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Theorem dalemrotps 34487
Description: Lemma for dath 34532. 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 34479 . . . 4  |-  ( ps 
->  c  e.  A
)
31dalemddea 34480 . . . 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 34481 . . . 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 34444 . . . . . 6  |-  ( ph  ->  ( ( Q  .\/  R )  .\/  P )  =  ( ( P 
.\/  Q )  .\/  R ) )
12 dalemrotps.y . . . . . 6  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
1311, 12syl6reqr 2527 . . . . 5  |-  ( ph  ->  Y  =  ( ( Q  .\/  R ) 
.\/  P ) )
1413breq2d 4459 . . . 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 34483 . . . . 5  |-  ( ps 
->  c  =/=  d
)
1817necomd 2738 . . . 4  |-  ( ps 
->  d  =/=  c
)
1918adantl 466 . . 3  |-  ( (
ph  /\  ps )  ->  d  =/=  c )
201dalem-ddly 34482 . . . . 5  |-  ( ps 
->  -.  d  .<_  Y )
2120adantl 466 . . . 4  |-  ( (
ph  /\  ps )  ->  -.  d  .<_  Y )
2213breq2d 4459 . . . . 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 34484 . . . 4  |-  ( ps 
->  C  .<_  ( c 
.\/  d ) )
2625adantl 466 . . 3  |-  ( (
ph  /\  ps )  ->  C  .<_  ( c  .\/  d ) )
2719, 24, 263jca 1176 . 2  |-  ( (
ph  /\  ps )  ->  ( d  =/=  c  /\  -.  d  .<_  ( ( Q  .\/  R ) 
.\/  P )  /\  C  .<_  ( c  .\/  d ) ) )
285, 16, 273jca 1176 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14486   lecple 14558   joincjn 15427   Atomscatm 34060   HLchlt 34147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-poset 15429  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-lat 15529  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148
This theorem is referenced by:  dalem29  34497  dalem30  34498  dalem31N  34499  dalem32  34500  dalem33  34501  dalem34  34502  dalem35  34503  dalem36  34504  dalem37  34505  dalem40  34508  dalem46  34514  dalem47  34515  dalem49  34517  dalem50  34518  dalem58  34526  dalem59  34527
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