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Theorem cdleme0ex2N 30706
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 957 . . 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 983 . . 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 1026 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  Q  e.  A )
4 simp3 959 . . 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 30705 . . 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 1189 . 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 1068 . . . . . . . 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 29842 . . . . . . . 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 1024 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  P  e.  A )
17163ad2ant1 978 . . . . . . 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 978 . . . . . . 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 958 . . . . . . 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 989 . . . . . . 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 29826 . . . . . . 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 1197 . . . . . 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 959 . . . . . . . . . 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 1025 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  -.  P  .<_  W )
25243ad2ant1 978 . . . . . . . . . 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 4190 . . . . . . . . . 10  |-  ( ( u  .<_  W  /\  -.  P  .<_  W )  ->  u  =/=  P
)
2723, 25, 26syl2anc 643 . . . . . . . . 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 1027 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  -.  Q  .<_  W )
29283ad2ant1 978 . . . . . . . . . 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 4190 . . . . . . . . . 10  |-  ( ( u  .<_  W  /\  -.  Q  .<_  W )  ->  u  =/=  Q
)
3123, 29, 30syl2anc 643 . . . . . . . . 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 519 . . . . . . . 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 495 . . . . . . 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 938 . . . . . . 7  |-  ( ( u  =/=  P  /\  u  =/=  Q  /\  u  .<_  ( P  .\/  Q
) )  <->  ( (
u  =/=  P  /\  u  =/=  Q )  /\  u  .<_  ( P  .\/  Q ) ) )
3533, 34syl6rbbr 256 . . . . . 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 245 . . . . 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 1155 . . . 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 622 . . 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 2683 . 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 224 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   CvLatclc 29748   HLchlt 29833   LHypclh 30466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-lhyp 30470
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