Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlema2N Structured version   Visualization version   Unicode version
Theorem cdlema2N 35078
Description: A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 9-May-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlema2.b |- B = ( Base ` K )
cdlema2.l |- .<_ = ( le ` K )
cdlema2.j |- .\/ = ( join ` K )
cdlema2.m |- ./\ = ( meet ` K )
cdlema2.z |- .0. = ( 0. ` K )
cdlema2.a |- A = ( Atoms ` K )
Assertion
Ref Expression
cdlema2N |- ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> ( R ./\ X ) = .0. )
Proof of Theorem cdlema2N
StepHypRef Expression
1 simp3ll 1132 . . 3 |- ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> R =/= P )
2 simp3rl 1134 . . . . . 6 |- ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> P .<_ X )
3 simp3rr 1135 . . . . . 6 |- ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> -. Q .<_ X )
4 simp3lr 1133 . . . . . 6 |- ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> R .<_ ( P .\/ Q ) )
52, 3, 43jca 1242 . . . . 5 |- ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) )
6 cdlema2.b . . . . . 6 |- B = ( Base ` K )
7 cdlema2.l . . . . . 6 |- .<_ = ( le ` K )
8 cdlema2.j . . . . . 6 |- .\/ = ( join ` K )
9 cdlema2.a . . . . . 6 |- A = ( Atoms ` K )
106, 7, 8, 9exatleN 34690 . . . . 5 |- ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) -> ( R .<_ X <-> R = P ) )
115, 10syld3an3 1371 . . . 4 |- ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> ( R .<_ X <-> R = P ) )
1211necon3bbid 2831 . . 3 |- ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> ( -. R .<_ X <-> R =/= P ) )
131, 12mpbird 247 . 2 |- ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> -. R .<_ X )
14 simp1l 1085 . . . 4 |- ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> K e. HL )
15 hlatl 34647 . . . 4 |- ( K e. HL -> K e. AtLat )
1614, 15syl 17 . . 3 |- ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> K e. AtLat )
17 simp23 1096 . . 3 |- ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> R e. A )
18 simp1r 1086 . . 3 |- ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> X e. B )
19 cdlema2.m . . . 4 |- ./\ = ( meet ` K )
20 cdlema2.z . . . 4 |- .0. = ( 0. ` K )
216, 7, 19, 20, 9atnle 34604 . . 3 |- ( ( K e. AtLat /\ R e. A /\ X e. B ) -> ( -. R .<_ X <-> ( R ./\ X ) = .0. ) )
2216, 17, 18, 21syl3anc 1326 . 2 |- ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> ( -. R .<_ X <-> ( R ./\ X ) = .0. ) )
2313, 22mpbid 222 1 |- ( ( ( K e. HL /\ X e. B ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( ( R =/= P /\ R .<_ ( P .\/ Q ) ) /\ ( P .<_ X /\ -. Q .<_ X ) ) ) -> ( R ./\ X ) = .0. )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ 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   0.cp0 17037   Atomscatm 34550   AtLatcal 34551   HLchlt 34637
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-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-lat 17046  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator