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Theorem 4atexlemntlpq 30939
Description: Lemma for 4atexlem7 30946. (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 30938 . 2  |-  ( ph  ->  T  .<_  W )
1014atexlemkc 30929 . . . . . 6  |-  ( ph  ->  K  e.  CvLat )
111, 2, 3, 4, 5, 6, 74atexlemu 30935 . . . . . 6  |-  ( ph  ->  U  e.  A )
121, 2, 3, 4, 5, 6, 7, 84atexlemv 30936 . . . . . 6  |-  ( ph  ->  V  e.  A )
1314atexlemt 30924 . . . . . 6  |-  ( ph  ->  T  e.  A )
141, 2, 3, 4, 5, 6, 7, 84atexlemunv 30937 . . . . . 6  |-  ( ph  ->  U  =/=  V )
1514atexlemutvt 30925 . . . . . 6  |-  ( ph  ->  ( U  .\/  T
)  =  ( V 
.\/  T ) )
165, 3cvlsupr5 30218 . . . . . 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 1203 . . . . 5  |-  ( ph  ->  T  =/=  U )
1817adantr 453 . . . 4  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  T  =/=  U )
1914atexlemk 30918 . . . . . . 7  |-  ( ph  ->  K  e.  HL )
2014atexlemw 30919 . . . . . . 7  |-  ( ph  ->  W  e.  H )
2119, 20jca 520 . . . . . 6  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
2221adantr 453 . . . . 5  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2314atexlempw 30920 . . . . . 6  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
2423adantr 453 . . . . 5  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
2514atexlemq 30922 . . . . . 6  |-  ( ph  ->  Q  e.  A )
2625adantr 453 . . . . 5  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  Q  e.  A )
2713adantr 453 . . . . 5  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  T  e.  A )
2814atexlempnq 30926 . . . . . 6  |-  ( ph  ->  P  =/=  Q )
2928adantr 453 . . . . 5  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  P  =/=  Q )
30 simpr 449 . . . . 5  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  T  .<_  ( P  .\/  Q ) )
312, 3, 4, 5, 6, 7lhpat3 30917 . . . . 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 1201 . . . 4  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  ( -.  T  .<_  W  <->  T  =/=  U ) )
3318, 32mpbird 225 . . 3  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  -.  T  .<_  W )
3433ex 425 . 2  |-  ( ph  ->  ( T  .<_  ( P 
.\/  Q )  ->  -.  T  .<_  W ) )
359, 34mt2d 112 1  |-  ( ph  ->  -.  T  .<_  ( P 
.\/  Q ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   lecple 13541   joincjn 14406   meetcmee 14407   Atomscatm 30135   CvLatclc 30137   HLchlt 30222   LHypclh 30855
This theorem is referenced by:  4atexlemc  30940  4atexlemex2  30942  4atexlemcnd  30943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-p1 14474  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-lhyp 30859
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