Theorem paddatclN 35235

Description: The projective sum of a closed subspace and an atom is a closed projective subspace. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
paddatcl.a	|- A = ( Atoms ` K )
paddatcl.p	|- .+ = ( +P ` K )
paddatcl.c	|- C = ( PSubCl ` K )

Assertion

Ref	Expression
paddatclN	|- ( ( K e. HL /\ X e. C /\ Q e. A ) -> ( X .+ { Q } ) e. C )

Proof of Theorem paddatclN

Step	Hyp	Ref	Expression
1		hlclat 34645	. . . . . 6 |- ( K e. HL -> K e. CLat )
2	1	3ad2ant1 1082	. . . . 5 |- ( ( K e. HL /\ X e. C /\ Q e. A ) -> K e. CLat )
3		paddatcl.a	. . . . . . . 8 |- A = ( Atoms ` K )
4		paddatcl.c	. . . . . . . 8 |- C = ( PSubCl ` K )
5	3, 4	psubclssatN 35227	. . . . . . 7 |- ( ( K e. HL /\ X e. C ) -> X C_ A )
6		eqid 2622	. . . . . . . 8 |- ( Base ` K ) = ( Base ` K )
7	6, 3	atssbase 34577	. . . . . . 7 |- A C_ ( Base ` K )
8	5, 7	syl6ss 3615	. . . . . 6 |- ( ( K e. HL /\ X e. C ) -> X C_ ( Base ` K ) )
9	8	3adant3 1081	. . . . 5 |- ( ( K e. HL /\ X e. C /\ Q e. A ) -> X C_ ( Base ` K ) )
10		eqid 2622	. . . . . 6 |- ( lub ` K ) = ( lub ` K )
11	6, 10	clatlubcl 17112	. . . . 5 |- ( ( K e. CLat /\ X C_ ( Base ` K ) ) -> ( ( lub ` K ) ` X ) e. ( Base ` K ) )
12	2, 9, 11	syl2anc 693	. . . 4 |- ( ( K e. HL /\ X e. C /\ Q e. A ) -> ( ( lub ` K ) ` X ) e. ( Base ` K ) )
13		eqid 2622	. . . . 5 |- ( join ` K ) = ( join ` K )
14		eqid 2622	. . . . 5 |- ( pmap ` K ) = ( pmap ` K )
15		paddatcl.p	. . . . 5 |- .+ = ( +P ` K )
16	6, 13, 3, 14, 15	pmapjat1 35139	. . . 4 |- ( ( K e. HL /\ ( ( lub ` K ) ` X ) e. ( Base ` K ) /\ Q e. A ) -> ( ( pmap ` K ) ` ( ( ( lub ` K ) ` X ) ( join ` K ) Q ) ) = ( ( ( pmap ` K ) ` ( ( lub ` K ) ` X ) ) .+ ( ( pmap ` K ) ` Q ) ) )
17	12, 16	syld3an2 1373	. . 3 |- ( ( K e. HL /\ X e. C /\ Q e. A ) -> ( ( pmap ` K ) ` ( ( ( lub ` K ) ` X ) ( join ` K ) Q ) ) = ( ( ( pmap ` K ) ` ( ( lub ` K ) ` X ) ) .+ ( ( pmap ` K ) ` Q ) ) )
18	10, 14, 4	pmapidclN 35228	. . . . 5 |- ( ( K e. HL /\ X e. C ) -> ( ( pmap ` K ) ` ( ( lub ` K ) ` X ) ) = X )
19	18	3adant3 1081	. . . 4 |- ( ( K e. HL /\ X e. C /\ Q e. A ) -> ( ( pmap ` K ) ` ( ( lub ` K ) ` X ) ) = X )
20	3, 14	pmapat 35049	. . . . 5 |- ( ( K e. HL /\ Q e. A ) -> ( ( pmap ` K ) ` Q ) = { Q } )
21	20	3adant2 1080	. . . 4 |- ( ( K e. HL /\ X e. C /\ Q e. A ) -> ( ( pmap ` K ) ` Q ) = { Q } )
22	19, 21	oveq12d 6668	. . 3 |- ( ( K e. HL /\ X e. C /\ Q e. A ) -> ( ( ( pmap ` K ) ` ( ( lub ` K ) ` X ) ) .+ ( ( pmap ` K ) ` Q ) ) = ( X .+ { Q } ) )
23	17, 22	eqtr2d 2657	. 2 |- ( ( K e. HL /\ X e. C /\ Q e. A ) -> ( X .+ { Q } ) = ( ( pmap ` K ) ` ( ( ( lub ` K ) ` X ) ( join ` K ) Q ) ) )
24		simp1 1061	. . 3 |- ( ( K e. HL /\ X e. C /\ Q e. A ) -> K e. HL )
25		hllat 34650	. . . . 5 |- ( K e. HL -> K e. Lat )
26	25	3ad2ant1 1082	. . . 4 |- ( ( K e. HL /\ X e. C /\ Q e. A ) -> K e. Lat )
27	6, 3	atbase 34576	. . . . 5 |- ( Q e. A -> Q e. ( Base ` K ) )
28	27	3ad2ant3 1084	. . . 4 |- ( ( K e. HL /\ X e. C /\ Q e. A ) -> Q e. ( Base ` K ) )
29	6, 13	latjcl 17051	. . . 4 |- ( ( K e. Lat /\ ( ( lub ` K ) ` X ) e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( ( ( lub ` K ) ` X ) ( join ` K ) Q ) e. ( Base ` K ) )
30	26, 12, 28, 29	syl3anc 1326	. . 3 |- ( ( K e. HL /\ X e. C /\ Q e. A ) -> ( ( ( lub ` K ) ` X ) ( join ` K ) Q ) e. ( Base ` K ) )
31	6, 14, 4	pmapsubclN 35232	. . 3 |- ( ( K e. HL /\ ( ( ( lub ` K ) ` X ) ( join ` K ) Q ) e. ( Base ` K ) ) -> ( ( pmap ` K ) ` ( ( ( lub ` K ) ` X ) ( join ` K ) Q ) ) e. C )
32	24, 30, 31	syl2anc 693	. 2 |- ( ( K e. HL /\ X e. C /\ Q e. A ) -> ( ( pmap ` K ) ` ( ( ( lub ` K ) ` X ) ( join ` K ) Q ) ) e. C )
33	23, 32	eqeltrd 2701	1 |- ( ( K e. HL /\ X e. C /\ Q e. A ) -> ( X .+ { Q } ) e. C )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   {csn 4177   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lubclub 16942   joincjn 16944   Latclat 17045   CLatccla 17107   Atomscatm 34550   HLchlt 34637   pmapcpmap 34783   +Pcpadd 35081   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-mpt2 6655  df-1st 7168  df-2nd 7169  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-padd 35082  df-polarityN 35189  df-psubclN 35221

This theorem is referenced by:  pclfinclN  35236  osumcllem9N  35250  pexmidlem6N  35261