Theorem snatpsubN 33752

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 4128	. . . . . 6 |- ( P e. A -> { P } C_ A )
2	1	adantl 466	. . . . 5 |- ( ( K e. AtLat /\ P e. A ) -> { P } C_ A )
3		atllat 33303	. . . . . . . . . . . . . . 15 |- ( K e. AtLat -> K e. Lat )
4		eqid 2454	. . . . . . . . . . . . . . . 16 |- ( Base ` K ) = ( Base ` K )
5		snpsub.a	. . . . . . . . . . . . . . . 16 |- A = ( Atoms ` K )
6	4, 5	atbase 33292	. . . . . . . . . . . . . . 15 |- ( P e. A -> P e. ( Base ` K ) )
7		eqid 2454	. . . . . . . . . . . . . . . 16 |- ( join ` K ) = ( join ` K )
8	4, 7	latjidm 15366	. . . . . . . . . . . . . . 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 4415	. . . . . . . . . . . 12 |- ( ( ( K e. AtLat /\ P e. A ) /\ r e. A ) -> ( r ( le ` K ) ( P ( join ` K ) P ) <-> r ( le ` K ) P ) )
12		eqid 2454	. . . . . . . . . . . . . . . 16 |- ( le ` K ) = ( le ` K )
13	12, 5	atcmp 33314	. . . . . . . . . . . . . . 15 |- ( ( K e. AtLat /\ r e. A /\ P e. A ) -> ( r ( le ` K ) P <-> r = P ) )
14	13	3com23 1194	. . . . . . . . . . . . . 14 |- ( ( K e. AtLat /\ P e. A /\ r e. A ) -> ( r ( le ` K ) P <-> r = P ) )
15	14	3expa 1188	. . . . . . . . . . . . 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 4002	. . . . . . . . . . . . 13 |- ( p e. { P } <-> p = P )
20		elsn 4002	. . . . . . . . . . . . 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 6212	. . . . . . . . . . . . 13 |- ( ( p = P /\ q = P ) -> ( p ( join ` K ) q ) = ( P ( join ` K ) P ) )
24	23	breq2d 4415	. . . . . . . . . . . 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 4002	. . . . . . . . . 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 2911	. . . . . 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 2913	. . . . 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 33747	. . 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 1370   e. wcel 1758   A.wral 2799   C_ wss 3439   {csn 3988   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14295   lecple 14367   joincjn 15236   Latclat 15337   Atomscatm 33266   AtLatcal 33267   PSubSpcpsubsp 33498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-poset 15238  df-plt 15250  df-lub 15266  df-glb 15267  df-join 15268  df-meet 15269  df-p0 15331  df-lat 15338  df-covers 33269  df-ats 33270  df-atl 33301  df-psubsp 33505

This theorem is referenced by:  pointpsubN  33753  pclfinN  33902  pclfinclN  33952