⊆ 𝐴) → ( ‘∅) = (𝐴 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))))
71, 6mpan2 690 . 2 (𝐾𝐵 → ( ‘∅) = (𝐴 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))))
8 0iin 5067 . . . 4 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)) = V
98ineq2i 4209 . . 3 (𝐴 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))) = (𝐴 ∩ V)
10 inv1 4395 . . 3 (𝐴 ∩ V) = 𝐴
119, 10eqtri 2756 . 2 (𝐴 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))) = 𝐴
127, 11eqtrdi 2784 1 (𝐾𝐵 → ( ‘∅) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  Vcvv 3471  cin 3946  wss 3947  c0 4323   ciin 4997  cfv 6548  occoc 17241  Atomscatm 38735  pmapcpmap 38970  𝑃cpolN 39375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-polarityN 39376
This theorem is referenced by:  2pol0N  39384  1psubclN  39417  osumcllem9N  39437  pexmidN  39442  pexmidlem6N  39448  pexmidALTN  39451
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