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Theorem polatN 34035
Description: The polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
polat.o = (oc‘𝐾)
polat.a 𝐴 = (Atoms‘𝐾)
polat.m 𝑀 = (pmap‘𝐾)
polat.p 𝑃 = (⊥𝑃𝐾)
Assertion
Ref Expression
polatN ((𝐾 ∈ OL ∧ 𝑄𝐴) → (𝑃‘{𝑄}) = (𝑀‘( 𝑄)))

Proof of Theorem polatN
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 snssi 4276 . . 3 (𝑄𝐴 → {𝑄} ⊆ 𝐴)
2 polat.o . . . 4 = (oc‘𝐾)
3 polat.a . . . 4 𝐴 = (Atoms‘𝐾)
4 polat.m . . . 4 𝑀 = (pmap‘𝐾)
5 polat.p . . . 4 𝑃 = (⊥𝑃𝐾)
62, 3, 4, 5polvalN 34009 . . 3 ((𝐾 ∈ OL ∧ {𝑄} ⊆ 𝐴) → (𝑃‘{𝑄}) = (𝐴 𝑝 ∈ {𝑄} (𝑀‘( 𝑝))))
71, 6sylan2 489 . 2 ((𝐾 ∈ OL ∧ 𝑄𝐴) → (𝑃‘{𝑄}) = (𝐴 𝑝 ∈ {𝑄} (𝑀‘( 𝑝))))
8 fveq2 6085 . . . . . 6 (𝑝 = 𝑄 → ( 𝑝) = ( 𝑄))
98fveq2d 6089 . . . . 5 (𝑝 = 𝑄 → (𝑀‘( 𝑝)) = (𝑀‘( 𝑄)))
109iinxsng 4527 . . . 4 (𝑄𝐴 𝑝 ∈ {𝑄} (𝑀‘( 𝑝)) = (𝑀‘( 𝑄)))
1110adantl 480 . . 3 ((𝐾 ∈ OL ∧ 𝑄𝐴) → 𝑝 ∈ {𝑄} (𝑀‘( 𝑝)) = (𝑀‘( 𝑄)))
1211ineq2d 3772 . 2 ((𝐾 ∈ OL ∧ 𝑄𝐴) → (𝐴 𝑝 ∈ {𝑄} (𝑀‘( 𝑝))) = (𝐴 ∩ (𝑀‘( 𝑄))))
13 olop 33319 . . . . 5 (𝐾 ∈ OL → 𝐾 ∈ OP)
14 eqid 2606 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
1514, 3atbase 33394 . . . . 5 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
1614, 2opoccl 33299 . . . . 5 ((𝐾 ∈ OP ∧ 𝑄 ∈ (Base‘𝐾)) → ( 𝑄) ∈ (Base‘𝐾))
1713, 15, 16syl2an 492 . . . 4 ((𝐾 ∈ OL ∧ 𝑄𝐴) → ( 𝑄) ∈ (Base‘𝐾))
1814, 3, 4pmapssat 33863 . . . 4 ((𝐾 ∈ OL ∧ ( 𝑄) ∈ (Base‘𝐾)) → (𝑀‘( 𝑄)) ⊆ 𝐴)
1917, 18syldan 485 . . 3 ((𝐾 ∈ OL ∧ 𝑄𝐴) → (𝑀‘( 𝑄)) ⊆ 𝐴)
20 sseqin2 3775 . . 3 ((𝑀‘( 𝑄)) ⊆ 𝐴 ↔ (𝐴 ∩ (𝑀‘( 𝑄))) = (𝑀‘( 𝑄)))
2119, 20sylib 206 . 2 ((𝐾 ∈ OL ∧ 𝑄𝐴) → (𝐴 ∩ (𝑀‘( 𝑄))) = (𝑀‘( 𝑄)))
227, 12, 213eqtrd 2644 1 ((𝐾 ∈ OL ∧ 𝑄𝐴) → (𝑃‘{𝑄}) = (𝑀‘( 𝑄)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  cin 3535  wss 3536  {csn 4121   ciin 4447  cfv 5787  Basecbs 15638  occoc 15719  OPcops 33277  OLcol 33279  Atomscatm 33368  pmapcpmap 33601  𝑃cpolN 34006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-iun 4448  df-iin 4449  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-ov 6527  df-oposet 33281  df-ol 33283  df-ats 33372  df-pmap 33608  df-polarityN 34007
This theorem is referenced by:  2polatN  34036
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