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Theorem cdleme0ex2N 29543
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 960 . . 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 986 . . 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 1029 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  Q  e.  A )
4 simp3 962 . . 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 29542 . . 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 1192 . 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 1071 . . . . . . . 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 28679 . . . . . . . 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 1027 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  P  e.  A )
17163ad2ant1 981 . . . . . . 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 981 . . . . . . 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 961 . . . . . . 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 992 . . . . . . 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 28663 . . . . . . 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 1200 . . . . . 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 962 . . . . . . . . . 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 1028 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  -.  P  .<_  W )
25243ad2ant1 981 . . . . . . . . . 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 3981 . . . . . . . . . 10  |-  ( ( u  .<_  W  /\  -.  P  .<_  W )  ->  u  =/=  P
)
2723, 25, 26syl2anc 645 . . . . . . . . 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 1030 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  -.  Q  .<_  W )
29283ad2ant1 981 . . . . . . . . . 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 3981 . . . . . . . . . 10  |-  ( ( u  .<_  W  /\  -.  Q  .<_  W )  ->  u  =/=  Q
)
3123, 29, 30syl2anc 645 . . . . . . . . 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 941 . . . . . . 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 1158 . . . 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 624 . . 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 2531 . 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 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   E.wrex 2517   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   lecple 13142   joincjn 14005   meetcmee 14006   Atomscatm 28583   CvLatclc 28585   HLchlt 28670   LHypclh 29303
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-glb 14036  df-join 14037  df-meet 14038  df-p0 14072  df-p1 14073  df-lat 14079  df-clat 14141  df-oposet 28496  df-ol 28498  df-oml 28499  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-lhyp 29307
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