Theorem cdleme8 30744

Description: Part of proof of Lemma E in [Crawley] p. 113, 2nd paragraph on p. 114. C represents s₁. In their notation, we prove p \/ s₁ = p \/ s. (Contributed by NM, 9-Jun-2012.)

Hypotheses

Ref	Expression
cdleme8.l	|- .<_ = ( le ` K )
cdleme8.j	|- .\/ = ( join ` K )
cdleme8.m	|- ./\ = ( meet ` K )
cdleme8.a	|- A = ( Atoms ` K )
cdleme8.h	|- H = ( LHyp ` K )
cdleme8.4	|- C = ( ( P .\/ S ) ./\ W )

Assertion

Ref	Expression
cdleme8	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> ( P .\/ C ) = ( P .\/ S ) )

Proof of Theorem cdleme8

Step	Hyp	Ref	Expression
1		cdleme8.4	. . 3 |- C = ( ( P .\/ S ) ./\ W )
2	1	oveq2i 6059	. 2 |- ( P .\/ C ) = ( P .\/ ( ( P .\/ S ) ./\ W ) )
3		simp1l 981	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> K e. HL )
4		simp2l 983	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> P e. A )
5		hllat 29858	. . . . . 6 |- ( K e. HL -> K e. Lat )
6	3, 5	syl 16	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> K e. Lat )
7		eqid 2412	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
8		cdleme8.a	. . . . . . 7 |- A = ( Atoms ` K )
9	7, 8	atbase 29784	. . . . . 6 |- ( P e. A -> P e. ( Base ` K ) )
10	4, 9	syl 16	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> P e. ( Base ` K ) )
11	7, 8	atbase 29784	. . . . . 6 |- ( S e. A -> S e. ( Base ` K ) )
12	11	3ad2ant3 980	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> S e. ( Base ` K ) )
13		cdleme8.j	. . . . . 6 |- .\/ = ( join ` K )
14	7, 13	latjcl 14442	. . . . 5 |- ( ( K e. Lat /\ P e. ( Base ` K ) /\ S e. ( Base ` K ) ) -> ( P .\/ S ) e. ( Base ` K ) )
15	6, 10, 12, 14	syl3anc 1184	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> ( P .\/ S ) e. ( Base ` K ) )
16		simp1r 982	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> W e. H )
17		cdleme8.h	. . . . . 6 |- H = ( LHyp ` K )
18	7, 17	lhpbase 30492	. . . . 5 |- ( W e. H -> W e. ( Base ` K ) )
19	16, 18	syl 16	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> W e. ( Base ` K ) )
20		cdleme8.l	. . . . . 6 |- .<_ = ( le ` K )
21	7, 20, 13	latlej1 14452	. . . . 5 |- ( ( K e. Lat /\ P e. ( Base ` K ) /\ S e. ( Base ` K ) ) -> P .<_ ( P .\/ S ) )
22	6, 10, 12, 21	syl3anc 1184	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> P .<_ ( P .\/ S ) )
23		cdleme8.m	. . . . 5 |- ./\ = ( meet ` K )
24	7, 20, 13, 23, 8	atmod3i1 30358	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ ( P .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) /\ P .<_ ( P .\/ S ) ) -> ( P .\/ ( ( P .\/ S ) ./\ W ) ) = ( ( P .\/ S ) ./\ ( P .\/ W ) ) )
25	3, 4, 15, 19, 22, 24	syl131anc 1197	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> ( P .\/ ( ( P .\/ S ) ./\ W ) ) = ( ( P .\/ S ) ./\ ( P .\/ W ) ) )
26		eqid 2412	. . . . . 6 |- ( 1. ` K ) = ( 1. ` K )
27	20, 13, 26, 8, 17	lhpjat2 30515	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ W ) = ( 1. ` K ) )
28	27	3adant3 977	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> ( P .\/ W ) = ( 1. ` K ) )
29	28	oveq2d 6064	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> ( ( P .\/ S ) ./\ ( P .\/ W ) ) = ( ( P .\/ S ) ./\ ( 1. ` K ) ) )
30		hlol 29856	. . . . 5 |- ( K e. HL -> K e. OL )
31	3, 30	syl 16	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> K e. OL )
32	7, 23, 26	olm11 29722	. . . 4 |- ( ( K e. OL /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ ( 1. ` K ) ) = ( P .\/ S ) )
33	31, 15, 32	syl2anc 643	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> ( ( P .\/ S ) ./\ ( 1. ` K ) ) = ( P .\/ S ) )
34	25, 29, 33	3eqtrd 2448	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> ( P .\/ ( ( P .\/ S ) ./\ W ) ) = ( P .\/ S ) )
35	2, 34	syl5eq 2456	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ S e. A ) -> ( P .\/ C ) = ( P .\/ S ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   Basecbs 13432   lecple 13499   joincjn 14364   meetcmee 14365   1.cp1 14430   Latclat 14437   OLcol 29669   Atomscatm 29758   HLchlt 29845   LHypclh 30478

This theorem is referenced by:  cdleme8tN  30749  cdleme15a  30768  cdleme17b  30781  cdlemg3a  31091

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-p1 14432  df-lat 14438  df-clat 14500  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-psubsp 29997  df-pmap 29998  df-padd 30290  df-lhyp 30482