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Theorem pol0N 35979
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 4199 . . 3 ∅ ⊆ 𝐴
2 eqid 2825 . . . 4 (oc‘𝐾) = (oc‘𝐾)
3 polssat.a . . . 4 𝐴 = (Atoms‘𝐾)
4 eqid 2825 . . . 4 (pmap‘𝐾) = (pmap‘𝐾)
5 polssat.p . . . 4 = (⊥𝑃𝐾)
62, 3, 4, 5polvalN 35975 . . 3 ((𝐾𝐵 ∧ ∅ ⊆ 𝐴) → ( ‘∅) = (𝐴 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))))
71, 6mpan2 682 . 2 (𝐾𝐵 → ( ‘∅) = (𝐴 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))))
8 0iin 4800 . . . 4 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)) = V
98ineq2i 4040 . . 3 (𝐴 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))) = (𝐴 ∩ V)
10 inv1 4197 . . 3 (𝐴 ∩ V) = 𝐴
119, 10eqtri 2849 . 2 (𝐴 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))) = 𝐴
127, 11syl6eq 2877 1 (𝐾𝐵 → ( ‘∅) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1656  wcel 2164  Vcvv 3414  cin 3797  wss 3798  c0 4146   ciin 4743  cfv 6127  occoc 16320  Atomscatm 35333  pmapcpmap 35567  𝑃cpolN 35972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-iin 4745  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-polarityN 35973
This theorem is referenced by:  2pol0N  35981  1psubclN  36014  osumcllem9N  36034  pexmidN  36039  pexmidlem6N  36045  pexmidALTN  36048
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