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Theorem cdleme0ex2N 33760
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 1005 . . 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 1031 . . 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 1074 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  Q  e.  A )
4 simp3 1007 . . 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 33759 . . 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 1269 . 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 1116 . . . . . . . 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 32895 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  CvLat )
1513, 14syl 17 . . . . . . 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 1072 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  P  e.  A )
17163ad2ant1 1026 . . . . . . 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 1026 . . . . . . 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 1006 . . . . . . 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 1037 . . . . . . 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 32879 . . . . . . 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 1277 . . . . . 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 1007 . . . . . . . . . 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 1073 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  -.  P  .<_  W )
25243ad2ant1 1026 . . . . . . . . . 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 4442 . . . . . . . . . 10  |-  ( ( u  .<_  W  /\  -.  P  .<_  W )  ->  u  =/=  P
)
2723, 25, 26syl2anc 665 . . . . . . . . 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 1075 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  -.  Q  .<_  W )
29283ad2ant1 1026 . . . . . . . . . 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 4442 . . . . . . . . . 10  |-  ( ( u  .<_  W  /\  -.  Q  .<_  W )  ->  u  =/=  Q
)
3123, 29, 30syl2anc 665 . . . . . . . . 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 534 . . . . . . . 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 510 . . . . . . 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 984 . . . . . . 7  |-  ( ( u  =/=  P  /\  u  =/=  Q  /\  u  .<_  ( P  .\/  Q
) )  <->  ( (
u  =/=  P  /\  u  =/=  Q )  /\  u  .<_  ( P  .\/  Q ) ) )
3533, 34syl6rbbr 267 . . . . . 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 256 . . . . 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 1207 . . . 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 644 . . 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 2933 . 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 235 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   E.wrex 2772   class class class wbr 4423   ` cfv 5601  (class class class)co 6306   lecple 15197   joincjn 16189   meetcmee 16190   Atomscatm 32799   CvLatclc 32801   HLchlt 32886   LHypclh 33519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6268  df-ov 6309  df-oprab 6310  df-preset 16173  df-poset 16191  df-plt 16204  df-lub 16220  df-glb 16221  df-join 16222  df-meet 16223  df-p0 16285  df-p1 16286  df-lat 16292  df-clat 16354  df-oposet 32712  df-ol 32714  df-oml 32715  df-covers 32802  df-ats 32803  df-atl 32834  df-cvlat 32858  df-hlat 32887  df-lhyp 33523
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator