Theorem cdleme1b 35513

Description: Part of proof of Lemma E in [Crawley] p. 113. Utility lemma showing F is a lattice element. F represents their f(r). (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 ) ) )
cdleme1.b	|- B = ( Base ` K )

Assertion

Ref	Expression
cdleme1b	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> F e. B )

Proof of Theorem cdleme1b

Step	Hyp	Ref	Expression
1		cdleme1.f	. 2 |- F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
2		hllat 34650	. . . 4 |- ( K e. HL -> K e. Lat )
3	2	ad2antrr 762	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> K e. Lat )
4		simpr3 1069	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> R e. A )
5		cdleme1.b	. . . . . 6 |- B = ( Base ` K )
6		cdleme1.a	. . . . . 6 |- A = ( Atoms ` K )
7	5, 6	atbase 34576	. . . . 5 |- ( R e. A -> R e. B )
8	4, 7	syl 17	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> R e. B )
9		cdleme1.l	. . . . . 6 |- .<_ = ( le ` K )
10		cdleme1.j	. . . . . 6 |- .\/ = ( join ` K )
11		cdleme1.m	. . . . . 6 |- ./\ = ( meet ` K )
12		cdleme1.h	. . . . . 6 |- H = ( LHyp ` K )
13		cdleme1.u	. . . . . 6 |- U = ( ( P .\/ Q ) ./\ W )
14	9, 10, 11, 6, 12, 13, 5	cdleme0aa 35497	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) -> U e. B )
15	14	3adant3r3 1276	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> U e. B )
16	5, 10	latjcl 17051	. . . 4 |- ( ( K e. Lat /\ R e. B /\ U e. B ) -> ( R .\/ U ) e. B )
17	3, 8, 15, 16	syl3anc 1326	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( R .\/ U ) e. B )
18		simpr2 1068	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> Q e. A )
19	5, 6	atbase 34576	. . . . 5 |- ( Q e. A -> Q e. B )
20	18, 19	syl 17	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> Q e. B )
21		simpr1 1067	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> P e. A )
22	5, 6	atbase 34576	. . . . . . 7 |- ( P e. A -> P e. B )
23	21, 22	syl 17	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> P e. B )
24	5, 10	latjcl 17051	. . . . . 6 |- ( ( K e. Lat /\ P e. B /\ R e. B ) -> ( P .\/ R ) e. B )
25	3, 23, 8, 24	syl3anc 1326	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( P .\/ R ) e. B )
26	5, 12	lhpbase 35284	. . . . . 6 |- ( W e. H -> W e. B )
27	26	ad2antlr 763	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> W e. B )
28	5, 11	latmcl 17052	. . . . 5 |- ( ( K e. Lat /\ ( P .\/ R ) e. B /\ W e. B ) -> ( ( P .\/ R ) ./\ W ) e. B )
29	3, 25, 27, 28	syl3anc 1326	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( P .\/ R ) ./\ W ) e. B )
30	5, 10	latjcl 17051	. . . 4 |- ( ( K e. Lat /\ Q e. B /\ ( ( P .\/ R ) ./\ W ) e. B ) -> ( Q .\/ ( ( P .\/ R ) ./\ W ) ) e. B )
31	3, 20, 29, 30	syl3anc 1326	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( Q .\/ ( ( P .\/ R ) ./\ W ) ) e. B )
32	5, 11	latmcl 17052	. . 3 |- ( ( K e. Lat /\ ( R .\/ U ) e. B /\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) e. B ) -> ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) e. B )
33	3, 17, 31, 32	syl3anc 1326	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) e. B )
34	1, 33	syl5eqel 2705	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> F e. B )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` 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:  cdleme3c  35517  cdleme4a  35526  cdleme5  35527  cdleme7e  35534  cdleme11  35557  cdleme15  35565  cdleme22gb  35581  cdleme19b  35592  cdleme19e  35595  cdleme20d  35600  cdleme20j  35606  cdleme20k  35607  cdleme20l2  35609  cdleme20l  35610  cdleme20m  35611  cdleme22e  35632  cdleme22eALTN  35633  cdleme22f  35634  cdleme27cl  35654  cdlemefr27cl  35691  cdleme35fnpq  35737