Theorem polatN 35217

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 e. OL /\ Q e. A ) -> ( P ` { Q } ) = ( M ` ( ._|_ ` Q ) ) )

Proof of Theorem polatN

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		snssi 4339	. . 3 |- ( Q e. A -> { Q } C_ 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 35191	. . 3 |- ( ( K e. OL /\ { Q } C_ A ) -> ( P ` { Q } ) = ( A i^i |^|_ p e. { Q } ( M ` ( ._|_ ` p ) ) ) )
7	1, 6	sylan2 491	. 2 |- ( ( K e. OL /\ Q e. A ) -> ( P ` { Q } ) = ( A i^i |^|_ p e. { Q } ( M ` ( ._|_ ` p ) ) ) )
8		fveq2 6191	. . . . . 6 |- ( p = Q -> ( ._|_ ` p ) = ( ._|_ ` Q ) )
9	8	fveq2d 6195	. . . . 5 |- ( p = Q -> ( M ` ( ._|_ ` p ) ) = ( M ` ( ._|_ ` Q ) ) )
10	9	iinxsng 4600	. . . 4 |- ( Q e. A -> |^|_ p e. { Q } ( M ` ( ._|_ ` p ) ) = ( M ` ( ._|_ ` Q ) ) )
11	10	adantl 482	. . 3 |- ( ( K e. OL /\ Q e. A ) -> |^|_ p e. { Q } ( M ` ( ._|_ ` p ) ) = ( M ` ( ._|_ ` Q ) ) )
12	11	ineq2d 3814	. 2 |- ( ( K e. OL /\ Q e. A ) -> ( A i^i |^|_ p e. { Q } ( M ` ( ._|_ ` p ) ) ) = ( A i^i ( M ` ( ._|_ ` Q ) ) ) )
13		olop 34501	. . . . 5 |- ( K e. OL -> K e. OP )
14		eqid 2622	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
15	14, 3	atbase 34576	. . . . 5 |- ( Q e. A -> Q e. ( Base ` K ) )
16	14, 2	opoccl 34481	. . . . 5 |- ( ( K e. OP /\ Q e. ( Base ` K ) ) -> ( ._|_ ` Q ) e. ( Base ` K ) )
17	13, 15, 16	syl2an 494	. . . 4 |- ( ( K e. OL /\ Q e. A ) -> ( ._|_ ` Q ) e. ( Base ` K ) )
18	14, 3, 4	pmapssat 35045	. . . 4 |- ( ( K e. OL /\ ( ._|_ ` Q ) e. ( Base ` K ) ) -> ( M ` ( ._|_ ` Q ) ) C_ A )
19	17, 18	syldan 487	. . 3 |- ( ( K e. OL /\ Q e. A ) -> ( M ` ( ._|_ ` Q ) ) C_ A )
20		sseqin2 3817	. . 3 |- ( ( M ` ( ._|_ ` Q ) ) C_ A <-> ( A i^i ( M ` ( ._|_ ` Q ) ) ) = ( M ` ( ._|_ ` Q ) ) )
21	19, 20	sylib 208	. 2 |- ( ( K e. OL /\ Q e. A ) -> ( A i^i ( M ` ( ._|_ ` Q ) ) ) = ( M ` ( ._|_ ` Q ) ) )
22	7, 12, 21	3eqtrd 2660	1 |- ( ( K e. OL /\ Q e. A ) -> ( P ` { Q } ) = ( M ` ( ._|_ ` Q ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   {csn 4177   |^|_ciin 4521   ` cfv 5888   Basecbs 15857   occoc 15949   OPcops 34459   OLcol 34461   Atomscatm 34550   pmapcpmap 34783   _|_PcpolN 35188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oposet 34463  df-ol 34465  df-ats 34554  df-pmap 34790  df-polarityN 35189

This theorem is referenced by:  2polatN  35218