Theorem cdleme3c 35517

Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 35523 and cdleme3 35524. (Contributed by NM, 6-Jun-2012.)

Hypotheses

Ref	Expression
cdleme1.l	|- .<_ = ( le ` K )
cdleme1.j	|- .\/ = ( join ` K )
cdleme1.m	|- ./\ = ( meet ` K )
cdleme1.a	|- A = ( Atoms ` K )
cdleme1.h	|- H = ( LHyp ` K )
cdleme1.u	|- U = ( ( P .\/ Q ) ./\ W )
cdleme1.f	|- F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
cdleme3c.z	|- .0. = ( 0. ` K )

Assertion

Ref	Expression
cdleme3c	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> F =/= .0. )

Proof of Theorem cdleme3c

Step	Hyp	Ref	Expression
1		simpll 790	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> K e. HL )
2		hllat 34650	. . . . . 6 |- ( K e. HL -> K e. Lat )
3	2	ad2antrr 762	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> K e. Lat )
4		simpr3l 1122	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> R e. A )
5		eqid 2622	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
6		cdleme1.a	. . . . . . 7 |- A = ( Atoms ` K )
7	5, 6	atbase 34576	. . . . . 6 |- ( R e. A -> R e. ( Base ` K ) )
8	4, 7	syl 17	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> R e. ( Base ` K ) )
9		hlop 34649	. . . . . . 7 |- ( K e. HL -> K e. OP )
10	9	ad2antrr 762	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> K e. OP )
11		cdleme3c.z	. . . . . . 7 |- .0. = ( 0. ` K )
12	5, 11	op0cl 34471	. . . . . 6 |- ( K e. OP -> .0. e. ( Base ` K ) )
13	10, 12	syl 17	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> .0. e. ( Base ` K ) )
14		cdleme1.j	. . . . . 6 |- .\/ = ( join ` K )
15	5, 14	latjcl 17051	. . . . 5 |- ( ( K e. Lat /\ R e. ( Base ` K ) /\ .0. e. ( Base ` K ) ) -> ( R .\/ .0. ) e. ( Base ` K ) )
16	3, 8, 13, 15	syl3anc 1326	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ .0. ) e. ( Base ` K ) )
17		simpl 473	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( K e. HL /\ W e. H ) )
18		simpr1l 1118	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> P e. A )
19		simpr2l 1120	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> Q e. A )
20		cdleme1.l	. . . . . . 7 |- .<_ = ( le ` K )
21		cdleme1.m	. . . . . . 7 |- ./\ = ( meet ` K )
22		cdleme1.h	. . . . . . 7 |- H = ( LHyp ` K )
23		cdleme1.u	. . . . . . 7 |- U = ( ( P .\/ Q ) ./\ W )
24		cdleme1.f	. . . . . . 7 |- F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
25	20, 14, 21, 6, 22, 23, 24, 5	cdleme1b 35513	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> F e. ( Base ` K ) )
26	17, 18, 19, 4, 25	syl13anc 1328	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> F e. ( Base ` K ) )
27	5, 14	latjcl 17051	. . . . 5 |- ( ( K e. Lat /\ R e. ( Base ` K ) /\ F e. ( Base ` K ) ) -> ( R .\/ F ) e. ( Base ` K ) )
28	3, 8, 26, 27	syl3anc 1326	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ F ) e. ( Base ` K ) )
29	5, 6	atbase 34576	. . . . . . . . . . . 12 |- ( P e. A -> P e. ( Base ` K ) )
30	18, 29	syl 17	. . . . . . . . . . 11 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> P e. ( Base ` K ) )
31	5, 6	atbase 34576	. . . . . . . . . . . 12 |- ( Q e. A -> Q e. ( Base ` K ) )
32	19, 31	syl 17	. . . . . . . . . . 11 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> Q e. ( Base ` K ) )
33	5, 14	latjcl 17051	. . . . . . . . . . 11 |- ( ( K e. Lat /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
34	3, 30, 32, 33	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
35	5, 22	lhpbase 35284	. . . . . . . . . . 11 |- ( W e. H -> W e. ( Base ` K ) )
36	35	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> W e. ( Base ` K ) )
37	5, 20, 21	latmle2 17077	. . . . . . . . . 10 |- ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
38	3, 34, 36, 37	syl3anc 1326	. . . . . . . . 9 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
39	23, 38	syl5eqbr 4688	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> U .<_ W )
40		simpr3r 1123	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> -. R .<_ W )
41		nbrne2 4673	. . . . . . . 8 |- ( ( U .<_ W /\ -. R .<_ W ) -> U =/= R )
42	39, 40, 41	syl2anc 693	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> U =/= R )
43	42	necomd 2849	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> R =/= U )
44	20, 14, 21, 6, 22, 23	lhpat2 35331	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> U e. A )
45	44	3adant3r3 1276	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> U e. A )
46		eqid 2622	. . . . . . . 8 |- ( <o ` K ) = ( <o ` K )
47	14, 46, 6	atcvr1 34703	. . . . . . 7 |- ( ( K e. HL /\ R e. A /\ U e. A ) -> ( R =/= U <-> R ( <o ` K ) ( R .\/ U ) ) )
48	1, 4, 45, 47	syl3anc 1326	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R =/= U <-> R ( <o ` K ) ( R .\/ U ) ) )
49	43, 48	mpbid 222	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> R ( <o ` K ) ( R .\/ U ) )
50		hlol 34648	. . . . . . 7 |- ( K e. HL -> K e. OL )
51	50	ad2antrr 762	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> K e. OL )
52	5, 14, 11	olj01 34512	. . . . . 6 |- ( ( K e. OL /\ R e. ( Base ` K ) ) -> ( R .\/ .0. ) = R )
53	51, 8, 52	syl2anc 693	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ .0. ) = R )
54		simpr3 1069	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R e. A /\ -. R .<_ W ) )
55	20, 14, 21, 6, 22, 23, 24	cdleme1 35514	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ F ) = ( R .\/ U ) )
56	17, 18, 19, 54, 55	syl13anc 1328	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ F ) = ( R .\/ U ) )
57	49, 53, 56	3brtr4d 4685	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ .0. ) ( <o ` K ) ( R .\/ F ) )
58	5, 46	cvrne 34568	. . . 4 |- ( ( ( K e. HL /\ ( R .\/ .0. ) e. ( Base ` K ) /\ ( R .\/ F ) e. ( Base ` K ) ) /\ ( R .\/ .0. ) ( <o ` K ) ( R .\/ F ) ) -> ( R .\/ .0. ) =/= ( R .\/ F ) )
59	1, 16, 28, 57, 58	syl31anc 1329	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ .0. ) =/= ( R .\/ F ) )
60		oveq2 6658	. . . 4 |- ( .0. = F -> ( R .\/ .0. ) = ( R .\/ F ) )
61	60	necon3i 2826	. . 3 |- ( ( R .\/ .0. ) =/= ( R .\/ F ) -> .0. =/= F )
62	59, 61	syl 17	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> .0. =/= F )
63	62	necomd 2849	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> F =/= .0. )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ 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   meetcmee 16945   0.cp0 17037   Latclat 17045   OPcops 34459   OLcol 34461   <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-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-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

This theorem is referenced by:  cdleme3h  35522