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Theorem cdleme0ex2N 31023
Description: Part of proof of Lemma E in [Crawley] p. 113. Note that  ( P  .\/  u )  =  ( Q  .\/  u ) is a shorter way to express  u  =/=  P  /\  u  =/=  Q  /\  u  .<_  ( P 
.\/  Q ). (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme0.l  |-  .<_  =  ( le `  K )
cdleme0.j  |-  .\/  =  ( join `  K )
cdleme0.m  |-  ./\  =  ( meet `  K )
cdleme0.a  |-  A  =  ( Atoms `  K )
cdleme0.h  |-  H  =  ( LHyp `  K
)
cdleme0.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
cdleme0ex2N  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  E. u  e.  A  ( ( P  .\/  u )  =  ( Q  .\/  u
)  /\  u  .<_  W ) )
Distinct variable groups:    u, A    u, 
.\/    u,  .<_    u, P    u, Q    u, U    u, W    u, H    u, K
Allowed substitution hint:    ./\ ( u)

Proof of Theorem cdleme0ex2N
StepHypRef Expression
1 simp1 958 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp2l 984 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3 simp2rl 1027 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  Q  e.  A )
4 simp3 960 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  P  =/=  Q )
5 cdleme0.l . . . 4  |-  .<_  =  ( le `  K )
6 cdleme0.j . . . 4  |-  .\/  =  ( join `  K )
7 cdleme0.m . . . 4  |-  ./\  =  ( meet `  K )
8 cdleme0.a . . . 4  |-  A  =  ( Atoms `  K )
9 cdleme0.h . . . 4  |-  H  =  ( LHyp `  K
)
10 cdleme0.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
115, 6, 7, 8, 9, 10cdleme0ex1N 31022 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  E. u  e.  A  ( u  .<_  ( P  .\/  Q
)  /\  u  .<_  W ) )
121, 2, 3, 4, 11syl121anc 1190 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  E. u  e.  A  ( u  .<_  ( P  .\/  Q
)  /\  u  .<_  W ) )
13 simp11l 1069 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  /\  u  e.  A  /\  u  .<_  W )  ->  K  e.  HL )
14 hlcvl 30159 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  CvLat )
1513, 14syl 16 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  /\  u  e.  A  /\  u  .<_  W )  ->  K  e.  CvLat )
16 simp2ll 1025 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  P  e.  A )
17163ad2ant1 979 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  /\  u  e.  A  /\  u  .<_  W )  ->  P  e.  A )
1833ad2ant1 979 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  /\  u  e.  A  /\  u  .<_  W )  ->  Q  e.  A )
19 simp2 959 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  /\  u  e.  A  /\  u  .<_  W )  ->  u  e.  A )
20 simp13 990 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  /\  u  e.  A  /\  u  .<_  W )  ->  P  =/=  Q )
218, 5, 6cvlsupr2 30143 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  u  e.  A )  /\  P  =/=  Q
)  ->  ( ( P  .\/  u )  =  ( Q  .\/  u
)  <->  ( u  =/= 
P  /\  u  =/=  Q  /\  u  .<_  ( P 
.\/  Q ) ) ) )
2215, 17, 18, 19, 20, 21syl131anc 1198 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  /\  u  e.  A  /\  u  .<_  W )  -> 
( ( P  .\/  u )  =  ( Q  .\/  u )  <-> 
( u  =/=  P  /\  u  =/=  Q  /\  u  .<_  ( P 
.\/  Q ) ) ) )
23 simp3 960 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  /\  u  e.  A  /\  u  .<_  W )  ->  u  .<_  W )
24 simp2lr 1026 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  -.  P  .<_  W )
25243ad2ant1 979 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  /\  u  e.  A  /\  u  .<_  W )  ->  -.  P  .<_  W )
26 nbrne2 4232 . . . . . . . . . 10  |-  ( ( u  .<_  W  /\  -.  P  .<_  W )  ->  u  =/=  P
)
2723, 25, 26syl2anc 644 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  /\  u  e.  A  /\  u  .<_  W )  ->  u  =/=  P )
28 simp2rr 1028 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  -.  Q  .<_  W )
29283ad2ant1 979 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  /\  u  e.  A  /\  u  .<_  W )  ->  -.  Q  .<_  W )
30 nbrne2 4232 . . . . . . . . . 10  |-  ( ( u  .<_  W  /\  -.  Q  .<_  W )  ->  u  =/=  Q
)
3123, 29, 30syl2anc 644 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  /\  u  e.  A  /\  u  .<_  W )  ->  u  =/=  Q )
3227, 31jca 520 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  /\  u  e.  A  /\  u  .<_  W )  -> 
( u  =/=  P  /\  u  =/=  Q
) )
3332biantrurd 496 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  /\  u  e.  A  /\  u  .<_  W )  -> 
( u  .<_  ( P 
.\/  Q )  <->  ( (
u  =/=  P  /\  u  =/=  Q )  /\  u  .<_  ( P  .\/  Q ) ) ) )
34 df-3an 939 . . . . . . 7  |-  ( ( u  =/=  P  /\  u  =/=  Q  /\  u  .<_  ( P  .\/  Q
) )  <->  ( (
u  =/=  P  /\  u  =/=  Q )  /\  u  .<_  ( P  .\/  Q ) ) )
3533, 34syl6rbbr 257 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  /\  u  e.  A  /\  u  .<_  W )  -> 
( ( u  =/= 
P  /\  u  =/=  Q  /\  u  .<_  ( P 
.\/  Q ) )  <-> 
u  .<_  ( P  .\/  Q ) ) )
3622, 35bitrd 246 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  /\  u  e.  A  /\  u  .<_  W )  -> 
( ( P  .\/  u )  =  ( Q  .\/  u )  <-> 
u  .<_  ( P  .\/  Q ) ) )
37363expia 1156 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  /\  u  e.  A )  ->  ( u  .<_  W  -> 
( ( P  .\/  u )  =  ( Q  .\/  u )  <-> 
u  .<_  ( P  .\/  Q ) ) ) )
3837pm5.32rd 623 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  /\  u  e.  A )  ->  ( ( ( P 
.\/  u )  =  ( Q  .\/  u
)  /\  u  .<_  W )  <->  ( u  .<_  ( P  .\/  Q )  /\  u  .<_  W ) ) )
3938rexbidva 2724 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  ( E. u  e.  A  ( ( P  .\/  u )  =  ( Q  .\/  u )  /\  u  .<_  W )  <->  E. u  e.  A  ( u  .<_  ( P 
.\/  Q )  /\  u  .<_  W ) ) )
4012, 39mpbird 225 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  E. u  e.  A  ( ( P  .\/  u )  =  ( Q  .\/  u
)  /\  u  .<_  W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   lecple 13538   joincjn 14403   meetcmee 14404   Atomscatm 30063   CvLatclc 30065   HLchlt 30150   LHypclh 30783
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-oposet 29976  df-ol 29978  df-oml 29979  df-covers 30066  df-ats 30067  df-atl 30098  df-cvlat 30122  df-hlat 30151  df-lhyp 30787
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