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Theorem cdleme0nex 29758
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 29679- which is under w, so the only 2 left not under w are p and q themselves.) Note that by cvlsupr2 28812, 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 984 . . . 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 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  .<_  ( P  .\/  Q ) )
31, 2jca 518 . . 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 983 . . . . . 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 987 . . . . . . 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 2554 . . . . . . 7  |-  ( A. r  e.  A  -.  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  <->  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) )
75, 6sylibr 203 . . . . . 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 4027 . . . . . . . . . 10  |-  ( r  =  R  ->  (
r  .<_  W  <->  R  .<_  W ) )
98notbid 285 . . . . . . . . 9  |-  ( r  =  R  ->  ( -.  r  .<_  W  <->  -.  R  .<_  W ) )
10 oveq2 5828 . . . . . . . . . 10  |-  ( r  =  R  ->  ( P  .\/  r )  =  ( P  .\/  R
) )
11 oveq2 5828 . . . . . . . . . 10  |-  ( r  =  R  ->  ( Q  .\/  r )  =  ( Q  .\/  R
) )
1210, 11eqeq12d 2298 . . . . . . . . 9  |-  ( r  =  R  ->  (
( P  .\/  r
)  =  ( Q 
.\/  r )  <->  ( P  .\/  R )  =  ( Q  .\/  R ) ) )
139, 12anbi12d 691 . . . . . . . 8  |-  ( r  =  R  ->  (
( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  <->  ( -.  R  .<_  W  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) ) ) )
1413notbid 285 . . . . . . 7  |-  ( r  =  R  ->  ( -.  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  <->  -.  ( -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R
) ) ) )
1514rspcva 2883 . . . . . 6  |-  ( ( R  e.  A  /\  A. r  e.  A  -.  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  ->  -.  ( -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q 
.\/  R ) ) )
164, 7, 15syl2anc 642 . . . . 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 985 . . . . . . . 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 28828 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  CvLat )
1917, 18syl 15 . . . . . . 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 988 . . . . . . 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 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 ) )  ->  Q  e.  A )
22 simp23 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  =/=  Q )
23 cdleme0nex.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
24 cdleme0nex.l . . . . . . . 8  |-  .<_  =  ( le `  K )
25 cdleme0nex.j . . . . . . . 8  |-  .\/  =  ( join `  K )
2623, 24, 25cvlsupr2 28812 . . . . . . 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 1195 . . . . . 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 684 . . . . 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 291 . . . 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 474 . . . . 5  |-  ( -.  ( ( R  =/= 
P  /\  R  =/=  Q )  /\  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) )  <->  ( -.  ( R  =/=  P  /\  R  =/=  Q
)  \/  -.  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) ) )
31 df-3an 936 . . . . . . . 8  |-  ( ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q
) )  <->  ( ( R  =/=  P  /\  R  =/=  Q )  /\  R  .<_  ( P  .\/  Q
) ) )
3231anbi2i 675 . . . . . . 7  |-  ( ( -.  R  .<_  W  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P 
.\/  Q ) ) )  <->  ( -.  R  .<_  W  /\  ( ( R  =/=  P  /\  R  =/=  Q )  /\  R  .<_  ( P  .\/  Q ) ) ) )
33 an12 772 . . . . . . 7  |-  ( ( -.  R  .<_  W  /\  ( ( R  =/= 
P  /\  R  =/=  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  <->  ( ( R  =/=  P  /\  R  =/=  Q )  /\  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) ) )
3432, 33bitri 240 . . . . . 6  |-  ( ( -.  R  .<_  W  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P 
.\/  Q ) ) )  <->  ( ( R  =/=  P  /\  R  =/=  Q )  /\  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) ) )
3534notbii 287 . . . . 5  |-  ( -.  ( -.  R  .<_  W  /\  ( R  =/= 
P  /\  R  =/=  Q  /\  R  .<_  ( P 
.\/  Q ) ) )  <->  -.  ( ( R  =/=  P  /\  R  =/=  Q )  /\  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) ) )
36 pm4.62 408 . . . . 5  |-  ( ( ( R  =/=  P  /\  R  =/=  Q
)  ->  -.  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) )  <->  ( -.  ( R  =/=  P  /\  R  =/=  Q
)  \/  -.  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) ) )
3730, 35, 363bitr4ri 269 . . . 4  |-  ( ( ( R  =/=  P  /\  R  =/=  Q
)  ->  -.  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) )  <->  -.  ( -.  R  .<_  W  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P 
.\/  Q ) ) ) )
3829, 37sylibr 203 . . 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 109 . 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 2532 . . 3  |-  ( ( R  =/=  P  /\  R  =/=  Q )  <->  -.  ( R  =  P  \/  R  =  Q )
)
4140con2bii 322 . 2  |-  ( ( R  =  P  \/  R  =  Q )  <->  -.  ( R  =/=  P  /\  R  =/=  Q
) )
4239, 41sylibr 203 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1685    =/= wne 2447   A.wral 2544   E.wrex 2545   class class class wbr 4024   ` cfv 5221  (class class class)co 5820   lecple 13211   joincjn 14074   Atomscatm 28732   CvLatclc 28734   HLchlt 28819
This theorem is referenced by:  cdleme18c  29761  cdleme18d  29763  cdlemg17b  30130  cdlemg17h  30136
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-poset 14076  df-plt 14088  df-lub 14104  df-join 14106  df-lat 14148  df-covers 28735  df-ats 28736  df-atl 28767  df-cvlat 28791  df-hlat 28820
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