Theorem polatN 40908

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 4745	. . 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 40882	. . 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 6878	. . . . 5 ⊢ (𝑝 = 𝑄 → (𝑀‘( ⊥ ‘𝑝)) = (𝑀‘( ⊥ ‘𝑄)))
9	8	iinxsng 5047	. . . 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 4165	. 2 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝐴 ∩ ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝))) = (𝐴 ∩ (𝑀‘( ⊥ ‘𝑄))))
12		olop 40191	. . . . 5 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP)
13		eqid 2760	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
14	13, 3	atbase 40266	. . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
15	13, 2	opoccl 40171	. . . . 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 40736	. . . 4 ⊢ ((𝐾 ∈ OL ∧ ( ⊥ ‘𝑄) ∈ (Base‘𝐾)) → (𝑀‘( ⊥ ‘𝑄)) ⊆ 𝐴)
18	16, 17	syldan 603	. . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑀‘( ⊥ ‘𝑄)) ⊆ 𝐴)
19		sseqin2 4168	. . 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 2799	1 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑃‘{𝑄}) = (𝑀‘( ⊥ ‘𝑄)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∩ cin 3897   ⊆ wss 3898  {csn 4583  ∩ ciin 4951  ‘cfv 6527  Basecbs 17348  occoc 17397  OPcops 40149  OLcol 40151  Atomscatm 40240  pmapcpmap 40474  ⊥_𝑃cpolN 40879

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-oposet 40153  df-ol 40155  df-ats 40244  df-pmap 40481  df-polarityN 40880

This theorem is used by:  2polatN  40909