Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalem4 Structured version   Unicode version

Theorem dalem4 33672
Description: Lemma for dalemdnee 33673. (Contributed by NM, 10-Aug-2012.)
Hypotheses
Ref Expression
dalema.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 ) ) ) ) )
dalemc.l  |-  .<_  =  ( le `  K )
dalemc.j  |-  .\/  =  ( join `  K )
dalemc.a  |-  A  =  ( Atoms `  K )
dalem3.m  |-  ./\  =  ( meet `  K )
dalem3.o  |-  O  =  ( LPlanes `  K )
dalem3.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem3.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem3.d  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
dalem3.e  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
Assertion
Ref Expression
dalem4  |-  ( (
ph  /\  D  =/=  T )  ->  D  =/=  E )

Proof of Theorem dalem4
StepHypRef Expression
1 dalema.ph . . . . 5  |-  ( 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 ) ) ) ) )
2 dalemc.l . . . . 5  |-  .<_  =  ( le `  K )
3 dalemc.j . . . . 5  |-  .\/  =  ( join `  K )
4 dalemc.a . . . . 5  |-  A  =  ( Atoms `  K )
51, 2, 3, 4dalemswapyz 33663 . . . 4  |-  ( ph  ->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  ( Z  e.  O  /\  Y  e.  O )  /\  ( ( -.  C  .<_  ( S  .\/  T
)  /\  -.  C  .<_  ( T  .\/  U
)  /\  -.  C  .<_  ( U  .\/  S
) )  /\  ( -.  C  .<_  ( P 
.\/  Q )  /\  -.  C  .<_  ( Q 
.\/  R )  /\  -.  C  .<_  ( R 
.\/  P ) )  /\  ( C  .<_  ( S  .\/  P )  /\  C  .<_  ( T 
.\/  Q )  /\  C  .<_  ( U  .\/  R ) ) ) ) )
65adantr 465 . . 3  |-  ( (
ph  /\  D  =/=  T )  ->  ( (
( K  e.  HL  /\  C  e.  ( Base `  K ) )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
)  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  /\  ( Z  e.  O  /\  Y  e.  O
)  /\  ( ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( C  .<_  ( S  .\/  P )  /\  C  .<_  ( T  .\/  Q )  /\  C  .<_  ( U 
.\/  R ) ) ) ) )
7 dalem3.d . . . . . 6  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
81dalemkelat 33631 . . . . . . 7  |-  ( ph  ->  K  e.  Lat )
91, 3, 4dalempjqeb 33652 . . . . . . 7  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
101, 3, 4dalemsjteb 33653 . . . . . . 7  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
11 eqid 2454 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
12 dalem3.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
1311, 12latmcom 15368 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  =  ( ( S  .\/  T ) 
./\  ( P  .\/  Q ) ) )
148, 9, 10, 13syl3anc 1219 . . . . . 6  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  =  ( ( S  .\/  T )  ./\  ( P  .\/  Q ) ) )
157, 14syl5eq 2507 . . . . 5  |-  ( ph  ->  D  =  ( ( S  .\/  T ) 
./\  ( P  .\/  Q ) ) )
1615neeq1d 2729 . . . 4  |-  ( ph  ->  ( D  =/=  T  <->  ( ( S  .\/  T
)  ./\  ( P  .\/  Q ) )  =/= 
T ) )
1716biimpa 484 . . 3  |-  ( (
ph  /\  D  =/=  T )  ->  ( ( S  .\/  T )  ./\  ( P  .\/  Q ) )  =/=  T )
18 biid 236 . . . 4  |-  ( ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  ( Z  e.  O  /\  Y  e.  O )  /\  ( ( -.  C  .<_  ( S  .\/  T
)  /\  -.  C  .<_  ( T  .\/  U
)  /\  -.  C  .<_  ( U  .\/  S
) )  /\  ( -.  C  .<_  ( P 
.\/  Q )  /\  -.  C  .<_  ( Q 
.\/  R )  /\  -.  C  .<_  ( R 
.\/  P ) )  /\  ( C  .<_  ( S  .\/  P )  /\  C  .<_  ( T 
.\/  Q )  /\  C  .<_  ( U  .\/  R ) ) ) )  <-> 
( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  ( Z  e.  O  /\  Y  e.  O )  /\  ( ( -.  C  .<_  ( S  .\/  T
)  /\  -.  C  .<_  ( T  .\/  U
)  /\  -.  C  .<_  ( U  .\/  S
) )  /\  ( -.  C  .<_  ( P 
.\/  Q )  /\  -.  C  .<_  ( Q 
.\/  R )  /\  -.  C  .<_  ( R 
.\/  P ) )  /\  ( C  .<_  ( S  .\/  P )  /\  C  .<_  ( T 
.\/  Q )  /\  C  .<_  ( U  .\/  R ) ) ) ) )
19 dalem3.o . . . 4  |-  O  =  ( LPlanes `  K )
20 dalem3.z . . . 4  |-  Z  =  ( ( S  .\/  T )  .\/  U )
21 dalem3.y . . . 4  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
22 eqid 2454 . . . 4  |-  ( ( S  .\/  T ) 
./\  ( P  .\/  Q ) )  =  ( ( S  .\/  T
)  ./\  ( P  .\/  Q ) )
23 eqid 2454 . . . 4  |-  ( ( T  .\/  U ) 
./\  ( Q  .\/  R ) )  =  ( ( T  .\/  U
)  ./\  ( Q  .\/  R ) )
2418, 2, 3, 4, 12, 19, 20, 21, 22, 23dalem3 33671 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  ( Z  e.  O  /\  Y  e.  O )  /\  ( ( -.  C  .<_  ( S  .\/  T
)  /\  -.  C  .<_  ( T  .\/  U
)  /\  -.  C  .<_  ( U  .\/  S
) )  /\  ( -.  C  .<_  ( P 
.\/  Q )  /\  -.  C  .<_  ( Q 
.\/  R )  /\  -.  C  .<_  ( R 
.\/  P ) )  /\  ( C  .<_  ( S  .\/  P )  /\  C  .<_  ( T 
.\/  Q )  /\  C  .<_  ( U  .\/  R ) ) ) )  /\  ( ( S 
.\/  T )  ./\  ( P  .\/  Q ) )  =/=  T )  ->  ( ( S 
.\/  T )  ./\  ( P  .\/  Q ) )  =/=  ( ( T  .\/  U ) 
./\  ( Q  .\/  R ) ) )
256, 17, 24syl2anc 661 . 2  |-  ( (
ph  /\  D  =/=  T )  ->  ( ( S  .\/  T )  ./\  ( P  .\/  Q ) )  =/=  ( ( T  .\/  U ) 
./\  ( Q  .\/  R ) ) )
2615adantr 465 . 2  |-  ( (
ph  /\  D  =/=  T )  ->  D  =  ( ( S  .\/  T )  ./\  ( P  .\/  Q ) ) )
27 dalem3.e . . . 4  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
281dalemkehl 33630 . . . . . 6  |-  ( ph  ->  K  e.  HL )
291dalemqea 33634 . . . . . 6  |-  ( ph  ->  Q  e.  A )
301dalemrea 33635 . . . . . 6  |-  ( ph  ->  R  e.  A )
3111, 3, 4hlatjcl 33374 . . . . . 6  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
3228, 29, 30, 31syl3anc 1219 . . . . 5  |-  ( ph  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
331, 3, 4dalemtjueb 33654 . . . . 5  |-  ( ph  ->  ( T  .\/  U
)  e.  ( Base `  K ) )
3411, 12latmcom 15368 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  ( T  .\/  U )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  =  ( ( T  .\/  U ) 
./\  ( Q  .\/  R ) ) )
358, 32, 33, 34syl3anc 1219 . . . 4  |-  ( ph  ->  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  =  ( ( T  .\/  U )  ./\  ( Q  .\/  R ) ) )
3627, 35syl5eq 2507 . . 3  |-  ( ph  ->  E  =  ( ( T  .\/  U ) 
./\  ( Q  .\/  R ) ) )
3736adantr 465 . 2  |-  ( (
ph  /\  D  =/=  T )  ->  E  =  ( ( T  .\/  U )  ./\  ( Q  .\/  R ) ) )
3825, 26, 373netr4d 2757 1  |-  ( (
ph  /\  D  =/=  T )  ->  D  =/=  E )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14296   lecple 14368   joincjn 15237   meetcmee 15238   Latclat 15338   Atomscatm 33271   HLchlt 33358   LPlanesclpl 33499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-poset 15239  df-plt 15251  df-lub 15267  df-glb 15268  df-join 15269  df-meet 15270  df-p0 15332  df-lat 15339  df-clat 15401  df-oposet 33184  df-ol 33186  df-oml 33187  df-covers 33274  df-ats 33275  df-atl 33306  df-cvlat 33330  df-hlat 33359  df-llines 33505  df-lplanes 33506
This theorem is referenced by:  dalemdnee  33673
  Copyright terms: Public domain W3C validator