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Theorem 4atexlemntlpq 30550
Description: Lemma for 4atexlem7 30557. (Contributed by NM, 24-Nov-2012.)
Hypotheses
Ref Expression
4thatlem.ph  |-  ( ph  <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) ) )
4thatlem0.l  |-  .<_  =  ( le `  K )
4thatlem0.j  |-  .\/  =  ( join `  K )
4thatlem0.m  |-  ./\  =  ( meet `  K )
4thatlem0.a  |-  A  =  ( Atoms `  K )
4thatlem0.h  |-  H  =  ( LHyp `  K
)
4thatlem0.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
4thatlem0.v  |-  V  =  ( ( P  .\/  S )  ./\  W )
Assertion
Ref Expression
4atexlemntlpq  |-  ( ph  ->  -.  T  .<_  ( P 
.\/  Q ) )

Proof of Theorem 4atexlemntlpq
StepHypRef Expression
1 4thatlem.ph . . 3  |-  ( ph  <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) ) )
2 4thatlem0.l . . 3  |-  .<_  =  ( le `  K )
3 4thatlem0.j . . 3  |-  .\/  =  ( join `  K )
4 4thatlem0.m . . 3  |-  ./\  =  ( meet `  K )
5 4thatlem0.a . . 3  |-  A  =  ( Atoms `  K )
6 4thatlem0.h . . 3  |-  H  =  ( LHyp `  K
)
7 4thatlem0.u . . 3  |-  U  =  ( ( P  .\/  Q )  ./\  W )
8 4thatlem0.v . . 3  |-  V  =  ( ( P  .\/  S )  ./\  W )
91, 2, 3, 4, 5, 6, 7, 84atexlemtlw 30549 . 2  |-  ( ph  ->  T  .<_  W )
1014atexlemkc 30540 . . . . . 6  |-  ( ph  ->  K  e.  CvLat )
111, 2, 3, 4, 5, 6, 74atexlemu 30546 . . . . . 6  |-  ( ph  ->  U  e.  A )
121, 2, 3, 4, 5, 6, 7, 84atexlemv 30547 . . . . . 6  |-  ( ph  ->  V  e.  A )
1314atexlemt 30535 . . . . . 6  |-  ( ph  ->  T  e.  A )
141, 2, 3, 4, 5, 6, 7, 84atexlemunv 30548 . . . . . 6  |-  ( ph  ->  U  =/=  V )
1514atexlemutvt 30536 . . . . . 6  |-  ( ph  ->  ( U  .\/  T
)  =  ( V 
.\/  T ) )
165, 3cvlsupr5 29829 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( U  e.  A  /\  V  e.  A  /\  T  e.  A )  /\  ( U  =/=  V  /\  ( U  .\/  T
)  =  ( V 
.\/  T ) ) )  ->  T  =/=  U )
1710, 11, 12, 13, 14, 15, 16syl132anc 1202 . . . . 5  |-  ( ph  ->  T  =/=  U )
1817adantr 452 . . . 4  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  T  =/=  U )
1914atexlemk 30529 . . . . . . 7  |-  ( ph  ->  K  e.  HL )
2014atexlemw 30530 . . . . . . 7  |-  ( ph  ->  W  e.  H )
2119, 20jca 519 . . . . . 6  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
2221adantr 452 . . . . 5  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2314atexlempw 30531 . . . . . 6  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
2423adantr 452 . . . . 5  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
2514atexlemq 30533 . . . . . 6  |-  ( ph  ->  Q  e.  A )
2625adantr 452 . . . . 5  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  Q  e.  A )
2713adantr 452 . . . . 5  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  T  e.  A )
2814atexlempnq 30537 . . . . . 6  |-  ( ph  ->  P  =/=  Q )
2928adantr 452 . . . . 5  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  P  =/=  Q )
30 simpr 448 . . . . 5  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  T  .<_  ( P  .\/  Q ) )
312, 3, 4, 5, 6, 7lhpat3 30528 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  T  e.  A )  /\  ( P  =/=  Q  /\  T  .<_  ( P  .\/  Q
) ) )  -> 
( -.  T  .<_  W  <-> 
T  =/=  U ) )
3222, 24, 26, 27, 29, 30, 31syl222anc 1200 . . . 4  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  ( -.  T  .<_  W  <->  T  =/=  U ) )
3318, 32mpbird 224 . . 3  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  -.  T  .<_  W )
3433ex 424 . 2  |-  ( ph  ->  ( T  .<_  ( P 
.\/  Q )  ->  -.  T  .<_  W ) )
359, 34mt2d 111 1  |-  ( ph  ->  -.  T  .<_  ( P 
.\/  Q ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   CvLatclc 29748   HLchlt 29833   LHypclh 30466
This theorem is referenced by:  4atexlemc  30551  4atexlemex2  30553  4atexlemcnd  30554
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-lhyp 30470
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