Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme0nex Structured version   Unicode version

Theorem cdleme0nex 31024
Description: Part of proof of Lemma E in [Crawley] p. 114, 4th line of 4th paragraph. Whenever (in their terminology) p  \/ q/0 (i.e. the sublattice from 0 to p  \/ q) contains precisely three atoms, any atom not under w must equal either p or q. (In case of 3 atoms, one of them must be u - see cdleme0a 30945- which is under w, so the only 2 left not under w are p and q themselves.) Note that by cvlsupr2 30078, our  ( P  .\/  r )  =  ( Q  .\/  r ) is a shorter way to express  r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ). Thus, the negated existential condition states there are no atoms different from p or q that are also not under w. (Contributed by NM, 12-Nov-2012.)
Hypotheses
Ref Expression
cdleme0nex.l  |-  .<_  =  ( le `  K )
cdleme0nex.j  |-  .\/  =  ( join `  K )
cdleme0nex.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cdleme0nex  |-  ( ( ( K  e.  HL  /\  R  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/= 
Q )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( R  =  P  \/  R  =  Q ) )
Distinct variable groups:    A, r    .\/ , r    .<_ , r    P, r    Q, r    R, r    W, r
Allowed substitution hint:    K( r)

Proof of Theorem cdleme0nex
StepHypRef Expression
1 simp3r 986 . . . 4  |-  ( ( ( K  e.  HL  /\  R  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/= 
Q )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  -.  R  .<_  W )
2 simp12 988 . . . 4  |-  ( ( ( K  e.  HL  /\  R  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/= 
Q )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  R  .<_  ( P  .\/  Q ) )
31, 2jca 519 . . 3  |-  ( ( ( K  e.  HL  /\  R  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/= 
Q )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q
) ) )
4 simp3l 985 . . . . . 6  |-  ( ( ( K  e.  HL  /\  R  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/= 
Q )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  R  e.  A )
5 simp13 989 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  R  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/= 
Q )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) )
6 ralnex 2707 . . . . . . 7  |-  ( A. r  e.  A  -.  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  <->  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) )
75, 6sylibr 204 . . . . . 6  |-  ( ( ( K  e.  HL  /\  R  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/= 
Q )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  A. r  e.  A  -.  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )
8 breq1 4207 . . . . . . . . . 10  |-  ( r  =  R  ->  (
r  .<_  W  <->  R  .<_  W ) )
98notbid 286 . . . . . . . . 9  |-  ( r  =  R  ->  ( -.  r  .<_  W  <->  -.  R  .<_  W ) )
10 oveq2 6081 . . . . . . . . . 10  |-  ( r  =  R  ->  ( P  .\/  r )  =  ( P  .\/  R
) )
11 oveq2 6081 . . . . . . . . . 10  |-  ( r  =  R  ->  ( Q  .\/  r )  =  ( Q  .\/  R
) )
1210, 11eqeq12d 2449 . . . . . . . . 9  |-  ( r  =  R  ->  (
( P  .\/  r
)  =  ( Q 
.\/  r )  <->  ( P  .\/  R )  =  ( Q  .\/  R ) ) )
139, 12anbi12d 692 . . . . . . . 8  |-  ( r  =  R  ->  (
( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  <->  ( -.  R  .<_  W  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) ) ) )
1413notbid 286 . . . . . . 7  |-  ( r  =  R  ->  ( -.  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  <->  -.  ( -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R
) ) ) )
1514rspcva 3042 . . . . . 6  |-  ( ( R  e.  A  /\  A. r  e.  A  -.  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  ->  -.  ( -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q 
.\/  R ) ) )
164, 7, 15syl2anc 643 . . . . 5  |-  ( ( ( K  e.  HL  /\  R  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/= 
Q )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  -.  ( -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) ) )
17 simp11 987 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  R  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/= 
Q )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  K  e.  HL )
18 hlcvl 30094 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  CvLat )
1917, 18syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  R  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/= 
Q )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  K  e.  CvLat
)
20 simp21 990 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  R  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/= 
Q )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  P  e.  A )
