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Theorem 4atexlemntlpq 30326
Description: Lemma for 4atexlem7 30333. (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 30325 . 2  |-  ( ph  ->  T  .<_  W )
1014atexlemkc 30316 . . . . . 6  |-  ( ph  ->  K  e.  CvLat )
111, 2, 3, 4, 5, 6, 74atexlemu 30322 . . . . . 6  |-  ( ph  ->  U  e.  A )
121, 2, 3, 4, 5, 6, 7, 84atexlemv 30323 . . . . . 6  |-  ( ph  ->  V  e.  A )
1314atexlemt 30311 . . . . . 6  |-  ( ph  ->  T  e.  A )
141, 2, 3, 4, 5, 6, 7, 84atexlemunv 30324 . . . . . 6  |-  ( ph  ->  U  =/=  V )
1514atexlemutvt 30312 . . . . . 6  |-  ( ph  ->  ( U  .\/  T
)  =  ( V 
.\/  T ) )
165, 3cvlsupr5 29605 . . . . . 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 1200 . . . . 5  |-  ( ph  ->  T  =/=  U )
1817adantr 451 . . . 4  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  T  =/=  U )
1914atexlemk 30305 . . . . . . 7  |-  ( ph  ->  K  e.  HL )
2014atexlemw 30306 . . . . . . 7  |-  ( ph  ->  W  e.  H )
2119, 20jca 518 . . . . . 6  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
2221adantr 451 . . . . 5  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2314atexlempw 30307 . . . . . 6  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
2423adantr 451 . . . . 5  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
2514atexlemq 30309 . . . . . 6  |-  ( ph  ->  Q  e.  A )
2625adantr 451 . . . . 5  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  Q  e.  A )
2713adantr 451 . . . . 5  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  T  e.  A )
2814atexlempnq 30313 . . . . . 6  |-  ( ph  ->  P  =/=  Q )
2928adantr 451 . . . . 5  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  P  =/=  Q )
30 simpr 447 . . . . 5  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  T  .<_  ( P  .\/  Q ) )
312, 3, 4, 5, 6, 7lhpat3 30304 . . . . 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 1198 . . . 4  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  ( -.  T  .<_  W  <->  T  =/=  U ) )
3318, 32mpbird 223 . . 3  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  -.  T  .<_  W )
3433ex 423 . 2  |-  ( ph  ->  ( T  .<_  ( P 
.\/  Q )  ->  -.  T  .<_  W ) )
359, 34mt2d 109 1  |-  ( ph  ->  -.  T  .<_  ( P 
.\/  Q ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   class class class wbr 4104   ` cfv 5337  (class class class)co 5945   lecple 13312   joincjn 14177   meetcmee 14178   Atomscatm 29522   CvLatclc 29524   HLchlt 29609   LHypclh 30242
This theorem is referenced by:  4atexlemc  30327  4atexlemex2  30329  4atexlemcnd  30330
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-undef 6385  df-riota 6391  df-poset 14179  df-plt 14191  df-lub 14207  df-glb 14208  df-join 14209  df-meet 14210  df-p0 14244  df-p1 14245  df-lat 14251  df-clat 14313  df-oposet 29435  df-ol 29437  df-oml 29438  df-covers 29525  df-ats 29526  df-atl 29557  df-cvlat 29581  df-hlat 29610  df-lhyp 30246
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