Theorem polatN 34235

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 = (⊥𝑃‘𝐾)

Assertion

Ref	Expression
polatN	⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑃‘{𝑄}) = (𝑀‘( ⊥ ‘𝑄)))

Proof of Theorem polatN

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		snssi 4280	. . 3 ⊢ (𝑄 ∈ 𝐴 → {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 = (⊥𝑃‘𝐾)
6	2, 3, 4, 5	polvalN 34209	. . 3 ⊢ ((𝐾 ∈ OL ∧ {𝑄} ⊆ 𝐴) → (𝑃‘{𝑄}) = (𝐴 ∩ ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝))))
7	1, 6	sylan2 490	. 2 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑃‘{𝑄}) = (𝐴 ∩ ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝))))
8		fveq2 6103	. . . . . 6 ⊢ (𝑝 = 𝑄 → ( ⊥ ‘𝑝) = ( ⊥ ‘𝑄))
9	8	fveq2d 6107	. . . . 5 ⊢ (𝑝 = 𝑄 → (𝑀‘( ⊥ ‘𝑝)) = (𝑀‘( ⊥ ‘𝑄)))
10	9	iinxsng 4536	. . . 4 ⊢ (𝑄 ∈ 𝐴 → ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝)) = (𝑀‘( ⊥ ‘𝑄)))
11	10	adantl 481	. . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝)) = (𝑀‘( ⊥ ‘𝑄)))
12	11	ineq2d 3776	. 2 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝐴 ∩ ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝))) = (𝐴 ∩ (𝑀‘( ⊥ ‘𝑄))))
13		olop 33519	. . . . 5 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP)
14		eqid 2610	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
15	14, 3	atbase 33594	. . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
16	14, 2	opoccl 33499	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑄 ∈ (Base‘𝐾)) → ( ⊥ ‘𝑄) ∈ (Base‘𝐾))
17	13, 15, 16	syl2an 493	. . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → ( ⊥ ‘𝑄) ∈ (Base‘𝐾))
18	14, 3, 4	pmapssat 34063	. . . 4 ⊢ ((𝐾 ∈ OL ∧ ( ⊥ ‘𝑄) ∈ (Base‘𝐾)) → (𝑀‘( ⊥ ‘𝑄)) ⊆ 𝐴)
19	17, 18	syldan 486	. . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑀‘( ⊥ ‘𝑄)) ⊆ 𝐴)
20		sseqin2 3779	. . 3 ⊢ ((𝑀‘( ⊥ ‘𝑄)) ⊆ 𝐴 ↔ (𝐴 ∩ (𝑀‘( ⊥ ‘𝑄))) = (𝑀‘( ⊥ ‘𝑄)))
21	19, 20	sylib 207	. 2 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝐴 ∩ (𝑀‘( ⊥ ‘𝑄))) = (𝑀‘( ⊥ ‘𝑄)))
22	7, 12, 21	3eqtrd 2648	1 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑃‘{𝑄}) = (𝑀‘( ⊥ ‘𝑄)))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∩ cin 3539   ⊆ wss 3540  {csn 4125  ∩ ciin 4456  ‘cfv 5804  Basecbs 15695  occoc 15776  OPcops 33477  OLcol 33479  Atomscatm 33568  pmapcpmap 33801  ⊥_𝑃cpolN 34206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oposet 33481  df-ol 33483  df-ats 33572  df-pmap 33808  df-polarityN 34207

This theorem is referenced by:  2polatN  34236