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Theorem polatN 37100
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 4715 . . 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 37074 . . 3 ((𝐾 ∈ OL ∧ {𝑄} ⊆ 𝐴) → (𝑃‘{𝑄}) = (𝐴 𝑝 ∈ {𝑄} (𝑀‘( 𝑝))))
71, 6sylan2 594 . 2 ((𝐾 ∈ OL ∧ 𝑄𝐴) → (𝑃‘{𝑄}) = (𝐴 𝑝 ∈ {𝑄} (𝑀‘( 𝑝))))
8 2fveq3 6649 . . . . 5 (𝑝 = 𝑄 → (𝑀‘( 𝑝)) = (𝑀‘( 𝑄)))
98iinxsng 4984 . . . 4 (𝑄𝐴 𝑝 ∈ {𝑄} (𝑀‘( 𝑝)) = (𝑀‘( 𝑄)))
109adantl 484 . . 3 ((𝐾 ∈ OL ∧ 𝑄𝐴) → 𝑝 ∈ {𝑄} (𝑀‘( 𝑝)) = (𝑀‘( 𝑄)))
1110ineq2d 4165 . 2 ((𝐾 ∈ OL ∧ 𝑄𝐴) → (𝐴 𝑝 ∈ {𝑄} (𝑀‘( 𝑝))) = (𝐴 ∩ (𝑀‘( 𝑄))))
12 olop 36383 . . . . 5 (𝐾 ∈ OL → 𝐾 ∈ OP)
13 eqid 2820 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
1413, 3atbase 36458 . . . . 5 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
1513, 2opoccl 36363 . . . . 5 ((𝐾 ∈ OP ∧ 𝑄 ∈ (Base‘𝐾)) → ( 𝑄) ∈ (Base‘𝐾))
1612, 14, 15syl2an 597 . . . 4 ((𝐾 ∈ OL ∧ 𝑄𝐴) → ( 𝑄) ∈ (Base‘𝐾))
1713, 3, 4pmapssat 36928 . . . 4 ((𝐾 ∈ OL ∧ ( 𝑄) ∈ (Base‘𝐾)) → (𝑀‘( 𝑄)) ⊆ 𝐴)
1816, 17syldan 593 . . 3 ((𝐾 ∈ OL ∧ 𝑄𝐴) → (𝑀‘( 𝑄)) ⊆ 𝐴)
19 sseqin2 4168 . . 3 ((𝑀‘( 𝑄)) ⊆ 𝐴 ↔ (𝐴 ∩ (𝑀‘( 𝑄))) = (𝑀‘( 𝑄)))
2018, 19sylib 220 . 2 ((𝐾 ∈ OL ∧ 𝑄𝐴) → (𝐴 ∩ (𝑀‘( 𝑄))) = (𝑀‘( 𝑄)))
217, 11, 203eqtrd 2859 1 ((𝐾 ∈ OL ∧ 𝑄𝐴) → (𝑃‘{𝑄}) = (𝑀‘( 𝑄)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  cin 3911  wss 3912  {csn 4541   ciin 4894  cfv 6329  Basecbs 16459  occoc 16549  OPcops 36341  OLcol 36343  Atomscatm 36432  pmapcpmap 36666  𝑃cpolN 37071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5240  ax-pr 5304
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3752  df-csb 3860  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-nul 4268  df-if 4442  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4813  df-iun 4895  df-iin 4896  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5434  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-ov 7134  df-oposet 36345  df-ol 36347  df-ats 36436  df-pmap 36673  df-polarityN 37072
This theorem is referenced by:  2polatN  37101
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