Theorem lhpj1 35308

Description: The join of a co-atom (hyperplane) and an element not under it is the lattice unit. (Contributed by NM, 7-Dec-2012.)

Hypotheses

Ref	Expression
lhpj1.b	|- B = ( Base ` K )
lhpj1.l	|- .<_ = ( le ` K )
lhpj1.j	|- .\/ = ( join ` K )
lhpj1.u	|- .1. = ( 1. ` K )
lhpj1.h	|- H = ( LHyp ` K )

Assertion

Ref	Expression
lhpj1	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( W .\/ X ) = .1. )

Proof of Theorem lhpj1

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 790	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> K e. HL )
2		simpr 477	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> X e. B )
3		lhpj1.b	. . . . . 6 |- B = ( Base ` K )
4		lhpj1.h	. . . . . 6 |- H = ( LHyp ` K )
5	3, 4	lhpbase 35284	. . . . 5 |- ( W e. H -> W e. B )
6	5	ad2antlr 763	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> W e. B )
7		lhpj1.l	. . . . 5 |- .<_ = ( le ` K )
8		eqid 2622	. . . . 5 |- ( Atoms ` K ) = ( Atoms ` K )
9	3, 7, 8	hlrelat2 34689	. . . 4 |- ( ( K e. HL /\ X e. B /\ W e. B ) -> ( -. X .<_ W <-> E. p e. ( Atoms ` K ) ( p .<_ X /\ -. p .<_ W ) ) )
10	1, 2, 6, 9	syl3anc 1326	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( -. X .<_ W <-> E. p e. ( Atoms ` K ) ( p .<_ X /\ -. p .<_ W ) ) )
11		simp1l 1085	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> ( K e. HL /\ W e. H ) )
12		simp2 1062	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> p e. ( Atoms ` K ) )
13		simp3r 1090	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> -. p .<_ W )
14		lhpj1.j	. . . . . . . 8 |- .\/ = ( join ` K )
15		lhpj1.u	. . . . . . . 8 |- .1. = ( 1. ` K )
16	7, 14, 15, 8, 4	lhpjat1 35306	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( p e. ( Atoms ` K ) /\ -. p .<_ W ) ) -> ( W .\/ p ) = .1. )
17	11, 12, 13, 16	syl12anc 1324	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> ( W .\/ p ) = .1. )
18		simp3l 1089	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> p .<_ X )
19		simp1ll 1124	. . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> K e. HL )
20		hllat 34650	. . . . . . . . 9 |- ( K e. HL -> K e. Lat )
21	19, 20	syl 17	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> K e. Lat )
22	3, 8	atbase 34576	. . . . . . . . 9 |- ( p e. ( Atoms ` K ) -> p e. B )
23	22	3ad2ant2 1083	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> p e. B )
24		simp1r 1086	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> X e. B )
25	6	3ad2ant1 1082	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> W e. B )
26	3, 7, 14	latjlej2 17066	. . . . . . . 8 |- ( ( K e. Lat /\ ( p e. B /\ X e. B /\ W e. B ) ) -> ( p .<_ X -> ( W .\/ p ) .<_ ( W .\/ X ) ) )
27	21, 23, 24, 25, 26	syl13anc 1328	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> ( p .<_ X -> ( W .\/ p ) .<_ ( W .\/ X ) ) )
28	18, 27	mpd 15	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> ( W .\/ p ) .<_ ( W .\/ X ) )
29	17, 28	eqbrtrrd 4677	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> .1. .<_ ( W .\/ X ) )
30		hlop 34649	. . . . . . 7 |- ( K e. HL -> K e. OP )
31	19, 30	syl 17	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> K e. OP )
32	3, 14	latjcl 17051	. . . . . . 7 |- ( ( K e. Lat /\ W e. B /\ X e. B ) -> ( W .\/ X ) e. B )
33	21, 25, 24, 32	syl3anc 1326	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> ( W .\/ X ) e. B )
34	3, 7, 15	op1le 34479	. . . . . 6 |- ( ( K e. OP /\ ( W .\/ X ) e. B ) -> ( .1. .<_ ( W .\/ X ) <-> ( W .\/ X ) = .1. ) )
35	31, 33, 34	syl2anc 693	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> ( .1. .<_ ( W .\/ X ) <-> ( W .\/ X ) = .1. ) )
36	29, 35	mpbid 222	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> ( W .\/ X ) = .1. )
37	36	rexlimdv3a 3033	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( E. p e. ( Atoms ` K ) ( p .<_ X /\ -. p .<_ W ) -> ( W .\/ X ) = .1. ) )
38	10, 37	sylbid 230	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( -. X .<_ W -> ( W .\/ X ) = .1. ) )
39	38	impr 649	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( W .\/ X ) = .1. )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   1.cp1 17038   Latclat 17045   OPcops 34459   Atomscatm 34550   HLchlt 34637   LHypclh 35270

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-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-lhyp 35274

This theorem is referenced by:  lhpmcvr  35309  cdleme30a  35666