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

Theorem dalemrotps 30489
Description: Lemma for dath 30534. 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 30481 . . . 4  |-  ( ps 
->  c  e.  A
)
31dalemddea 30482 . . . 4  |-  ( ps 
->  d  e.  A
)
42, 3jca 520 . . 3  |-  ( ps 
->  ( c  e.  A  /\  d  e.  A
) )
54adantl 454 . 2  |-  ( (
ph  /\  ps )  ->  ( c  e.  A  /\  d  e.  A
) )
61dalem-ccly 30483 . . . 4  |-  ( ps 
->  -.  c  .<_  Y )
76adantl 454 . . 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 30446 . . . . . 6  |-  ( ph  ->  ( ( Q  .\/  R )  .\/  P )  =  ( ( P 
.\/  Q )  .\/  R ) )
12 dalemrotps.y . . . . . 6  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
1311, 12syl6reqr 2488 . . . . 5  |-  ( ph  ->  Y  =  ( ( Q  .\/  R ) 
.\/  P ) )
1413breq2d 4225 . . . 4  |-  ( ph  ->  ( c  .<_  Y  <->  c  .<_  ( ( Q  .\/  R
)  .\/  P )
) )
1514adantr 453 . . 3  |-  ( (
ph  /\  ps )  ->  ( c  .<_  Y  <->  c  .<_  ( ( Q  .\/  R
)  .\/  P )
) )
167, 15mtbid 293 . 2  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  ( ( Q  .\/  R ) 
.\/  P ) )
171dalemccnedd 30485 . . . . 5  |-  ( ps 
->  c  =/=  d
)
1817necomd 2688 . . . 4  |-  ( ps 
->  d  =/=  c
)
1918adantl 454 . . 3  |-  ( (
ph  /\  ps )  ->  d  =/=  c )
201dalem-ddly 30484 . . . . 5  |-  ( ps 
->  -.  d  .<_  Y )
2120adantl 454 . . . 4  |-  ( (
ph  /\  ps )  ->  -.  d  .<_  Y )
2213breq2d 4225 . . . . 5  |-  ( ph  ->  ( d  .<_  Y  <->  d  .<_  ( ( Q  .\/  R
)  .\/  P )
) )
2322adantr 453 . . . 4  |-  ( (
ph  /\  ps )  ->  ( d  .<_  Y  <->  d  .<_  ( ( Q  .\/  R
)  .\/  P )
) )
2421, 23mtbid 293 . . 3  |-  ( (
ph  /\  ps )  ->  -.  d  .<_  ( ( Q  .\/  R ) 
.\/  P ) )
251dalemclccjdd 30486 . . . 4  |-  ( ps 
->  C  .<_  ( c 
.\/  d ) )
2625adantl 454 . . 3  |-  ( (
ph  /\  ps )  ->  C  .<_  ( c  .\/  d ) )
2719, 24, 263jca 1135 . 2  |-  ( (
ph  /\  ps )  ->  ( d  =/=  c  /\  -.  d  .<_  ( ( Q  .\/  R ) 
.\/  P )  /\  C  .<_  ( c  .\/  d ) ) )
285, 16, 273jca 1135 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   Basecbs 13470   lecple 13537   joincjn 14402   Atomscatm 30062   HLchlt 30149
This theorem is referenced by:  dalem29  30499  dalem30  30500  dalem31N  30501  dalem32  30502  dalem33  30503  dalem34  30504  dalem35  30505  dalem36  30506  dalem37  30507  dalem40  30510  dalem46  30516  dalem47  30517  dalem49  30519  dalem50  30520  dalem58  30528  dalem59  30529
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-undef 6544  df-riota 6550  df-poset 14404  df-lub 14432  df-join 14434  df-lat 14476  df-ats 30066  df-atl 30097  df-cvlat 30121  df-hlat 30150
  Copyright terms: Public domain W3C validator