Theorem polatN 39932

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 4775	. . 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 39906	. . 3 ⊢ ((𝐾 ∈ OL ∧ {𝑄} ⊆ 𝐴) → (𝑃‘{𝑄}) = (𝐴 ∩ ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝))))
7	1, 6	sylan2 593	. 2 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑃‘{𝑄}) = (𝐴 ∩ ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝))))
8		2fveq3 6866	. . . . 5 ⊢ (𝑝 = 𝑄 → (𝑀‘( ⊥ ‘𝑝)) = (𝑀‘( ⊥ ‘𝑄)))
9	8	iinxsng 5055	. . . 4 ⊢ (𝑄 ∈ 𝐴 → ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝)) = (𝑀‘( ⊥ ‘𝑄)))
10	9	adantl 481	. . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝)) = (𝑀‘( ⊥ ‘𝑄)))
11	10	ineq2d 4186	. 2 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝐴 ∩ ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝))) = (𝐴 ∩ (𝑀‘( ⊥ ‘𝑄))))
12		olop 39214	. . . . 5 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP)
13		eqid 2730	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
14	13, 3	atbase 39289	. . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
15	13, 2	opoccl 39194	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑄 ∈ (Base‘𝐾)) → ( ⊥ ‘𝑄) ∈ (Base‘𝐾))
16	12, 14, 15	syl2an 596	. . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → ( ⊥ ‘𝑄) ∈ (Base‘𝐾))
17	13, 3, 4	pmapssat 39760	. . . 4 ⊢ ((𝐾 ∈ OL ∧ ( ⊥ ‘𝑄) ∈ (Base‘𝐾)) → (𝑀‘( ⊥ ‘𝑄)) ⊆ 𝐴)
18	16, 17	syldan 591	. . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑀‘( ⊥ ‘𝑄)) ⊆ 𝐴)
19		sseqin2 4189	. . 3 ⊢ ((𝑀‘( ⊥ ‘𝑄)) ⊆ 𝐴 ↔ (𝐴 ∩ (𝑀‘( ⊥ ‘𝑄))) = (𝑀‘( ⊥ ‘𝑄)))
20	18, 19	sylib 218	. 2 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝐴 ∩ (𝑀‘( ⊥ ‘𝑄))) = (𝑀‘( ⊥ ‘𝑄)))
21	7, 11, 20	3eqtrd 2769	1 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑃‘{𝑄}) = (𝑀‘( ⊥ ‘𝑄)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∩ cin 3916   ⊆ wss 3917  {csn 4592  ∩ ciin 4959  ‘cfv 6514  Basecbs 17186  occoc 17235  OPcops 39172  OLcol 39174  Atomscatm 39263  pmapcpmap 39498  ⊥_𝑃cpolN 39903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oposet 39176  df-ol 39178  df-ats 39267  df-pmap 39505  df-polarityN 39904

This theorem is referenced by:  2polatN  39933