Theorem snatpsubN 34947

Description: The singleton of an atom is a projective subspace. (Contributed by NM, 9-Sep-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
snpsub.a	|- A = ( Atoms ` K )
snpsub.s	|- S = ( PSubSp ` K )

Assertion

Ref	Expression
snatpsubN	|- ( ( K e. AtLat /\ P e. A ) -> { P } e. S )

Proof of Theorem snatpsubN

Dummy variables q p r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snssi 4177	. . . . . 6 |- ( P e. A -> { P } C_ A )
2	1	adantl 466	. . . . 5 |- ( ( K e. AtLat /\ P e. A ) -> { P } C_ A )
3		atllat 34498	. . . . . . . . . . . . . . 15 |- ( K e. AtLat -> K e. Lat )
4		eqid 2467	. . . . . . . . . . . . . . . 16 |- ( Base ` K ) = ( Base ` K )
5		snpsub.a	. . . . . . . . . . . . . . . 16 |- A = ( Atoms ` K )
6	4, 5	atbase 34487	. . . . . . . . . . . . . . 15 |- ( P e. A -> P e. ( Base ` K ) )
7		eqid 2467	. . . . . . . . . . . . . . . 16 |- ( join ` K ) = ( join ` K )
8	4, 7	latjidm 15578	. . . . . . . . . . . . . . 15 |- ( ( K e. Lat /\ P e. ( Base ` K ) ) -> ( P ( join ` K ) P ) = P )
9	3, 6, 8	syl2an 477	. . . . . . . . . . . . . 14 |- ( ( K e. AtLat /\ P e. A ) -> ( P ( join ` K ) P ) = P )
10	9	adantr 465	. . . . . . . . . . . . 13 |- ( ( ( K e. AtLat /\ P e. A ) /\ r e. A ) -> ( P ( join ` K ) P ) = P )
11	10	breq2d 4465	. . . . . . . . . . . 12 |- ( ( ( K e. AtLat /\ P e. A ) /\ r e. A ) -> ( r ( le ` K ) ( P ( join ` K ) P ) <-> r ( le ` K ) P ) )
12		eqid 2467	. . . . . . . . . . . . . . . 16 |- ( le ` K ) = ( le ` K )
13	12, 5	atcmp 34509	. . . . . . . . . . . . . . 15 |- ( ( K e. AtLat /\ r e. A /\ P e. A ) -> ( r ( le ` K ) P <-> r = P ) )
14	13	3com23 1202	. . . . . . . . . . . . . 14 |- ( ( K e. AtLat /\ P e. A /\ r e. A ) -> ( r ( le ` K ) P <-> r = P ) )
15	14	3expa 1196	. . . . . . . . . . . . 13 |- ( ( ( K e. AtLat /\ P e. A ) /\ r e. A ) -> ( r ( le ` K ) P <-> r = P ) )
16	15	biimpd 207	. . . . . . . . . . . 12 |- ( ( ( K e. AtLat /\ P e. A ) /\ r e. A ) -> ( r ( le ` K ) P -> r = P ) )
17	11, 16	sylbid 215	. . . . . . . . . . 11 |- ( ( ( K e. AtLat /\ P e. A ) /\ r e. A ) -> ( r ( le ` K ) ( P ( join ` K ) P ) -> r = P ) )
18	17	adantld 467	. . . . . . . . . 10 |- ( ( ( K e. AtLat /\ P e. A ) /\ r e. A ) -> ( ( ( p = P /\ q = P ) /\ r ( le ` K ) ( P ( join ` K ) P ) ) -> r = P ) )
19		elsn 4047	. . . . . . . . . . . . 13 |- ( p e. { P } <-> p = P )
20		elsn 4047	. . . . . . . . . . . . 13 |- ( q e. { P } <-> q = P )
21	19, 20	anbi12i 697	. . . . . . . . . . . 12 |- ( ( p e. { P } /\ q e. { P } ) <-> ( p = P /\ q = P ) )
22	21	anbi1i 695	. . . . . . . . . . 11 |- ( ( ( p e. { P } /\ q e. { P } ) /\ r ( le ` K ) ( p ( join ` K ) q ) ) <-> ( ( p = P /\ q = P ) /\ r ( le ` K ) ( p ( join ` K ) q ) ) )
