Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  4atexlemntlpq Structured version   Unicode version

Theorem 4atexlemntlpq 35978
Description: Lemma for 4atexlem7 35985. (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 35977 . 2  |-  ( ph  ->  T  .<_  W )
1014atexlemkc 35968 . . . . . 6  |-  ( ph  ->  K  e.  CvLat )
111, 2, 3, 4, 5, 6, 74atexlemu 35974 . . . . . 6  |-  ( ph  ->  U  e.  A )
121, 2, 3, 4, 5, 6, 7, 84atexlemv 35975 . . . . . 6  |-  ( ph  ->  V  e.  A )
1314atexlemt 35963 . . . . . 6  |-  ( ph  ->  T  e.  A )
141, 2, 3, 4, 5, 6, 7, 84atexlemunv 35976 . . . . . 6  |-  ( ph  ->  U  =/=  V )
1514atexlemutvt 35964 . . . . . 6  |-  ( ph  ->  ( U  .\/  T
)  =  ( V 
.\/  T ) )
165, 3cvlsupr5 35257 . . . . . 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 1246 . . . . 5  |-  ( ph  ->  T  =/=  U )
1817adantr 465 . . . 4  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  T  =/=  U )
1914atexlemk 35957 . . . . . . 7  |-  ( ph  ->  K  e.  HL )
2014atexlemw 35958 . . . . . . 7  |-  ( ph  ->  W  e.  H )
2119, 20jca 532 . . . . . 6  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
2221adantr 465 . . . . 5  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2314atexlempw 35959 . . . . . 6  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
2423adantr 465 . . . . 5  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
2514atexlemq 35961 . . . . . 6  |-  ( ph  ->  Q  e.  A )
2625adantr 465 . . . . 5  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  Q  e.  A )
2713adantr 465 . . . . 5  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  T  e.  A )
2814atexlempnq 35965 . . . . . 6  |-  ( ph  ->  P  =/=  Q )
2928adantr 465 . . . . 5  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  P  =/=  Q )
30 simpr 461 . . . . 5  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  T  .<_  ( P  .\/  Q ) )
312, 3, 4, 5, 6, 7lhpat3 35956 . . . . 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 1244 . . . 4  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  ( -.  T  .<_  W  <->  T  =/=  U ) )
3318, 32mpbird 232 . . 3  |-  ( (
ph  /\  T  .<_  ( P  .\/  Q ) )  ->  -.  T  .<_  W )
3433ex 434 . 2  |-  ( ph  ->  ( T  .<_  ( P 
.\/  Q )  ->  -.  T  .<_  W ) )
359, 34mt2d 117 1  |-  ( ph  ->  -.  T  .<_  ( P 
.\/  Q ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   lecple 14810   joincjn 15791   meetcmee 15792   Atomscatm 35174   CvLatclc 35176   HLchlt 35261   LHypclh 35894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-preset 15775  df-poset 15793  df-plt 15806  df-lub 15822  df-glb 15823  df-join 15824  df-meet 15825  df-p0 15887  df-p1 15888  df-lat 15894  df-clat 15956  df-oposet 35087  df-ol 35089  df-oml 35090  df-covers 35177  df-ats 35178  df-atl 35209  df-cvlat 35233  df-hlat 35262  df-lhyp 35898
This theorem is referenced by:  4atexlemc  35979  4atexlemex2  35981  4atexlemcnd  35982
  Copyright terms: Public domain W3C validator