Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme17c Structured version   Visualization version   Unicode version
Theorem cdleme17c 35575
Description: Part of proof of Lemma E in [Crawley] p. 114, first part of 4th paragraph. C represents s1. We show, in their notation, (p \/ q) /\ (q \/ s1)=q. (Contributed by NM, 11-Oct-2012.)
Hypotheses
Ref Expression
cdleme17.l |- .<_ = ( le ` K )
cdleme17.j |- .\/ = ( join ` K )
cdleme17.m |- ./\ = ( meet ` K )
cdleme17.a |- A = ( Atoms ` K )
cdleme17.h |- H = ( LHyp ` K )
cdleme17.u |- U = ( ( P .\/ Q ) ./\ W )
cdleme17.f |- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme17.g |- G = ( ( P .\/ Q ) ./\ ( F .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme17.c |- C = ( ( P .\/ S ) ./\ W )
Assertion
Ref Expression
cdleme17c |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ Q ) ./\ ( Q .\/ C ) ) = Q )
Proof of Theorem cdleme17c
StepHypRef Expression
1 simp1l 1085 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> K e. HL )
2 simp2l 1087 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> P e. A )
3 simp31 1097 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> Q e. A )
4 cdleme17.j . . . . 5 |- .\/ = ( join ` K )
5 cdleme17.a . . . . 5 |- A = ( Atoms ` K )
64, 5hlatjcom 34654 . . . 4 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) = ( Q .\/ P ) )
71, 2, 3, 6syl3anc 1326 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> ( P .\/ Q ) = ( Q .\/ P ) )
87oveq1d 6665 . 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ Q ) ./\ ( Q .\/ C ) ) = ( ( Q .\/ P ) ./\ ( Q .\/ C ) ) )
9 simp1r 1086 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> W e. H )
10 simp2r 1088 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> -. P .<_ W )
11 simp32 1098 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> S e. A )
12 hllat 34650 . . . . . 6 |- ( K e. HL -> K e. Lat )
131, 12syl 17 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> K e. Lat )
14 eqid 2622 . . . . . . 7 |- ( Base ` K ) = ( Base ` K )
1514, 5atbase 34576 . . . . . 6 |- ( S e. A -> S e. ( Base ` K ) )
1611, 15syl 17 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> S e. ( Base ` K ) )
1714, 5atbase 34576 . . . . . 6 |- ( P e. A -> P e. ( Base ` K ) )
182, 17syl 17 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> P e. ( Base ` K ) )
1914, 5atbase 34576 . . . . . 6 |- ( Q e. A -> Q e. ( Base ` K ) )
203, 19syl 17 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> Q e. ( Base ` K ) )
21 simp33 1099 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> -. S .<_ ( P .\/ Q ) )
22 cdleme17.l . . . . . . 7 |- .<_ = ( le ` K )
2314, 22, 4latnlej1l 17069 . . . . . 6 |- ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. S .<_ ( P .\/ Q ) ) -> S =/= P )
2423necomd 2849 . . . . 5 |- ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. S .<_ ( P .\/ Q ) ) -> P =/= S )
2513, 16, 18, 20, 21, 24syl131anc 1339 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> P =/= S )
26 cdleme17.m . . . . 5 |- ./\ = ( meet ` K )
27 cdleme17.h . . . . 5 |- H = ( LHyp ` K )
28 cdleme17.c . . . . 5 |- C = ( ( P .\/ S ) ./\ W )
2922, 4, 26, 5, 27, 28cdleme9a 35538 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( S e. A /\ P =/= S ) ) -> C e. A )
301, 9, 2, 10, 11, 25, 29syl222anc 1342 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> C e. A )
31 cdleme17.u . . . 4 |- U = ( ( P .\/ Q ) ./\ W )
32 cdleme17.f . . . 4 |- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
33 cdleme17.g . . . 4 |- G = ( ( P .\/ Q ) ./\ ( F .\/ ( ( P .\/ S ) ./\ W ) ) )
3422, 4, 26, 5, 27, 31, 32, 33, 28cdleme17b 35574 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> -. C .<_ ( P .\/ Q ) )
3522, 4, 26, 52llnma1 35073 . . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ C e. A ) /\ -. C .<_ ( P .\/ Q ) ) -> ( ( Q .\/ P ) ./\ ( Q .\/ C ) ) = Q )
361, 2, 3, 30, 34, 35syl131anc 1339 . 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( Q .\/ P ) ./\ ( Q .\/ C ) ) = Q )
378, 36eqtrd 2656 1 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ Q ) ./\ ( Q .\/ C ) ) = Q )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ 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   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-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:  cdleme17d1  35576
  Copyright terms: Public domain W3C validator