Theorem polatN 41180

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 4746	. . 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 41154	. . 3 ⊢ ((𝐾 ∈ OL ∧ {𝑄} ⊆ 𝐴) → (𝑃‘{𝑄}) = (𝐴 ∩ ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝))))
7	1, 6	sylan2 605	. 2 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑃‘{𝑄}) = (𝐴 ∩ ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝))))
8		2fveq3 6913	. . . . 5 ⊢ (𝑝 = 𝑄 → (𝑀‘( ⊥ ‘𝑝)) = (𝑀‘( ⊥ ‘𝑄)))
9	8	iinxsng 5051	. . . 4 ⊢ (𝑄 ∈ 𝐴 → ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝)) = (𝑀‘( ⊥ ‘𝑄)))
10	9	adantl 487	. . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝)) = (𝑀‘( ⊥ ‘𝑄)))
11	10	ineq2d 4120	. 2 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝐴 ∩ ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝))) = (𝐴 ∩ (𝑀‘( ⊥ ‘𝑄))))
12		olop 40463	. . . . 5 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP)
13		eqid 2763	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
14	13, 3	atbase 40538	. . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
15	13, 2	opoccl 40443	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑄 ∈ (Base‘𝐾)) → ( ⊥ ‘𝑄) ∈ (Base‘𝐾))
16	12, 14, 15	syl2an 608	. . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → ( ⊥ ‘𝑄) ∈ (Base‘𝐾))
17	13, 3, 4	pmapssat 41008	. . . 4 ⊢ ((𝐾 ∈ OL ∧ ( ⊥ ‘𝑄) ∈ (Base‘𝐾)) → (𝑀‘( ⊥ ‘𝑄)) ⊆ 𝐴)
18	16, 17	syldan 603	. . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑀‘( ⊥ ‘𝑄)) ⊆ 𝐴)
19		sseqin2 4123	. . 3 ⊢ ((𝑀‘( ⊥ ‘𝑄)) ⊆ 𝐴 ↔ (𝐴 ∩ (𝑀‘( ⊥ ‘𝑄))) = (𝑀‘( ⊥ ‘𝑄)))
20	18, 19	sylib 221	. 2 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝐴 ∩ (𝑀‘( ⊥ ‘𝑄))) = (𝑀‘( ⊥ ‘𝑄)))
21	7, 11, 20	3eqtrd 2802	1 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑃‘{𝑄}) = (𝑀‘( ⊥ ‘𝑄)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   ∩ cin 4092  {csn 4588  ∩ ciin 4955  ‘cfv 6558  Basecbs 17416  occoc 17465  OPcops 40421  OLcol 40423  Atomscatm 40512  pmapcpmap 40746  ⊥_𝑃cpolN 41151

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oposet 40425  df-ol 40427  df-ats 40516  df-pmap 40753  df-polarityN 41152

This theorem is used by:  2polatN  41181