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Theorem cdleme0nex 35487
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 35408- which is under w, so the only 2 left not under w are p and q themselves.) Note that by cvlsupr2 34541, 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 1025 . . . 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 1027 . . . 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 532 . . 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 1024 . . . . . 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 1028 . . . . . . 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 2913 . . . . . . 7  |-  ( A. r  e.  A  -.  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  <->  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) )
75, 6sylibr 212 . . . . . 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 4456 . . . . . . . . . 10  |-  ( r  =  R  ->  (
r  .<_  W  <->  R  .<_  W ) )
98notbid 294 . . . . . . . . 9  |-  ( r  =  R  ->  ( -.  r  .<_  W  <->  -.  R  .<_  W ) )
10 oveq2 6303 . . . . . . . . . 10  |-  ( r  =  R  ->  ( P  .\/  r )  =  ( P  .\/  R
) )
11 oveq2 6303 . . . . . . . . . 10  |-  ( r  =  R  ->  ( Q  .\/  r )  =  ( Q  .\/  R
) )
1210, 11eqeq12d 2489 . . . . . . . . 9  |-  ( r  =  R  ->  (
( P  .\/  r
)  =  ( Q 
.\/  r )  <->  ( P  .\/  R )  =  ( Q  .\/  R ) ) )
139, 12anbi12d 710 . . . . . . . 8  |-  ( r  =  R  ->  (
( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  <->  ( -.  R  .<_  W  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) ) ) )
1413notbid 294 . . . . . . 7  |-  ( r  =  R  ->  ( -.  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  <->  -.  ( -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R
) ) ) )
1514rspcva 3217 . . . . . 6  |-  ( ( R  e.  A  /\  A. r  e.  A  -.  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  ->  -.  ( -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q 
.\/  R ) ) )
164, 7, 15syl2anc 661 . . . . 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 1026 . . . . . . . 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 34557 . . . . . . . 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 1029 . . . . . . 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 1030 . . . . . . 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 1031 . . . . . . 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 34541 . . . . . . 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 1241 . . . . . 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 703 . . . . 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 300 . . . 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 488 . . . . 5  |-  ( -.  ( ( R  =/= 
P  /\  R  =/=  Q )  /\  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) )  <->  ( -.  ( R  =/=  P  /\  R  =/=  Q
)  \/  -.  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) ) )
31 df-3an 975 . . . . . . . 8  |-  ( ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q
) )  <->  ( ( R  =/=  P  /\  R  =/=  Q )  /\  R  .<_  ( P  .\/  Q
) ) )
3231anbi2i 694 . . . . . . 7  |-  ( ( -.  R  .<_  W  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P 
.\/  Q ) ) )  <->  ( -.  R  .<_  W  /\  ( ( R  =/=  P  /\  R  =/=  Q )  /\  R  .<_  ( P  .\/  Q ) ) ) )
33 an12 795 . . . . . . 7  |-  ( ( -.  R  .<_  W  /\  ( ( R  =/= 
P  /\  R  =/=  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  <->  ( ( R  =/=  P  /\  R  =/=  Q )  /\  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) ) )
3432, 33bitri 249 . . . . . 6  |-  ( ( -.  R  .<_  W  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P 
.\/  Q ) ) )  <->  ( ( R  =/=  P  /\  R  =/=  Q )  /\  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) ) )
3534notbii 296 . . . . 5  |-  ( -.  ( -.  R  .<_  W  /\  ( R  =/= 
P  /\  R  =/=  Q  /\  R  .<_  ( P 
.\/  Q ) ) )  <->  -.  ( ( R  =/=  P  /\  R  =/=  Q )  /\  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) ) )
36 pm4.62 419 . . . . 5  |-  ( ( ( R  =/=  P  /\  R  =/=  Q
)  ->  -.  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) )  <->  ( -.  ( R  =/=  P  /\  R  =/=  Q
)  \/  -.  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) ) )
3730, 35, 363bitr4ri 278 . . . 4  |-  ( ( ( R  =/=  P  /\  R  =/=  Q
)  ->  -.  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) )  <->  -.  ( -.  R  .<_  W  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P 
.\/  Q ) ) ) )
3829, 37sylibr 212 . . 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 117 . 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 2792 . . 3  |-  ( ( R  =/=  P  /\  R  =/=  Q )  <->  -.  ( R  =  P  \/  R  =  Q )
)
4140con2bii 332 . 2  |-  ( ( R  =  P  \/  R  =  Q )  <->  -.  ( R  =/=  P  /\  R  =/=  Q
) )
4239, 41sylibr 212 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   lecple 14579   joincjn 15448   Atomscatm 34461   CvLatclc 34463   HLchlt 34548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-lat 15550  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549
This theorem is referenced by:  cdleme18c  35490  cdleme18d  35492  cdlemg17b  35859  cdlemg17h  35865
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