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Theorem pol0N 39376
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 4392 . . 3 βˆ… βŠ† 𝐴
2 eqid 2728 . . . 4 (ocβ€˜πΎ) = (ocβ€˜πΎ)
3 polssat.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
4 eqid 2728 . . . 4 (pmapβ€˜πΎ) = (pmapβ€˜πΎ)
5 polssat.p . . . 4 βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
62, 3, 4, 5polvalN 39372 . . 3 ((𝐾 ∈ 𝐡 ∧ βˆ… βŠ† 𝐴) β†’ ( βŠ₯ β€˜βˆ…) = (𝐴 ∩ ∩ 𝑝 ∈ βˆ… ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘))))
71, 6mpan2 690 . 2 (𝐾 ∈ 𝐡 β†’ ( βŠ₯ β€˜βˆ…) = (𝐴 ∩ ∩ 𝑝 ∈ βˆ… ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘))))
8 0iin 5061 . . . 4 ∩ 𝑝 ∈ βˆ… ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘)) = V
98ineq2i 4205 . . 3 (𝐴 ∩ ∩ 𝑝 ∈ βˆ… ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘))) = (𝐴 ∩ V)
10 inv1 4390 . . 3 (𝐴 ∩ V) = 𝐴
119, 10eqtri 2756 . 2 (𝐴 ∩ ∩ 𝑝 ∈ βˆ… ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘))) = 𝐴
127, 11eqtrdi 2784 1 (𝐾 ∈ 𝐡 β†’ ( βŠ₯ β€˜βˆ…) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1534   ∈ wcel 2099  Vcvv 3470   ∩ cin 3944   βŠ† wss 3945  βˆ…c0 4318  βˆ© ciin 4992  β€˜cfv 6542  occoc 17234  Atomscatm 38729  pmapcpmap 38964  βŠ₯𝑃cpolN 39369
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 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423
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 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-polarityN 39370
This theorem is referenced by:  2pol0N  39378  1psubclN  39411  osumcllem9N  39431  pexmidN  39436  pexmidlem6N  39442  pexmidALTN  39445
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