Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  atbtwn Structured version   Visualization version   Unicode version
Theorem atbtwn 34732
Description: Property of a 3rd atom R on a line P .\/ Q intersecting element X at P. (Contributed by NM, 30-Jul-2012.)
Hypotheses
Ref Expression
atbtwn.b |- B = ( Base ` K )
atbtwn.l |- .<_ = ( le ` K )
atbtwn.j |- .\/ = ( join ` K )
atbtwn.a |- A = ( Atoms ` K )
Assertion
Ref Expression
atbtwn |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) -> ( R =/= P <-> -. R .<_ X ) )
Proof of Theorem atbtwn
StepHypRef Expression
1 simpl33 1144 . . . . . . 7 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) /\ R .<_ X ) -> R .<_ ( P .\/ Q ) )
2 simpr 477 . . . . . . 7 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) /\ R .<_ X ) -> R .<_ X )
3 simpl11 1136 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) /\ R .<_ X ) -> K e. HL )
4 hllat 34650 . . . . . . . . 9 |- ( K e. HL -> K e. Lat )
53, 4syl 17 . . . . . . . 8 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) /\ R .<_ X ) -> K e. Lat )
6 simpl2l 1114 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) /\ R .<_ X ) -> R e. A )
7 atbtwn.b . . . . . . . . . 10 |- B = ( Base ` K )
8 atbtwn.a . . . . . . . . . 10 |- A = ( Atoms ` K )
97, 8atbase 34576 . . . . . . . . 9 |- ( R e. A -> R e. B )
106, 9syl 17 . . . . . . . 8 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) /\ R .<_ X ) -> R e. B )
11 simpl1 1064 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) /\ R .<_ X ) -> ( K e. HL /\ P e. A /\ Q e. A ) )
12 atbtwn.j . . . . . . . . . 10 |- .\/ = ( join ` K )
137, 12, 8hlatjcl 34653 . . . . . . . . 9 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. B )
1411, 13syl 17 . . . . . . . 8 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) /\ R .<_ X ) -> ( P .\/ Q ) e. B )
15 simpl2r 1115 . . . . . . . 8 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) /\ R .<_ X ) -> X e. B )
16 atbtwn.l . . . . . . . . 9 |- .<_ = ( le ` K )
17 eqid 2622 . . . . . . . . 9 |- ( meet ` K ) = ( meet ` K )
187, 16, 17latlem12 17078 . . . . . . . 8 |- ( ( K e. Lat /\ ( R e. B /\ ( P .\/ Q ) e. B /\ X e. B ) ) -> ( ( R .<_ ( P .\/ Q ) /\ R .<_ X ) <-> R .<_ ( ( P .\/ Q ) ( meet ` K ) X ) ) )
195, 10, 14, 15, 18syl13anc 1328 . . . . . . 7 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) /\ R .<_ X ) -> ( ( R .<_ ( P .\/ Q ) /\ R .<_ X ) <-> R .<_ ( ( P .\/ Q ) ( meet ` K ) X ) ) )
201, 2, 19mpbi2and 956 . . . . . 6 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) /\ R .<_ X ) -> R .<_ ( ( P .\/ Q ) ( meet ` K ) X ) )
21 simpl12 1137 . . . . . . 7 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) /\ R .<_ X ) -> P e. A )
22 simpl13 1138 . . . . . . 7 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) /\ R .<_ X ) -> Q e. A )
23 simpl31 1142 . . . . . . 7 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) /\ R .<_ X ) -> P .<_ X )
24 simpl32 1143 . . . . . . 7 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) /\ R .<_ X ) -> -. Q .<_ X )
257, 16, 12, 17, 82atjm 34731 . . . . . . 7 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X ) ) -> ( ( P .\/ Q ) ( meet ` K ) X ) = P )
263, 21, 22, 15, 23, 24, 25syl132anc 1344 . . . . . 6 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) /\ R .<_ X ) -> ( ( P .\/ Q ) ( meet ` K ) X ) = P )
2720, 26breqtrd 4679 . . . . 5 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) /\ R .<_ X ) -> R .<_ P )
28 hlatl 34647 . . . . . . 7 |- ( K e. HL -> K e. AtLat )
293, 28syl 17 . . . . . 6 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) /\ R .<_ X ) -> K e. AtLat )
3016, 8atcmp 34598 . . . . . 6 |- ( ( K e. AtLat /\ R e. A /\ P e. A ) -> ( R .<_ P <-> R = P ) )
3129, 6, 21, 30syl3anc 1326 . . . . 5 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) /\ R .<_ X ) -> ( R .<_ P <-> R = P ) )
3227, 31mpbid 222 . . . 4 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) /\ R .<_ X ) -> R = P )
3332ex 450 . . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) -> ( R .<_ X -> R = P ) )
3433necon3ad 2807 . 2 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) -> ( R =/= P -> -. R .<_ X ) )
35 simp31 1097 . . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) -> P .<_ X )
36 nbrne2 4673 . . . . 5 |- ( ( P .<_ X /\ -. R .<_ X ) -> P =/= R )
3736necomd 2849 . . . 4 |- ( ( P .<_ X /\ -. R .<_ X ) -> R =/= P )
3837ex 450 . . 3 |- ( P .<_ X -> ( -. R .<_ X -> R =/= P ) )
3935, 38syl 17 . 2 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) -> ( -. R .<_ X -> R =/= P ) )
4034, 39impbid 202 1 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ X e. B ) /\ ( P .<_ X /\ -. Q .<_ X /\ R .<_ ( P .\/ Q ) ) ) -> ( R =/= P <-> -. R .<_ X ) )
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   Latclat 17045   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-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
This theorem is referenced by:  atbtwnexOLDN  34733  atbtwnex  34734
  Copyright terms: Public domain W3C validator