Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  4noncolr3 Structured version   Unicode version

Theorem 4noncolr3 33455
Description: A way to express 4 non-colinear atoms (rotated right 3 places). (Contributed by NM, 11-Jul-2012.)
Hypotheses
Ref Expression
3noncol.l  |-  .<_  =  ( le `  K )
3noncol.j  |-  .\/  =  ( join `  K )
3noncol.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
4noncolr3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R )  /\  -.  P  .<_  ( ( Q  .\/  R )  .\/  S ) ) )

Proof of Theorem 4noncolr3
StepHypRef Expression
1 simp11 1018 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  K  e.  HL )
2 hllat 33366 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  K  e.  Lat )
4 simp2l 1014 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  R  e.  A )
5 eqid 2454 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
6 3noncol.a . . . . . 6  |-  A  =  ( Atoms `  K )
75, 6atbase 33292 . . . . 5  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
84, 7syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  R  e.  ( Base `  K
) )
9 simp12 1019 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  P  e.  A )
105, 6atbase 33292 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
119, 10syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  P  e.  ( Base `  K
) )
12 simp13 1020 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  Q  e.  A )
135, 6atbase 33292 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1412, 13syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  Q  e.  ( Base `  K
) )
15 simp32 1025 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  -.  R  .<_  ( P  .\/  Q ) )
16 3noncol.l . . . . 5  |-  .<_  =  ( le `  K )
17 3noncol.j . . . . 5  |-  .\/  =  ( join `  K )
185, 16, 17latnlej1r 15362 . . . 4  |-  ( ( K  e.  Lat  /\  ( R  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  /\  -.  R  .<_  ( P  .\/  Q ) )  ->  R  =/=  Q )
193, 8, 11, 14, 15, 18syl131anc 1232 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  R  =/=  Q )
2019necomd 2723 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  Q  =/=  R )
21 simp2r 1015 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  S  e.  A )
225, 6atbase 33292 . . . 4  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
2321, 22syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  S  e.  ( Base `  K
) )
245, 17latjcl 15343 . . . 4  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( Q  .\/  R )  e.  ( Base `  K
) )
253, 14, 8, 24syl3anc 1219 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  ( Q  .\/  R )  e.  ( Base `  K
) )
26 simp33 1026 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  -.  S  .<_  ( ( P 
.\/  Q )  .\/  R ) )
2717, 6hlatjass 33372 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( P  .\/  ( Q  .\/  R ) ) )
281, 9, 12, 4, 27syl13anc 1221 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( P  .\/  ( Q  .\/  R ) ) )
2928breq2d 4415 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  ( S  .<_  ( ( P 
.\/  Q )  .\/  R )  <->  S  .<_  ( P 
.\/  ( Q  .\/  R ) ) ) )
3026, 29mtbid 300 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  -.  S  .<_  ( P  .\/  ( Q  .\/  R ) ) )
315, 16, 17latnlej2r 15365 . . 3  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
) )  /\  -.  S  .<_  ( P  .\/  ( Q  .\/  R ) ) )  ->  -.  S  .<_  ( Q  .\/  R ) )
323, 23, 11, 25, 30, 31syl131anc 1232 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  -.  S  .<_  ( Q  .\/  R ) )
33 simp31 1024 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  P  =/=  Q )
3416, 17, 6hlatexch1 33397 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  R  e.  A  /\  Q  e.  A
)  /\  P  =/=  Q )  ->  ( P  .<_  ( Q  .\/  R
)  ->  R  .<_  ( Q  .\/  P ) ) )
351, 9, 4, 12, 33, 34syl131anc 1232 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  ( P  .<_  ( Q  .\/  R )  ->  R  .<_  ( Q  .\/  P ) ) )
365, 17latjcom 15351 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
373, 11, 14, 36syl3anc 1219 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
3837breq2d 4415 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  ( R  .<_  ( P  .\/  Q )  <->  R  .<_  ( Q 
.\/  P ) ) )
3935, 38sylibrd 234 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  ( P  .<_  ( Q  .\/  R )  ->  R  .<_  ( P  .\/  Q ) ) )
4015, 39mtod 177 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  -.  P  .<_  ( Q  .\/  R ) )
415, 16, 17, 6hlexch1 33384 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  S  e.  A  /\  ( Q  .\/  R
)  e.  ( Base `  K ) )  /\  -.  P  .<_  ( Q 
.\/  R ) )  ->  ( P  .<_  ( ( Q  .\/  R
)  .\/  S )  ->  S  .<_  ( ( Q  .\/  R )  .\/  P ) ) )
421, 9, 21, 25, 40, 41syl131anc 1232 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  ( P  .<_  ( ( Q 
.\/  R )  .\/  S )  ->  S  .<_  ( ( Q  .\/  R
)  .\/  P )
) )
435, 17latjcom 15351 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( Q  .\/  R )  =  ( R  .\/  Q
) )
443, 14, 8, 43syl3anc 1219 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  ( Q  .\/  R )  =  ( R  .\/  Q
) )
4544oveq1d 6218 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  (
( Q  .\/  R
)  .\/  P )  =  ( ( R 
.\/  Q )  .\/  P ) )
465, 17latj31 15391 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( R  e.  ( Base `  K )  /\  Q  e.  ( Base `  K )  /\  P  e.  ( Base `  K
) ) )  -> 
( ( R  .\/  Q )  .\/  P )  =  ( ( P 
.\/  Q )  .\/  R ) )
473, 8, 14, 11, 46syl13anc 1221 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  (
( R  .\/  Q
)  .\/  P )  =  ( ( P 
.\/  Q )  .\/  R ) )
4845, 47eqtrd 2495 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  (
( Q  .\/  R
)  .\/  P )  =  ( ( P 
.\/  Q )  .\/  R ) )
4948breq2d 4415 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  ( S  .<_  ( ( Q 
.\/  R )  .\/  P )  <->  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) ) )
5042, 49sylibd 214 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  ( P  .<_  ( ( Q 
.\/  R )  .\/  S )  ->  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )
5126, 50mtod 177 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  -.  P  .<_  ( ( Q 
.\/  R )  .\/  S ) )
5220, 32, 513jca 1168 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R )  /\  -.  P  .<_  ( ( Q  .\/  R )  .\/  S ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14295   lecple 14367   joincjn 15236   Latclat 15337   Atomscatm 33266   HLchlt 33353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-poset 15238  df-plt 15250  df-lub 15266  df-glb 15267  df-join 15268  df-meet 15269  df-p0 15331  df-lat 15338  df-covers 33269  df-ats 33270  df-atl 33301  df-cvlat 33325  df-hlat 33354
This theorem is referenced by:  4noncolr2  33456  4noncolr1  33457  4atlem12  33614
  Copyright terms: Public domain W3C validator