Theorem lhp2at0nle 35321

Description: Inequality for 2 different atoms (or an atom and zero) under co-atom W. (Contributed by NM, 28-Jul-2013.)

Hypotheses

Ref	Expression
lhp2at0nle.l	|- .<_ = ( le ` K )
lhp2at0nle.j	|- .\/ = ( join ` K )
lhp2at0nle.z	|- .0. = ( 0. ` K )
lhp2at0nle.a	|- A = ( Atoms ` K )
lhp2at0nle.h	|- H = ( LHyp ` K )

Assertion

Ref	Expression
lhp2at0nle	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> -. V .<_ ( P .\/ U ) )

Proof of Theorem lhp2at0nle

Step	Hyp	Ref	Expression
1		simpl1 1064	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U e. A ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) )
2		simpr 477	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U e. A ) -> U e. A )
3		simpl2r 1115	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U e. A ) -> U .<_ W )
4		simpl3 1066	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U e. A ) -> ( V e. A /\ V .<_ W ) )
5		lhp2at0nle.l	. . . 4 |- .<_ = ( le ` K )
6		lhp2at0nle.j	. . . 4 |- .\/ = ( join ` K )
7		lhp2at0nle.a	. . . 4 |- A = ( Atoms ` K )
8		lhp2at0nle.h	. . . 4 |- H = ( LHyp ` K )
9	5, 6, 7, 8	lhp2atnle 35319	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> -. V .<_ ( P .\/ U ) )
10	1, 2, 3, 4, 9	syl121anc 1331	. 2 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U e. A ) -> -. V .<_ ( P .\/ U ) )
11		simp3r 1090	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> V .<_ W )
12		simp12r 1175	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> -. P .<_ W )
13		nbrne2 4673	. . . . . . 7 |- ( ( V .<_ W /\ -. P .<_ W ) -> V =/= P )
14	11, 12, 13	syl2anc 693	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> V =/= P )
15	14	neneqd 2799	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> -. V = P )
16		simp11l 1172	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> K e. HL )
17		hlatl 34647	. . . . . . 7 |- ( K e. HL -> K e. AtLat )
18	16, 17	syl 17	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> K e. AtLat )
19		simp3l 1089	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> V e. A )
20		simp12l 1174	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> P e. A )
21	5, 7	atcmp 34598	. . . . . 6 |- ( ( K e. AtLat /\ V e. A /\ P e. A ) -> ( V .<_ P <-> V = P ) )
22	18, 19, 20, 21	syl3anc 1326	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> ( V .<_ P <-> V = P ) )
23	15, 22	mtbird 315	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> -. V .<_ P )
24	23	adantr 481	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U = .0. ) -> -. V .<_ P )
25		oveq2 6658	. . . . 5 |- ( U = .0. -> ( P .\/ U ) = ( P .\/ .0. ) )
26		hlol 34648	. . . . . . 7 |- ( K e. HL -> K e. OL )
27	16, 26	syl 17	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> K e. OL )
28		eqid 2622	. . . . . . . 8 |- ( Base ` K ) = ( Base ` K )
29	28, 7	atbase 34576	. . . . . . 7 |- ( P e. A -> P e. ( Base ` K ) )
30	20, 29	syl 17	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> P e. ( Base ` K ) )
31		lhp2at0nle.z	. . . . . . 7 |- .0. = ( 0. ` K )
32	28, 6, 31	olj01 34512	. . . . . 6 |- ( ( K e. OL /\ P e. ( Base ` K ) ) -> ( P .\/ .0. ) = P )
33	27, 30, 32	syl2anc 693	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> ( P .\/ .0. ) = P )
34	25, 33	sylan9eqr 2678	. . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U = .0. ) -> ( P .\/ U ) = P )
35	34	breq2d 4665	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U = .0. ) -> ( V .<_ ( P .\/ U ) <-> V .<_ P ) )
36	24, 35	mtbird 315	. 2 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ U = .0. ) -> -. V .<_ ( P .\/ U ) )
37		simp2l 1087	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> ( U e. A \/ U = .0. ) )
38	10, 36, 37	mpjaodan 827	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( ( U e. A \/ U = .0. ) /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> -. V .<_ ( P .\/ U ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   0.cp0 17037   OLcol 34461   Atomscatm 34550   AtLatcal 34551   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-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-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  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-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274

This theorem is referenced by:  lhp2at0ne  35322  cdlemkfid1N  36209