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Theorem pol0N 39274
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 = (Atomsβ€˜πΎ)
polssat.p βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
Assertion
Ref Expression
pol0N (𝐾 ∈ 𝐡 β†’ ( βŠ₯ β€˜βˆ…) = 𝐴)

Proof of Theorem pol0N
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 0ss 4389 . . 3 βˆ… βŠ† 𝐴
2 eqid 2724 . . . 4 (ocβ€˜πΎ) = (ocβ€˜πΎ)
3 polssat.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
4 eqid 2724 . . . 4 (pmapβ€˜πΎ) = (pmapβ€˜πΎ)
5 polssat.p . . . 4 βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
62, 3, 4, 5polvalN 39270 . . 3 ((𝐾 ∈ 𝐡 ∧ βˆ… βŠ† 𝐴) β†’ ( βŠ₯ β€˜βˆ…) = (𝐴 ∩ ∩ 𝑝 ∈ βˆ… ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘))))
71, 6mpan2 688 . 2 (𝐾 ∈ 𝐡 β†’ ( βŠ₯ β€˜βˆ…) = (𝐴 ∩ ∩ 𝑝 ∈ βˆ… ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘))))
8 0iin 5058 . . . 4 ∩ 𝑝 ∈ βˆ… ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘)) = V
98ineq2i 4202 . . 3 (𝐴 ∩ ∩ 𝑝 ∈ βˆ… ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘))) = (𝐴 ∩ V)
10 inv1 4387 . . 3 (𝐴 ∩ V) = 𝐴
119, 10eqtri 2752 . 2 (𝐴 ∩ ∩ 𝑝 ∈ βˆ… ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘))) = 𝐴
127, 11eqtrdi 2780 1 (𝐾 ∈ 𝐡 β†’ ( βŠ₯ β€˜βˆ…) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3466   ∩ cin 3940   βŠ† wss 3941  βˆ…c0 4315  βˆ© ciin 4989  β€˜cfv 6534  occoc 17206  Atomscatm 38627  pmapcpmap 38862  βŠ₯𝑃cpolN 39267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-iin 4991  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-polarityN 39268
This theorem is referenced by:  2pol0N  39276  1psubclN  39309  osumcllem9N  39329  pexmidN  39334  pexmidlem6N  39340  pexmidALTN  39343
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