Theorem lhp0lt 35289

Description: A co-atom is greater than zero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)

Hypotheses

Ref	Expression
lhp0lt.s	|- .< = ( lt ` K )
lhp0lt.z	|- .0. = ( 0. ` K )
lhp0lt.h	|- H = ( LHyp ` K )

Assertion

Ref	Expression
lhp0lt	|- ( ( K e. HL /\ W e. H ) -> .0. .< W )

Proof of Theorem lhp0lt

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lhp0lt.s	. . 3 |- .< = ( lt ` K )
2		eqid 2622	. . 3 |- ( Atoms ` K ) = ( Atoms ` K )
3		lhp0lt.h	. . 3 |- H = ( LHyp ` K )
4	1, 2, 3	lhpexlt 35288	. 2 |- ( ( K e. HL /\ W e. H ) -> E. p e. ( Atoms ` K ) p .< W )
5		simp1l 1085	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ p e. ( Atoms ` K ) /\ p .< W ) -> K e. HL )
6		hlop 34649	. . . . . 6 |- ( K e. HL -> K e. OP )
7		eqid 2622	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
8		lhp0lt.z	. . . . . . 7 |- .0. = ( 0. ` K )
9	7, 8	op0cl 34471	. . . . . 6 |- ( K e. OP -> .0. e. ( Base ` K ) )
10	5, 6, 9	3syl 18	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ p e. ( Atoms ` K ) /\ p .< W ) -> .0. e. ( Base ` K ) )
11	7, 2	atbase 34576	. . . . . 6 |- ( p e. ( Atoms ` K ) -> p e. ( Base ` K ) )
12	11	3ad2ant2 1083	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ p e. ( Atoms ` K ) /\ p .< W ) -> p e. ( Base ` K ) )
13		simp2 1062	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ p e. ( Atoms ` K ) /\ p .< W ) -> p e. ( Atoms ` K ) )
14		eqid 2622	. . . . . . 7 |- ( <o ` K ) = ( <o ` K )
15	8, 14, 2	atcvr0 34575	. . . . . 6 |- ( ( K e. HL /\ p e. ( Atoms ` K ) ) -> .0. ( <o ` K ) p )
16	5, 13, 15	syl2anc 693	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ p e. ( Atoms ` K ) /\ p .< W ) -> .0. ( <o ` K ) p )
17	7, 1, 14	cvrlt 34557	. . . . 5 |- ( ( ( K e. HL /\ .0. e. ( Base ` K ) /\ p e. ( Base ` K ) ) /\ .0. ( <o ` K ) p ) -> .0. .< p )
18	5, 10, 12, 16, 17	syl31anc 1329	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ p e. ( Atoms ` K ) /\ p .< W ) -> .0. .< p )
19		simp3 1063	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ p e. ( Atoms ` K ) /\ p .< W ) -> p .< W )
20		hlpos 34652	. . . . . 6 |- ( K e. HL -> K e. Poset )
21	5, 20	syl 17	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ p e. ( Atoms ` K ) /\ p .< W ) -> K e. Poset )
22		simp1r 1086	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ p e. ( Atoms ` K ) /\ p .< W ) -> W e. H )
23	7, 3	lhpbase 35284	. . . . . 6 |- ( W e. H -> W e. ( Base ` K ) )
24	22, 23	syl 17	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ p e. ( Atoms ` K ) /\ p .< W ) -> W e. ( Base ` K ) )
25	7, 1	plttr 16970	. . . . 5 |- ( ( K e. Poset /\ ( .0. e. ( Base ` K ) /\ p e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( .0. .< p /\ p .< W ) -> .0. .< W ) )
26	21, 10, 12, 24, 25	syl13anc 1328	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ p e. ( Atoms ` K ) /\ p .< W ) -> ( ( .0. .< p /\ p .< W ) -> .0. .< W ) )
27	18, 19, 26	mp2and 715	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ p e. ( Atoms ` K ) /\ p .< W ) -> .0. .< W )
28	27	rexlimdv3a 3033	. 2 |- ( ( K e. HL /\ W e. H ) -> ( E. p e. ( Atoms ` K ) p .< W -> .0. .< W ) )
29	4, 28	mpd 15	1 |- ( ( K e. HL /\ W e. H ) -> .0. .< W )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653   ` cfv 5888   Basecbs 15857   Posetcpo 16940   ltcplt 16941   0.cp0 17037   OPcops 34459   <o ccvr 34549   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:  lhpn0  35290