Theorem psubclinN 35234

Description: The intersection of two closed subspaces is closed. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)

Hypothesis

Ref	Expression
psubclin.c	|- C = ( PSubCl ` K )

Assertion

Ref	Expression
psubclinN	|- ( ( K e. HL /\ X e. C /\ Y e. C ) -> ( X i^i Y ) e. C )

Proof of Theorem psubclinN

Step	Hyp	Ref	Expression
1		simp1 1061	. . . 4 |- ( ( K e. HL /\ X e. C /\ Y e. C ) -> K e. HL )
2		hlclat 34645	. . . . . 6 |- ( K e. HL -> K e. CLat )
3	2	3ad2ant1 1082	. . . . 5 |- ( ( K e. HL /\ X e. C /\ Y e. C ) -> K e. CLat )
4		eqid 2622	. . . . . . . 8 |- ( Atoms ` K ) = ( Atoms ` K )
5		psubclin.c	. . . . . . . 8 |- C = ( PSubCl ` K )
6	4, 5	psubclssatN 35227	. . . . . . 7 |- ( ( K e. HL /\ X e. C ) -> X C_ ( Atoms ` K ) )
7	6	3adant3 1081	. . . . . 6 |- ( ( K e. HL /\ X e. C /\ Y e. C ) -> X C_ ( Atoms ` K ) )
8		eqid 2622	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
9	8, 4	atssbase 34577	. . . . . 6 |- ( Atoms ` K ) C_ ( Base ` K )
10	7, 9	syl6ss 3615	. . . . 5 |- ( ( K e. HL /\ X e. C /\ Y e. C ) -> X C_ ( Base ` K ) )
11		eqid 2622	. . . . . 6 |- ( lub ` K ) = ( lub ` K )
12	8, 11	clatlubcl 17112	. . . . 5 |- ( ( K e. CLat /\ X C_ ( Base ` K ) ) -> ( ( lub ` K ) ` X ) e. ( Base ` K ) )
13	3, 10, 12	syl2anc 693	. . . 4 |- ( ( K e. HL /\ X e. C /\ Y e. C ) -> ( ( lub ` K ) ` X ) e. ( Base ` K ) )
14	4, 5	psubclssatN 35227	. . . . . . 7 |- ( ( K e. HL /\ Y e. C ) -> Y C_ ( Atoms ` K ) )
15	14	3adant2 1080	. . . . . 6 |- ( ( K e. HL /\ X e. C /\ Y e. C ) -> Y C_ ( Atoms ` K ) )
16	15, 9	syl6ss 3615	. . . . 5 |- ( ( K e. HL /\ X e. C /\ Y e. C ) -> Y C_ ( Base ` K ) )
17	8, 11	clatlubcl 17112	. . . . 5 |- ( ( K e. CLat /\ Y C_ ( Base ` K ) ) -> ( ( lub ` K ) ` Y ) e. ( Base ` K ) )
18	3, 16, 17	syl2anc 693	. . . 4 |- ( ( K e. HL /\ X e. C /\ Y e. C ) -> ( ( lub ` K ) ` Y ) e. ( Base ` K ) )
19		eqid 2622	. . . . 5 |- ( meet ` K ) = ( meet ` K )
20		eqid 2622	. . . . 5 |- ( pmap ` K ) = ( pmap ` K )
21	8, 19, 4, 20	pmapmeet 35059	. . . 4 |- ( ( K e. HL /\ ( ( lub ` K ) ` X ) e. ( Base ` K ) /\ ( ( lub ` K ) ` Y ) e. ( Base ` K ) ) -> ( ( pmap ` K ) ` ( ( ( lub ` K ) ` X ) ( meet ` K ) ( ( lub ` K ) ` Y ) ) ) = ( ( ( pmap ` K ) ` ( ( lub ` K ) ` X ) ) i^i ( ( pmap ` K ) ` ( ( lub ` K ) ` Y ) ) ) )
22	1, 13, 18, 21	syl3anc 1326	. . 3 |- ( ( K e. HL /\ X e. C /\ Y e. C ) -> ( ( pmap ` K ) ` ( ( ( lub ` K ) ` X ) ( meet ` K ) ( ( lub ` K ) ` Y ) ) ) = ( ( ( pmap ` K ) ` ( ( lub ` K ) ` X ) ) i^i ( ( pmap ` K ) ` ( ( lub ` K ) ` Y ) ) ) )
