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

Theorem dalemrotps 33643
Description: Lemma for dath 33688. Rotate triangles  Y  =  P Q R and  Z  =  S T U to allow reuse of analogous proofs. (Contributed by NM, 15-Aug-2012.)
Hypotheses
Ref Expression
dalem.ph  |-  ( ph  <->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) ) )
dalem.l  |-  .<_  =  ( le `  K )
dalem.j  |-  .\/  =  ( join `  K )
dalem.a  |-  A  =  ( Atoms `  K )
dalem.ps  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
dalemrotps.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
Assertion
Ref Expression
dalemrotps  |-  ( (
ph  /\  ps )  ->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  ( ( Q 
.\/  R )  .\/  P )  /\  ( d  =/=  c  /\  -.  d  .<_  ( ( Q 
.\/  R )  .\/  P )  /\  C  .<_  ( c  .\/  d ) ) ) )

Proof of Theorem dalemrotps
StepHypRef Expression
1 dalem.ps . . . . 5  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
21dalemccea 33635 . . . 4  |-  ( ps 
->  c  e.  A
)
31dalemddea 33636 . . . 4  |-  ( ps 
->  d  e.  A
)
42, 3jca 532 . . 3  |-  ( ps 
->  ( c  e.  A  /\  d  e.  A
) )
54adantl 466 . 2  |-  ( (
ph  /\  ps )  ->  ( c  e.  A  /\  d  e.  A
) )
61dalem-ccly 33637 . . . 4  |-  ( ps 
->  -.  c  .<_  Y )
76adantl 466 . . 3  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  Y )
8 dalem.ph . . . . . . 7  |-  ( ph  <->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) ) )
9 dalem.j . . . . . . 7  |-  .\/  =  ( join `  K )
10 dalem.a . . . . . . 7  |-  A  =  ( Atoms `  K )
118, 9, 10dalemqrprot 33600 . . . . . 6  |-  ( ph  ->  ( ( Q  .\/  R )  .\/  P )  =  ( ( P 
.\/  Q )  .\/  R ) )
12 dalemrotps.y . . . . . 6  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
1311, 12syl6reqr 2511 . . . . 5  |-  ( ph  ->  Y  =  ( ( Q  .\/  R ) 
.\/  P ) )
1413breq2d 4404 . . . 4  |-  ( ph  ->  ( c  .<_  Y  <->  c  .<_  ( ( Q  .\/  R
)  .\/  P )
) )
1514adantr 465 . . 3  |-  ( (
ph  /\  ps )  ->  ( c  .<_  Y  <->  c  .<_  ( ( Q  .\/  R
)  .\/  P )
) )
167, 15mtbid 300 . 2  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  ( ( Q  .\/  R ) 
.\/  P ) )
171dalemccnedd 33639 . . . . 5  |-  ( ps 
->  c  =/=  d
)
1817necomd 2719 . . . 4  |-  ( ps 
->  d  =/=  c
)
1918adantl 466 . . 3  |-  ( (
ph  /\  ps )  ->  d  =/=  c )
201dalem-ddly 33638 . . . . 5  |-  ( ps 
->  -.  d  .<_  Y )
2120adantl 466 . . . 4  |-  ( (
ph  /\  ps )  ->  -.  d  .<_  Y )
2213breq2d 4404 . . . . 5  |-  ( ph  ->  ( d  .<_  Y  <->  d  .<_  ( ( Q  .\/  R
)  .\/  P )
) )
2322adantr 465 . . . 4  |-  ( (
ph  /\  ps )  ->  ( d  .<_  Y  <->  d  .<_  ( ( Q  .\/  R
)  .\/  P )
) )
2421, 23mtbid 300 . . 3  |-  ( (
ph  /\  ps )  ->  -.  d  .<_  ( ( Q  .\/  R ) 
.\/  P ) )
251dalemclccjdd 33640 . . . 4  |-  ( ps 
->  C  .<_  ( c 
.\/  d ) )
2625adantl 466 . . 3  |-  ( (
ph  /\  ps )  ->  C  .<_  ( c  .\/  d ) )
2719, 24, 263jca 1168 . 2  |-  ( (
ph  /\  ps )  ->  ( d  =/=  c  /\  -.  d  .<_  ( ( Q  .\/  R ) 
.\/  P )  /\  C  .<_  ( c  .\/  d ) ) )
285, 16, 273jca 1168 1  |-  ( (
ph  /\  ps )  ->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  ( ( Q 
.\/  R )  .\/  P )  /\  ( d  =/=  c  /\  -.  d  .<_  ( ( Q 
.\/  R )  .\/  P )  /\  C  .<_  ( c  .\/  d ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   Basecbs 14278   lecple 14349   joincjn 15218   Atomscatm 33216   HLchlt 33303
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 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-poset 15220  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-lat 15320  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304
This theorem is referenced by:  dalem29  33653  dalem30  33654  dalem31N  33655  dalem32  33656  dalem33  33657  dalem34  33658  dalem35  33659  dalem36  33660  dalem37  33661  dalem40  33664  dalem46  33670  dalem47  33671  dalem49  33673  dalem50  33674  dalem58  33682  dalem59  33683
  Copyright terms: Public domain W3C validator