Theorem snatpsubN 35036

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 4339	. . . . . 6 |- ( P e. A -> { P } C_ A )
2	1	adantl 482	. . . . 5 |- ( ( K e. AtLat /\ P e. A ) -> { P } C_ A )
3		atllat 34587	. . . . . . . . . . . . . . 15 |- ( K e. AtLat -> K e. Lat )
4		eqid 2622	. . . . . . . . . . . . . . . 16 |- ( Base ` K ) = ( Base ` K )
5		snpsub.a	. . . . . . . . . . . . . . . 16 |- A = ( Atoms ` K )
6	4, 5	atbase 34576	. . . . . . . . . . . . . . 15 |- ( P e. A -> P e. ( Base ` K ) )
7		eqid 2622	. . . . . . . . . . . . . . . 16 |- ( join ` K ) = ( join ` K )
8	4, 7	latjidm 17074	. . . . . . . . . . . . . . 15 |- ( ( K e. Lat /\ P e. ( Base ` K ) ) -> ( P ( join ` K ) P ) = P )
9	3, 6, 8	syl2an 494	. . . . . . . . . . . . . 14 |- ( ( K e. AtLat /\ P e. A ) -> ( P ( join ` K ) P ) = P )
10	9	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( K e. AtLat /\ P e. A ) /\ r e. A ) -> ( P ( join ` K ) P ) = P )
11	10	breq2d 4665	. . . . . . . . . . . 12 |- ( ( ( K e. AtLat /\ P e. A ) /\ r e. A ) -> ( r ( le ` K ) ( P ( join ` K ) P ) <-> r ( le ` K ) P ) )
12		eqid 2622	. . . . . . . . . . . . . . . 16 |- ( le ` K ) = ( le ` K )
13	12, 5	atcmp 34598	. . . . . . . . . . . . . . 15 |- ( ( K e. AtLat /\ r e. A /\ P e. A ) -> ( r ( le ` K ) P <-> r = P ) )
14	13	3com23 1271	. . . . . . . . . . . . . 14 |- ( ( K e. AtLat /\ P e. A /\ r e. A ) -> ( r ( le ` K ) P <-> r = P ) )
15	14	3expa 1265	. . . . . . . . . . . . 13 |- ( ( ( K e. AtLat /\ P e. A ) /\ r e. A ) -> ( r ( le ` K ) P <-> r = P ) )
16	15	biimpd 219	. . . . . . . . . . . 12 |- ( ( ( K e. AtLat /\ P e. A ) /\ r e. A ) -> ( r ( le ` K ) P -> r = P ) )
17	11, 16	sylbid 230	. . . . . . . . . . 11 |- ( ( ( K e. AtLat /\ P e. A ) /\ r e. A ) -> ( r ( le ` K ) ( P ( join ` K ) P ) -> r = P ) )
18	17	adantld 483	. . . . . . . . . 10 |- ( ( ( K e. AtLat /\ P e. A ) /\ r e. A ) -> ( ( ( p = P /\ q = P ) /\ r ( le ` K ) ( P ( join ` K ) P ) ) -> r = P ) )
19		velsn 4193	. . . . . . . . . . . . 13 |- ( p e. { P } <-> p = P )
20		velsn 4193	. . . . . . . . . . . . 13 |- ( q e. { P } <-> q = P )
21	19, 20	anbi12i 733	. . . . . . . . . . . 12 |- ( ( p e. { P } /\ q e. { P } ) <-> ( p = P /\ q = P ) )
22	21	anbi1i 731	. . . . . . . . . . 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 6659	. . . . . . . . . . . . 13 |- ( ( p = P /\ q = P ) -> ( p ( join ` K ) q ) = ( P ( join ` K ) P ) )
24	23	breq2d 4665	. . . . . . . . . . . 12 |- ( ( p = P /\ q = P ) -> ( r ( le ` K ) ( p ( join ` K ) q ) <-> r ( le ` K ) ( P ( join ` K ) P ) ) )
25	24	pm5.32i 669	. . . . . . . . . . 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 264	. . . . . . . . . 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		velsn 4193	. . . . . . . . . 10 |- ( r e. { P } <-> r = P )
28	18, 26, 27	3imtr4g 285	. . . . . . . . 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 632	. . . . . . . 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 86	. . . . . . 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 2968	. . . . . 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 2970	. . . . 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 554	. . . 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 450	. . 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 35031	. . 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 249	. 2 |- ( K e. AtLat -> ( P e. A -> { P } e. S ) )
38	37	imp 445	1 |- ( ( K e. AtLat /\ P e. A ) -> { P } e. S )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   {csn 4177   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   Latclat 17045   Atomscatm 34550   AtLatcal 34551   PSubSpcpsubsp 34782

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  ax-un 6949

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-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-riota 6611  df-ov 6653  df-oprab 6654  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-lat 17046  df-covers 34553  df-ats 34554  df-atl 34585  df-psubsp 34789

This theorem is referenced by:  pointpsubN  35037  pclfinN  35186  pclfinclN  35236