21 simp22 991 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  R  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/= 
Q )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  Q  e.  A )
22 simp23 992 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  R  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/= 
Q )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  P  =/=  Q )
23 cdleme0nex.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
24 cdleme0nex.l . . . . . . . 8  |-  .<_  =  ( le `  K )
25 cdleme0nex.j . . . . . . . 8  |-  .\/  =  ( join `  K )
2623, 24, 25cvlsupr2 30078 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  Q
)  ->  ( ( P  .\/  R )  =  ( Q  .\/  R
)  <->  ( R  =/= 
P  /\  R  =/=  Q  /\  R  .<_  ( P 
.\/  Q ) ) ) )
2719, 20, 21, 4, 22, 26syl131anc 1197 . . . . . 6  |-  ( ( ( K  e.  HL  /\  R  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/= 
Q )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( ( P  .\/  R )  =  ( Q  .\/  R
)  <->  ( R  =/= 
P  /\  R  =/=  Q  /\  R  .<_  ( P 
.\/  Q ) ) ) )
2827anbi2d 685 . . . . 5  |-  ( ( ( K  e.  HL  /\  R  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/= 
Q )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( ( -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  <->  ( -.  R  .<_  W  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q ) ) ) ) )
2916, 28mtbid 292 . . . 4  |-  ( ( ( K  e.  HL  /\  R  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/= 
Q )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  -.  ( -.  R  .<_  W  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P 
.\/  Q ) ) ) )
30 ianor 475 . . . . 5  |-  ( -.  ( ( R  =/= 
P  /\  R  =/=  Q )  /\  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) )  <->  ( -.  ( R  =/=  P  /\  R  =/=  Q
)  \/  -.  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) ) )
31 df-3an 938 . . . . . . . 8  |-  ( ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q
) )  <->  ( ( R  =/=  P  /\  R  =/=  Q )  /\  R  .<_  ( P  .\/  Q
) ) )
3231anbi2i 676 . . . . . . 7  |-  ( ( -.  R  .<_  W  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P 
.\/  Q ) ) )  <->  ( -.  R  .<_  W  /\  ( ( R  =/=  P  /\  R  =/=  Q )  /\  R  .<_  ( P  .\/  Q ) ) ) )
33 an12 773 . . . . . . 7  |-  ( ( -.  R  .<_  W  /\  ( ( R  =/= 
P  /\  R  =/=  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  <->  ( ( R  =/=  P  /\  R  =/=  Q )  /\  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) ) )
3432, 33bitri 241 . . . . . 6  |-  ( ( -.  R  .<_  W  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P 
.\/  Q ) ) )  <->  ( ( R  =/=  P  /\  R  =/=  Q )  /\  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) ) )
3534notbii 288 . . . . 5  |-  ( -.  ( -.  R  .<_  W  /\  ( R  =/= 
P  /\  R  =/=  Q  /\  R  .<_  ( P 
.\/  Q ) ) )  <->  -.  ( ( R  =/=  P  /\  R  =/=  Q )  /\  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) ) )
36 pm4.62 409 . . . . 5  |-  ( ( ( R  =/=  P  /\  R  =/=  Q
)  ->  -.  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) )  <->  ( -.  ( R  =/=  P  /\  R  =/=  Q
)  \/  -.  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) ) )
3730, 35, 363bitr4ri 270 . . . 4  |-  ( ( ( R  =/=  P  /\  R  =/=  Q
)  ->  -.  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) )  <->  -.  ( -.  R  .<_  W  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P 
.\/  Q ) ) ) )
3829, 37sylibr 204 . . 3  |-  ( ( ( K  e.  HL  /\  R  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/= 
Q )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( ( R  =/=  P  /\  R  =/=  Q )  ->  -.  ( -.  R  .<_  W  /\  R  .<_  ( P 
.\/  Q ) ) ) )
393, 38mt2d 111 . 2  |-  ( ( ( K  e.  HL  /\  R  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/= 
Q )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  -.  ( R  =/=  P  /\  R  =/=  Q ) )
40 neanior 2683 . . 3  |-  ( ( R  =/=  P  /\  R  =/=  Q )  <->  -.  ( R  =  P  \/  R  =  Q )
)
4140con2bii 323 . 2  |-  ( ( R  =  P  \/  R  =  Q )  <->  -.  ( R  =/=  P  /\  R  =/=  Q
) )
4239, 41sylibr 204 1  |-  ( ( ( K  e.  HL  /\  R  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/= 
Q )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( R  =  P  \/  R  =  Q ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   lecple 13528   joincjn 14393   Atomscatm 29998   CvLatclc 30000   HLchlt 30085
This theorem is referenced by:  cdleme18c  31027  cdleme18d  31029  cdlemg17b  31396  cdlemg17h  31402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-join 14425  df-lat 14467  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086
  Copyright terms: Public domain W3C validator