Theorem cdleme42a 35759

Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 3-Mar-2013.)

Hypotheses

Ref	Expression
cdleme42.b	|- B = ( Base ` K )
cdleme42.l	|- .<_ = ( le ` K )
cdleme42.j	|- .\/ = ( join ` K )
cdleme42.m	|- ./\ = ( meet ` K )
cdleme42.a	|- A = ( Atoms ` K )
cdleme42.h	|- H = ( LHyp ` K )
cdleme42.v	|- V = ( ( R .\/ S ) ./\ W )

Assertion

Ref	Expression
cdleme42a	|- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> ( R .\/ S ) = ( R .\/ V ) )

Proof of Theorem cdleme42a

Step	Hyp	Ref	Expression
1		cdleme42.l	. . . . 5 |- .<_ = ( le ` K )
2		cdleme42.j	. . . . 5 |- .\/ = ( join ` K )
3		eqid 2622	. . . . 5 |- ( 1. ` K ) = ( 1. ` K )
4		cdleme42.a	. . . . 5 |- A = ( Atoms ` K )
5		cdleme42.h	. . . . 5 |- H = ( LHyp ` K )
6	1, 2, 3, 4, 5	lhpjat2 35307	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( R .\/ W ) = ( 1. ` K ) )
7	6	3adant3 1081	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> ( R .\/ W ) = ( 1. ` K ) )
8	7	oveq2d 6666	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> ( ( R .\/ S ) ./\ ( R .\/ W ) ) = ( ( R .\/ S ) ./\ ( 1. ` K ) ) )
9		cdleme42.v	. . . 4 |- V = ( ( R .\/ S ) ./\ W )
10	9	oveq2i 6661	. . 3 |- ( R .\/ V ) = ( R .\/ ( ( R .\/ S ) ./\ W ) )
11		simp1l 1085	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> K e. HL )
12		simp2l 1087	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> R e. A )
13		simp3l 1089	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> S e. A )
14		cdleme42.b	. . . . . 6 |- B = ( Base ` K )
15	14, 2, 4	hlatjcl 34653	. . . . 5 |- ( ( K e. HL /\ R e. A /\ S e. A ) -> ( R .\/ S ) e. B )
16	11, 12, 13, 15	syl3anc 1326	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> ( R .\/ S ) e. B )
17		simp1r 1086	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> W e. H )
18	14, 5	lhpbase 35284	. . . . 5 |- ( W e. H -> W e. B )
19	17, 18	syl 17	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> W e. B )
20	1, 2, 4	hlatlej1 34661	. . . . 5 |- ( ( K e. HL /\ R e. A /\ S e. A ) -> R .<_ ( R .\/ S ) )
21	11, 12, 13, 20	syl3anc 1326	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> R .<_ ( R .\/ S ) )
22		cdleme42.m	. . . . 5 |- ./\ = ( meet ` K )
23	14, 1, 2, 22, 4	atmod3i1 35150	. . . 4 |- ( ( K e. HL /\ ( R e. A /\ ( R .\/ S ) e. B /\ W e. B ) /\ R .<_ ( R .\/ S ) ) -> ( R .\/ ( ( R .\/ S ) ./\ W ) ) = ( ( R .\/ S ) ./\ ( R .\/ W ) ) )
24	11, 12, 16, 19, 21, 23	syl131anc 1339	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> ( R .\/ ( ( R .\/ S ) ./\ W ) ) = ( ( R .\/ S ) ./\ ( R .\/ W ) ) )
25	10, 24	syl5req 2669	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> ( ( R .\/ S ) ./\ ( R .\/ W ) ) = ( R .\/ V ) )
26		hlol 34648	. . . 4 |- ( K e. HL -> K e. OL )
27	11, 26	syl 17	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> K e. OL )
28	14, 22, 3	olm11 34514	. . 3 |- ( ( K e. OL /\ ( R .\/ S ) e. B ) -> ( ( R .\/ S ) ./\ ( 1. ` K ) ) = ( R .\/ S ) )
29	27, 16, 28	syl2anc 693	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> ( ( R .\/ S ) ./\ ( 1. ` K ) ) = ( R .\/ S ) )
30	8, 25, 29	3eqtr3rd 2665	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> ( R .\/ S ) = ( R .\/ V ) )

Syntax hints:   -. wn 3   -> 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   1.cp1 17038   OLcol 34461   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:  cdleme42d  35761  cdleme42f  35768  cdleme42g  35769  cdleme42keg  35774  cdleme43cN  35779