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Theorem polatN 36006
 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 4557 . . 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 35980 . . 3 ((𝐾 ∈ OL ∧ {𝑄} ⊆ 𝐴) → (𝑃‘{𝑄}) = (𝐴 𝑝 ∈ {𝑄} (𝑀‘( 𝑝))))
71, 6sylan2 588 . 2 ((𝐾 ∈ OL ∧ 𝑄𝐴) → (𝑃‘{𝑄}) = (𝐴 𝑝 ∈ {𝑄} (𝑀‘( 𝑝))))
8 2fveq3 6438 . . . . 5 (𝑝 = 𝑄 → (𝑀‘( 𝑝)) = (𝑀‘( 𝑄)))
98iinxsng 4820 . . . 4 (𝑄𝐴 𝑝 ∈ {𝑄} (𝑀‘( 𝑝)) = (𝑀‘( 𝑄)))
109adantl 475 . . 3 ((𝐾 ∈ OL ∧ 𝑄𝐴) → 𝑝 ∈ {𝑄} (𝑀‘( 𝑝)) = (𝑀‘( 𝑄)))
1110ineq2d 4041 . 2 ((𝐾 ∈ OL ∧ 𝑄𝐴) → (𝐴 𝑝 ∈ {𝑄} (𝑀‘( 𝑝))) = (𝐴 ∩ (𝑀‘( 𝑄))))
12 olop 35289 . . . . 5 (𝐾 ∈ OL → 𝐾 ∈ OP)
13 eqid 2825 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
1413, 3atbase 35364 . . . . 5 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
1513, 2opoccl 35269 . . . . 5 ((𝐾 ∈ OP ∧ 𝑄 ∈ (Base‘𝐾)) → ( 𝑄) ∈ (Base‘𝐾))
1612, 14, 15syl2an 591 . . . 4 ((𝐾 ∈ OL ∧ 𝑄𝐴) → ( 𝑄) ∈ (Base‘𝐾))
1713, 3, 4pmapssat 35834 . . . 4 ((𝐾 ∈ OL ∧ ( 𝑄) ∈ (Base‘𝐾)) → (𝑀‘( 𝑄)) ⊆ 𝐴)
1816, 17syldan 587 . . 3 ((𝐾 ∈ OL ∧ 𝑄𝐴) → (𝑀‘( 𝑄)) ⊆ 𝐴)
19 sseqin2 4044 . . 3 ((𝑀‘( 𝑄)) ⊆ 𝐴 ↔ (𝐴 ∩ (𝑀‘( 𝑄))) = (𝑀‘( 𝑄)))
2018, 19sylib 210 . 2 ((𝐾 ∈ OL ∧ 𝑄𝐴) → (𝐴 ∩ (𝑀‘( 𝑄))) = (𝑀‘( 𝑄)))
217, 11, 203eqtrd 2865 1 ((𝐾 ∈ OL ∧ 𝑄𝐴) → (𝑃‘{𝑄}) = (𝑀‘( 𝑄)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166   ∩ cin 3797   ⊆ wss 3798  {csn 4397  ∩ ciin 4741  ‘cfv 6123  Basecbs 16222  occoc 16313  OPcops 35247  OLcol 35249  Atomscatm 35338  pmapcpmap 35572  ⊥𝑃cpolN 35977 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  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 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-iin 4743  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oposet 35251  df-ol 35253  df-ats 35342  df-pmap 35579  df-polarityN 35978 This theorem is referenced by:  2polatN  36007
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