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Theorem polatN 39933
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 4808 . . 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 39907 . . 3 ((𝐾 ∈ OL ∧ {𝑄} ⊆ 𝐴) → (𝑃‘{𝑄}) = (𝐴 𝑝 ∈ {𝑄} (𝑀‘( 𝑝))))
71, 6sylan2 593 . 2 ((𝐾 ∈ OL ∧ 𝑄𝐴) → (𝑃‘{𝑄}) = (𝐴 𝑝 ∈ {𝑄} (𝑀‘( 𝑝))))
8 2fveq3 6911 . . . . 5 (𝑝 = 𝑄 → (𝑀‘( 𝑝)) = (𝑀‘( 𝑄)))
98iinxsng 5088 . . . 4 (𝑄𝐴 𝑝 ∈ {𝑄} (𝑀‘( 𝑝)) = (𝑀‘( 𝑄)))
109adantl 481 . . 3 ((𝐾 ∈ OL ∧ 𝑄𝐴) → 𝑝 ∈ {𝑄} (𝑀‘( 𝑝)) = (𝑀‘( 𝑄)))
1110ineq2d 4220 . 2 ((𝐾 ∈ OL ∧ 𝑄𝐴) → (𝐴 𝑝 ∈ {𝑄} (𝑀‘( 𝑝))) = (𝐴 ∩ (𝑀‘( 𝑄))))
12 olop 39215 . . . . 5 (𝐾 ∈ OL → 𝐾 ∈ OP)
13 eqid 2737 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
1413, 3atbase 39290 . . . . 5 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
1513, 2opoccl 39195 . . . . 5 ((𝐾 ∈ OP ∧ 𝑄 ∈ (Base‘𝐾)) → ( 𝑄) ∈ (Base‘𝐾))
1612, 14, 15syl2an 596 . . . 4 ((𝐾 ∈ OL ∧ 𝑄𝐴) → ( 𝑄) ∈ (Base‘𝐾))
1713, 3, 4pmapssat 39761 . . . 4 ((𝐾 ∈ OL ∧ ( 𝑄) ∈ (Base‘𝐾)) → (𝑀‘( 𝑄)) ⊆ 𝐴)
1816, 17syldan 591 . . 3 ((𝐾 ∈ OL ∧ 𝑄𝐴) → (𝑀‘( 𝑄)) ⊆ 𝐴)
19 sseqin2 4223 . . 3 ((𝑀‘( 𝑄)) ⊆ 𝐴 ↔ (𝐴 ∩ (𝑀‘( 𝑄))) = (𝑀‘( 𝑄)))
2018, 19sylib 218 . 2 ((𝐾 ∈ OL ∧ 𝑄𝐴) → (𝐴 ∩ (𝑀‘( 𝑄))) = (𝑀‘( 𝑄)))
217, 11, 203eqtrd 2781 1 ((𝐾 ∈ OL ∧ 𝑄𝐴) → (𝑃‘{𝑄}) = (𝑀‘( 𝑄)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cin 3950  wss 3951  {csn 4626   ciin 4992  cfv 6561  Basecbs 17247  occoc 17305  OPcops 39173  OLcol 39175  Atomscatm 39264  pmapcpmap 39499  𝑃cpolN 39904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oposet 39177  df-ol 39179  df-ats 39268  df-pmap 39506  df-polarityN 39905
This theorem is referenced by:  2polatN  39934
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