Theorem cdleme4a 35526

Description: Part of proof of Lemma E in [Crawley] p. 114 top. G represents f_s(r). Auxiliary lemma derived from cdleme5 35527. We show f_s(r) <_ p \/ q. (Contributed by NM, 10-Nov-2012.)

Hypotheses

Ref	Expression
cdleme4.l	|- .<_ = ( le ` K )
cdleme4.j	|- .\/ = ( join ` K )
cdleme4.m	|- ./\ = ( meet ` K )
cdleme4.a	|- A = ( Atoms ` K )
cdleme4.h	|- H = ( LHyp ` K )
cdleme4.u	|- U = ( ( P .\/ Q ) ./\ W )
cdleme4.f	|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme4.g	|- G = ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) )

Assertion

Ref	Expression
cdleme4a	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> G .<_ ( P .\/ Q ) )

Proof of Theorem cdleme4a

Step	Hyp	Ref	Expression
1		cdleme4.g	. 2 |- G = ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) )
2		simp1l 1085	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> K e. HL )
3		hllat 34650	. . . 4 |- ( K e. HL -> K e. Lat )
4	2, 3	syl 17	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> K e. Lat )
5		simp21 1094	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> P e. A )
6		simp22 1095	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> Q e. A )
7		eqid 2622	. . . . 5 |- ( Base ` K ) = ( Base ` K )
8		cdleme4.j	. . . . 5 |- .\/ = ( join ` K )
9		cdleme4.a	. . . . 5 |- A = ( Atoms ` K )
10	7, 8, 9	hlatjcl 34653	. . . 4 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
11	2, 5, 6, 10	syl3anc 1326	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
12		simp1r 1086	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> W e. H )
13		simp3 1063	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> S e. A )
14		cdleme4.l	. . . . . 6 |- .<_ = ( le ` K )
15		cdleme4.m	. . . . . 6 |- ./\ = ( meet ` K )
16		cdleme4.h	. . . . . 6 |- H = ( LHyp ` K )
17		cdleme4.u	. . . . . 6 |- U = ( ( P .\/ Q ) ./\ W )
18		cdleme4.f	. . . . . 6 |- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
19	14, 8, 15, 9, 16, 17, 18, 7	cdleme1b 35513	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ S e. A ) ) -> F e. ( Base ` K ) )
20	2, 12, 5, 6, 13, 19	syl23anc 1333	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> F e. ( Base ` K ) )
21		simp23 1096	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> R e. A )
22	7, 8, 9	hlatjcl 34653	. . . . . 6 |- ( ( K e. HL /\ R e. A /\ S e. A ) -> ( R .\/ S ) e. ( Base ` K ) )
23	2, 21, 13, 22	syl3anc 1326	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> ( R .\/ S ) e. ( Base ` K ) )
24	7, 16	lhpbase 35284	. . . . . 6 |- ( W e. H -> W e. ( Base ` K ) )
25	12, 24	syl 17	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> W e. ( Base ` K ) )
26	7, 15	latmcl 17052	. . . . 5 |- ( ( K e. Lat /\ ( R .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( R .\/ S ) ./\ W ) e. ( Base ` K ) )
27	4, 23, 25, 26	syl3anc 1326	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> ( ( R .\/ S ) ./\ W ) e. ( Base ` K ) )
28	7, 8	latjcl 17051	. . . 4 |- ( ( K e. Lat /\ F e. ( Base ` K ) /\ ( ( R .\/ S ) ./\ W ) e. ( Base ` K ) ) -> ( F .\/ ( ( R .\/ S ) ./\ W ) ) e. ( Base ` K ) )
29	4, 20, 27, 28	syl3anc 1326	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> ( F .\/ ( ( R .\/ S ) ./\ W ) ) e. ( Base ` K ) )
30	7, 14, 15	latmle1 17076	. . 3 |- ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) ) .<_ ( P .\/ Q ) )
31	4, 11, 29, 30	syl3anc 1326	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) ) .<_ ( P .\/ Q ) )
32	1, 31	syl5eqbr 4688	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> G .<_ ( P .\/ Q ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   Latclat 17045   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-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-lat 17046  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-lhyp 35274

This theorem is referenced by:  cdleme18c  35580  cdleme41sn3a  35721