Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalem43 Structured version   Unicode version

Theorem dalem43 32696
Description: Lemma for dath 32717. Planes  G H I and  Y are different. (Contributed by NM, 8-Aug-2012.)
Hypotheses
Ref Expression
dalem.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 ) ) ) ) )
dalem.l  |-  .<_  =  ( le `  K )
dalem.j  |-  .\/  =  ( join `  K )
dalem.a  |-  A  =  ( Atoms `  K )
dalem.ps  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
dalem38.m  |-  ./\  =  ( meet `  K )
dalem38.o  |-  O  =  ( LPlanes `  K )
dalem38.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem38.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem38.g  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
dalem38.h  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
dalem38.i  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
Assertion
Ref Expression
dalem43  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  =/=  Y )

Proof of Theorem dalem43
StepHypRef Expression
1 dalem.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 ) ) ) ) )
21dalemkelat 32605 . . . 4  |-  ( ph  ->  K  e.  Lat )
323ad2ant1 1016 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
41dalemkehl 32604 . . . . 5  |-  ( ph  ->  K  e.  HL )
543ad2ant1 1016 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
6 dalem.l . . . . 5  |-  .<_  =  ( le `  K )
7 dalem.j . . . . 5  |-  .\/  =  ( join `  K )
8 dalem.a . . . . 5  |-  A  =  ( Atoms `  K )
9 dalem.ps . . . . 5  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
10 dalem38.m . . . . 5  |-  ./\  =  ( meet `  K )
11 dalem38.o . . . . 5  |-  O  =  ( LPlanes `  K )
12 dalem38.y . . . . 5  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
13 dalem38.z . . . . 5  |-  Z  =  ( ( S  .\/  T )  .\/  U )
14 dalem38.g . . . . 5  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
151, 6, 7, 8, 9, 10, 11, 12, 13, 14dalem23 32677 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
16 dalem38.h . . . . 5  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
171, 6, 7, 8, 9, 10, 11, 12, 13, 16dalem29 32682 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  A )
18 eqid 2400 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
1918, 7, 8hlatjcl 32348 . . . 4  |-  ( ( K  e.  HL  /\  G  e.  A  /\  H  e.  A )  ->  ( G  .\/  H
)  e.  ( Base `  K ) )
205, 15, 17, 19syl3anc 1228 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  e.  ( Base `  K ) )
21 dalem38.i . . . . 5  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
221, 6, 7, 8, 9, 10, 11, 12, 13, 21dalem34 32687 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  A )
2318, 8atbase 32271 . . . 4  |-  ( I  e.  A  ->  I  e.  ( Base `  K
) )
2422, 23syl 17 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  ( Base `  K ) )
2518, 6, 7latlej2 15905 . . 3  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  I  e.  ( Base `  K )
)  ->  I  .<_  ( ( G  .\/  H
)  .\/  I )
)
263, 20, 24, 25syl3anc 1228 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  .<_  ( ( G 
.\/  H )  .\/  I ) )
271, 6, 7, 8, 9, 10, 11, 12, 13, 21dalem35 32688 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  I  .<_  Y )
28 nbrne1 4409 . 2  |-  ( ( I  .<_  ( ( G  .\/  H )  .\/  I )  /\  -.  I  .<_  Y )  -> 
( ( G  .\/  H )  .\/  I )  =/=  Y )
2926, 27, 28syl2anc 659 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  =/=  Y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840    =/= wne 2596   class class class wbr 4392   ` cfv 5523  (class class class)co 6232   Basecbs 14731   lecple 14806   joincjn 15787   meetcmee 15788   Latclat 15889   Atomscatm 32245   HLchlt 32332   LPlanesclpl 32473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-preset 15771  df-poset 15789  df-plt 15802  df-lub 15818  df-glb 15819  df-join 15820  df-meet 15821  df-p0 15883  df-lat 15890  df-clat 15952  df-oposet 32158  df-ol 32160  df-oml 32161  df-covers 32248  df-ats 32249  df-atl 32280  df-cvlat 32304  df-hlat 32333  df-llines 32479  df-lplanes 32480
This theorem is referenced by:  dalem44  32697  dalem51  32704
  Copyright terms: Public domain W3C validator