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Theorem dalemrotps 32721
Description: Lemma for dath 32766. 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 32713 . . . 4  |-  ( ps 
->  c  e.  A
)
31dalemddea 32714 . . . 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 32715 . . . 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 32678 . . . . . 6  |-  ( ph  ->  ( ( Q  .\/  R )  .\/  P )  =  ( ( P 
.\/  Q )  .\/  R ) )
12 dalemrotps.y . . . . . 6  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
1311, 12syl6reqr 2464 . . . . 5  |-  ( ph  ->  Y  =  ( ( Q  .\/  R ) 
.\/  P ) )
1413breq2d 4409 . . . 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 32717 . . . . 5  |-  ( ps 
->  c  =/=  d
)
1817necomd 2676 . . . 4  |-  ( ps 
->  d  =/=  c
)
1918adantl 466 . . 3  |-  ( (
ph  /\  ps )  ->  d  =/=  c )
201dalem-ddly 32716 . . . . 5  |-  ( ps 
->  -.  d  .<_  Y )
2120adantl 466 . . . 4  |-  ( (
ph  /\  ps )  ->  -.  d  .<_  Y )
2213breq2d 4409 . . . . 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 32718 . . . 4  |-  ( ps 
->  C  .<_  ( c 
.\/  d ) )
2625adantl 466 . . 3  |-  ( (
ph  /\  ps )  ->  C  .<_  ( c  .\/  d ) )
2719, 24, 263jca 1179 . 2  |-  ( (
ph  /\  ps )  ->  ( d  =/=  c  /\  -.  d  .<_  ( ( Q  .\/  R ) 
.\/  P )  /\  C  .<_  ( c  .\/  d ) ) )
285, 16, 273jca 1179 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 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844    =/= wne 2600   class class class wbr 4397   ` cfv 5571  (class class class)co 6280   Basecbs 14843   lecple 14918   joincjn 15899   Atomscatm 32294   HLchlt 32381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-preset 15883  df-poset 15901  df-lub 15930  df-glb 15931  df-join 15932  df-meet 15933  df-lat 16002  df-ats 32298  df-atl 32329  df-cvlat 32353  df-hlat 32382
This theorem is referenced by:  dalem29  32731  dalem30  32732  dalem31N  32733  dalem32  32734  dalem33  32735  dalem34  32736  dalem35  32737  dalem36  32738  dalem37  32739  dalem40  32742  dalem46  32748  dalem47  32749  dalem49  32751  dalem50  32752  dalem58  32760  dalem59  32761
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