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Theorem cdleme0nex 29609
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 29530- which is under w, so the only 2 left not under w are p and q themselves.) Note that by cvlsupr2 28663, 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 989 . . . 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 991 . . . 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 520 . . 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 988 . . . . . 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 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 ) )  ->  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) )
6 ralnex 2524 . . . . . . 7  |-  ( A. r  e.  A  -.  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  <->  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) )
75, 6sylibr 205 . . . . . 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 3966 . . . . . . . . . 10  |-  ( r  =  R  ->  (
r  .<_  W  <->  R  .<_  W ) )
98notbid 287 . . . . . . . . 9  |-  ( r  =  R  ->  ( -.  r  .<_  W  <->  -.  R  .<_  W ) )
10 oveq2 5765 . . . . . . . . . 10  |-  ( r  =  R  ->  ( P  .\/  r )  =  ( P  .\/  R
) )
11 oveq2 5765 . . . . . . . . . 10  |-  ( r  =  R  ->  ( Q  .\/  r )  =  ( Q  .\/  R
) )
1210, 11eqeq12d 2270 . . . . . . . . 9  |-  ( r  =  R  ->  (
( P  .\/  r
)  =  ( Q 
.\/  r )  <->  ( P  .\/  R )  =  ( Q  .\/  R ) ) )
139, 12anbi12d 694 . . . . . . . 8  |-  ( r  =  R  ->  (
( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  <->  ( -.  R  .<_  W  /\  ( P 
.\/  R )  =  ( Q  .\/  R
) ) ) )
1413notbid 287 . . . . . . 7  |-  ( r  =  R  ->  ( -.  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  <->  -.  ( -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R
) ) ) )
1514rcla4va 2833 . . . . . 6  |-  ( ( R  e.  A  /\  A. r  e.  A  -.  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  ->  -.  ( -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q 
.\/  R ) ) )
164, 7, 15syl2anc 645 . . . . 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 990 . . . . . . . 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 28679 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  CvLat )
1917, 18syl 17 . . . . . . 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 993 . . . . . . 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 994 . . . . . . 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 995 . . . . . . 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 28663 . . . . . . 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 1200 . . . . . 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 687 . . . . 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 293 . . . 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 476 . . . . 5  |-  ( -.  ( ( R  =/= 
P  /\  R  =/=  Q )  /\  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) )  <->  ( -.  ( R  =/=  P  /\  R  =/=  Q
)  \/  -.  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) ) )
31 df-3an 941 . . . . . . . 8  |-  ( ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P  .\/  Q
) )  <->  ( ( R  =/=  P  /\  R  =/=  Q )  /\  R  .<_  ( P  .\/  Q
) ) )
3231anbi2i 678 . . . . . . 7  |-  ( ( -.  R  .<_  W  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P 
.\/  Q ) ) )  <->  ( -.  R  .<_  W  /\  ( ( R  =/=  P  /\  R  =/=  Q )  /\  R  .<_  ( P  .\/  Q ) ) ) )
33 an12 775 . . . . . . 7  |-  ( ( -.  R  .<_  W  /\  ( ( R  =/= 
P  /\  R  =/=  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  <->  ( ( R  =/=  P  /\  R  =/=  Q )  /\  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) ) )
3432, 33bitri 242 . . . . . 6  |-  ( ( -.  R  .<_  W  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P 
.\/  Q ) ) )  <->  ( ( R  =/=  P  /\  R  =/=  Q )  /\  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) ) )
3534notbii 289 . . . . 5  |-  ( -.  ( -.  R  .<_  W  /\  ( R  =/= 
P  /\  R  =/=  Q  /\  R  .<_  ( P 
.\/  Q ) ) )  <->  -.  ( ( R  =/=  P  /\  R  =/=  Q )  /\  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) ) )
36 pm4.62 410 . . . . 5  |-  ( ( ( R  =/=  P  /\  R  =/=  Q
)  ->  -.  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) )  <->  ( -.  ( R  =/=  P  /\  R  =/=  Q
)  \/  -.  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) ) )
3730, 35, 363bitr4ri 271 . . . 4  |-  ( ( ( R  =/=  P  /\  R  =/=  Q
)  ->  -.  ( -.  R  .<_  W  /\  R  .<_  ( P  .\/  Q ) ) )  <->  -.  ( -.  R  .<_  W  /\  ( R  =/=  P  /\  R  =/=  Q  /\  R  .<_  ( P 
.\/  Q ) ) ) )
3829, 37sylibr 205 . . 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 2504 . . 3  |-  ( ( R  =/=  P  /\  R  =/=  Q )  <->  -.  ( R  =  P  \/  R  =  Q )
)
4140con2bii 324 . 2  |-  ( ( R  =  P  \/  R  =  Q )  <->  -.  ( R  =/=  P  /\  R  =/=  Q
) )
4239, 41sylibr 205 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 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   A.wral 2516   E.wrex 2517   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   lecple 13142   joincjn 14005   Atomscatm 28583   CvLatclc 28585   HLchlt 28670
This theorem is referenced by:  cdleme18c  29612  cdleme18d  29614  cdlemg17b  29981  cdlemg17h  29987
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-undef 6229  df-riota 6237  df-poset 14007  df-plt 14019  df-lub 14035  df-join 14037  df-lat 14079  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671
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