23	11, 20, 5	pmapidclN 35228	. . . . 5 |- ( ( K e. HL /\ X e. C ) -> ( ( pmap ` K ) ` ( ( lub ` K ) ` X ) ) = X )
24	23	3adant3 1081	. . . 4 |- ( ( K e. HL /\ X e. C /\ Y e. C ) -> ( ( pmap ` K ) ` ( ( lub ` K ) ` X ) ) = X )
25	11, 20, 5	pmapidclN 35228	. . . . 5 |- ( ( K e. HL /\ Y e. C ) -> ( ( pmap ` K ) ` ( ( lub ` K ) ` Y ) ) = Y )
26	25	3adant2 1080	. . . 4 |- ( ( K e. HL /\ X e. C /\ Y e. C ) -> ( ( pmap ` K ) ` ( ( lub ` K ) ` Y ) ) = Y )
27	24, 26	ineq12d 3815	. . 3 |- ( ( K e. HL /\ X e. C /\ Y e. C ) -> ( ( ( pmap ` K ) ` ( ( lub ` K ) ` X ) ) i^i ( ( pmap ` K ) ` ( ( lub ` K ) ` Y ) ) ) = ( X i^i Y ) )
28	22, 27	eqtrd 2656	. 2 |- ( ( K e. HL /\ X e. C /\ Y e. C ) -> ( ( pmap ` K ) ` ( ( ( lub ` K ) ` X ) ( meet ` K ) ( ( lub ` K ) ` Y ) ) ) = ( X i^i Y ) )
29		hllat 34650	. . . . 5 |- ( K e. HL -> K e. Lat )
30	29	3ad2ant1 1082	. . . 4 |- ( ( K e. HL /\ X e. C /\ Y e. C ) -> K e. Lat )
31	8, 19	latmcl 17052	. . . 4 |- ( ( K e. Lat /\ ( ( lub ` K ) ` X ) e. ( Base ` K ) /\ ( ( lub ` K ) ` Y ) e. ( Base ` K ) ) -> ( ( ( lub ` K ) ` X ) ( meet ` K ) ( ( lub ` K ) ` Y ) ) e. ( Base ` K ) )
32	30, 13, 18, 31	syl3anc 1326	. . 3 |- ( ( K e. HL /\ X e. C /\ Y e. C ) -> ( ( ( lub ` K ) ` X ) ( meet ` K ) ( ( lub ` K ) ` Y ) ) e. ( Base ` K ) )
33	8, 20, 5	pmapsubclN 35232	. . 3 |- ( ( K e. HL /\ ( ( ( lub ` K ) ` X ) ( meet ` K ) ( ( lub ` K ) ` Y ) ) e. ( Base ` K ) ) -> ( ( pmap ` K ) ` ( ( ( lub ` K ) ` X ) ( meet ` K ) ( ( lub ` K ) ` Y ) ) ) e. C )
34	1, 32, 33	syl2anc 693	. 2 |- ( ( K e. HL /\ X e. C /\ Y e. C ) -> ( ( pmap ` K ) ` ( ( ( lub ` K ) ` X ) ( meet ` K ) ( ( lub ` K ) ` Y ) ) ) e. C )
35	28, 34	eqeltrrd 2702	1 |- ( ( K e. HL /\ X e. C /\ Y e. C ) -> ( X i^i Y ) e. C )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lubclub 16942   meetcmee 16945   Latclat 17045   CLatccla 17107   Atomscatm 34550   HLchlt 34637   pmapcpmap 34783   PSubClcpscN 35220

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  ax-riotaBAD 34239

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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-riota 6611  df-ov 6653  df-oprab 6654  df-undef 7399  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-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-pmap 34790  df-polarityN 35189  df-psubclN 35221

This theorem is referenced by:  osumcllem9N  35250