23		oveq12 6304	. . . . . . . . . . . . 13 |- ( ( p = P /\ q = P ) -> ( p ( join ` K ) q ) = ( P ( join ` K ) P ) )
24	23	breq2d 4465	. . . . . . . . . . . 12 |- ( ( p = P /\ q = P ) -> ( r ( le ` K ) ( p ( join ` K ) q ) <-> r ( le ` K ) ( P ( join ` K ) P ) ) )
25	24	pm5.32i 637	. . . . . . . . . . 11 |- ( ( ( p = P /\ q = P ) /\ r ( le ` K ) ( p ( join ` K ) q ) ) <-> ( ( p = P /\ q = P ) /\ r ( le ` K ) ( P ( join ` K ) P ) ) )
26	22, 25	bitri 249	. . . . . . . . . 10 |- ( ( ( p e. { P } /\ q e. { P } ) /\ r ( le ` K ) ( p ( join ` K ) q ) ) <-> ( ( p = P /\ q = P ) /\ r ( le ` K ) ( P ( join ` K ) P ) ) )
27		elsn 4047	. . . . . . . . . 10 |- ( r e. { P } <-> r = P )
28	18, 26, 27	3imtr4g 270	. . . . . . . . 9 |- ( ( ( K e. AtLat /\ P e. A ) /\ r e. A ) -> ( ( ( p e. { P } /\ q e. { P } ) /\ r ( le ` K ) ( p ( join ` K ) q ) ) -> r e. { P } ) )
29	28	exp4b 607	. . . . . . . 8 |- ( ( K e. AtLat /\ P e. A ) -> ( r e. A -> ( ( p e. { P } /\ q e. { P } ) -> ( r ( le ` K ) ( p ( join ` K ) q ) -> r e. { P } ) ) ) )
30	29	com23 78	. . . . . . 7 |- ( ( K e. AtLat /\ P e. A ) -> ( ( p e. { P } /\ q e. { P } ) -> ( r e. A -> ( r ( le ` K ) ( p ( join ` K ) q ) -> r e. { P } ) ) ) )
31	30	ralrimdv 2883	. . . . . 6 |- ( ( K e. AtLat /\ P e. A ) -> ( ( p e. { P } /\ q e. { P } ) -> A. r e. A ( r ( le ` K ) ( p ( join ` K ) q ) -> r e. { P } ) ) )
32	31	ralrimivv 2887	. . . . 5 |- ( ( K e. AtLat /\ P e. A ) -> A. p e. { P } A. q e. { P } A. r e. A ( r ( le ` K ) ( p ( join ` K ) q ) -> r e. { P } ) )
33	2, 32	jca 532	. . . 4 |- ( ( K e. AtLat /\ P e. A ) -> ( { P } C_ A /\ A. p e. { P } A. q e. { P } A. r e. A ( r ( le ` K ) ( p ( join ` K ) q ) -> r e. { P } ) ) )
34	33	ex 434	. . 3 |- ( K e. AtLat -> ( P e. A -> ( { P } C_ A /\ A. p e. { P } A. q e. { P } A. r e. A ( r ( le ` K ) ( p ( join ` K ) q ) -> r e. { P } ) ) ) )
35		snpsub.s	. . . 4 |- S = ( PSubSp ` K )
36	12, 7, 5, 35	ispsubsp 34942	. . 3 |- ( K e. AtLat -> ( { P } e. S <-> ( { P } C_ A /\ A. p e. { P } A. q e. { P } A. r e. A ( r ( le ` K ) ( p ( join ` K ) q ) -> r e. { P } ) ) ) )
37	34, 36	sylibrd 234	. 2 |- ( K e. AtLat -> ( P e. A -> { P } e. S ) )
38	37	imp 429	1 |- ( ( K e. AtLat /\ P e. A ) -> { P } e. S )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   A.wral 2817   C_ wss 3481   {csn 4033   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   lecple 14579   joincjn 15448   Latclat 15549   Atomscatm 34461   AtLatcal 34462   PSubSpcpsubsp 34693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-lat 15550  df-covers 34464  df-ats 34465  df-atl 34496  df-psubsp 34700

This theorem is referenced by:  pointpsubN  34948  pclfinN  35097  pclfinclN  35147