Theorem cdlemg31c 35987

Description: Show that when N is an atom, it is not under W. TODO: Is there a shorter direct proof? TODO: should we eliminate ( F ` P ) =/= P here? (Contributed by NM, 29-May-2013.)

Hypotheses

Ref	Expression
cdlemg12.l	|- .<_ = ( le ` K )
cdlemg12.j	|- .\/ = ( join ` K )
cdlemg12.m	|- ./\ = ( meet ` K )
cdlemg12.a	|- A = ( Atoms ` K )
cdlemg12.h	|- H = ( LHyp ` K )
cdlemg12.t	|- T = ( ( LTrn ` K ) ` W )
cdlemg12b.r	|- R = ( ( trL ` K ) ` W )
cdlemg31.n	|- N = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) )

Assertion

Ref	Expression
cdlemg31c	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> -. N .<_ W )

Proof of Theorem cdlemg31c

Step	Hyp	Ref	Expression
1		simp11l 1172	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> K e. HL )
2		simp11r 1173	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> W e. H )
3	1, 2	jca 554	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> ( K e. HL /\ W e. H ) )
4		simp13 1093	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> ( Q e. A /\ -. Q .<_ W ) )
5		simp31 1097	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> v =/= ( R ` F ) )
6	5	necomd 2849	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> ( R ` F ) =/= v )
7		simp12 1092	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> ( P e. A /\ -. P .<_ W ) )
8		simp2r 1088	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> F e. T )
9		simp32 1098	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> ( F ` P ) =/= P )
10		cdlemg12.l	. . . . 5 |- .<_ = ( le ` K )
11		cdlemg12.a	. . . . 5 |- A = ( Atoms ` K )
12		cdlemg12.h	. . . . 5 |- H = ( LHyp ` K )
13		cdlemg12.t	. . . . 5 |- T = ( ( LTrn ` K ) ` W )
14		cdlemg12b.r	. . . . 5 |- R = ( ( trL ` K ) ` W )
15	10, 11, 12, 13, 14	trlat 35456	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> ( R ` F ) e. A )
16	3, 7, 8, 9, 15	syl112anc 1330	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> ( R ` F ) e. A )
17	10, 12, 13, 14	trlle 35471	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) .<_ W )
18	3, 8, 17	syl2anc 693	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> ( R ` F ) .<_ W )
19		simp2l 1087	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> ( v e. A /\ v .<_ W ) )
20		cdlemg12.j	. . . 4 |- .\/ = ( join ` K )
21	10, 20, 11, 12	lhp2atnle 35319	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R ` F ) =/= v ) /\ ( ( R ` F ) e. A /\ ( R ` F ) .<_ W ) /\ ( v e. A /\ v .<_ W ) ) -> -. v .<_ ( Q .\/ ( R ` F ) ) )
22	3, 4, 6, 16, 18, 19, 21	syl321anc 1348	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> -. v .<_ ( Q .\/ ( R ` F ) ) )
23		simp12l 1174	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> P e. A )
24		simp13l 1176	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> Q e. A )
25		simp2ll 1128	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> v e. A )
26		cdlemg12.m	. . . . . . 7 |- ./\ = ( meet ` K )
27		cdlemg31.n	. . . . . . 7 |- N = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) )
28	10, 20, 26, 11, 12, 13, 14, 27	cdlemg31a 35985	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) /\ ( v e. A /\ F e. T ) ) -> N .<_ ( P .\/ v ) )
29	1, 2, 23, 24, 25, 8, 28	syl222anc 1342	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> N .<_ ( P .\/ v ) )
30	29	adantr 481	. . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) /\ N .<_ W ) -> N .<_ ( P .\/ v ) )
31		simp111 1190	. . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) /\ N .<_ W /\ N =/= v ) -> ( K e. HL /\ W e. H ) )
32		simp112 1191	. . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) /\ N .<_ W /\ N =/= v ) -> ( P e. A /\ -. P .<_ W ) )
33		simp3 1063	. . . . . . . 8 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) /\ N .<_ W /\ N =/= v ) -> N =/= v )
34	33	necomd 2849	. . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) /\ N .<_ W /\ N =/= v ) -> v =/= N )
35		simp12l 1174	. . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) /\ N .<_ W /\ N =/= v ) -> ( v e. A /\ v .<_ W ) )
36		simp133 1198	. . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) /\ N .<_ W /\ N =/= v ) -> N e. A )
37		simp2 1062	. . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) /\ N .<_ W /\ N =/= v ) -> N .<_ W )
38	10, 20, 11, 12	lhp2atnle 35319	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ v =/= N ) /\ ( v e. A /\ v .<_ W ) /\ ( N e. A /\ N .<_ W ) ) -> -. N .<_ ( P .\/ v ) )
39	31, 32, 34, 35, 36, 37, 38	syl312anc 1347	. . . . . 6 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) /\ N .<_ W /\ N =/= v ) -> -. N .<_ ( P .\/ v ) )
40	39	3expia 1267	. . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) /\ N .<_ W ) -> ( N =/= v -> -. N .<_ ( P .\/ v ) ) )
41	40	necon4ad 2813	. . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) /\ N .<_ W ) -> ( N .<_ ( P .\/ v ) -> N = v ) )
42	30, 41	mpd 15	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) /\ N .<_ W ) -> N = v )
43	10, 20, 26, 11, 12, 13, 14, 27	cdlemg31b 35986	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) /\ ( v e. A /\ F e. T ) ) -> N .<_ ( Q .\/ ( R ` F ) ) )
44	1, 2, 23, 24, 25, 8, 43	syl222anc 1342	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> N .<_ ( Q .\/ ( R ` F ) ) )
45	44	adantr 481	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) /\ N .<_ W ) -> N .<_ ( Q .\/ ( R ` F ) ) )
46	42, 45	eqbrtrrd 4677	. 2 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) /\ N .<_ W ) -> v .<_ ( Q .\/ ( R ` F ) ) )
47	22, 46	mtand 691	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> -. N .<_ W )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   lecple 15948   joincjn 16944   meetcmee 16945   Atomscatm 34550   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   trLctrl 35445

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-map 7859  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-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446

This theorem is referenced by:  cdlemg31d  35988