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Theorem dalawlem13 35085
Description: Lemma for dalaw 35088. Special case to eliminate the requirement  ( ( P  .\/  Q )  .\/  R )  e.  O in dalawlem1 35073. (Contributed by NM, 6-Oct-2012.)
Hypotheses
Ref Expression
dalawlem.l  |-  .<_  =  ( le `  K )
dalawlem.j  |-  .\/  =  ( join `  K )
dalawlem.m  |-  ./\  =  ( meet `  K )
dalawlem.a  |-  A  =  ( Atoms `  K )
dalawlem2.o  |-  O  =  ( LPlanes `  K )
Assertion
Ref Expression
dalawlem13  |-  ( ( ( K  e.  HL  /\ 
-.  ( ( P 
.\/  Q )  .\/  R )  e.  O  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( (
( Q  .\/  R
)  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )

Proof of Theorem dalawlem13
StepHypRef Expression
1 simp11 1026 . 2  |-  ( ( ( K  e.  HL  /\ 
-.  ( ( P 
.\/  Q )  .\/  R )  e.  O  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  K  e.  HL )
2 simp12 1027 . . 3  |-  ( ( ( K  e.  HL  /\ 
-.  ( ( P 
.\/  Q )  .\/  R )  e.  O  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  -.  (
( P  .\/  Q
)  .\/  R )  e.  O )
3 simp22 1030 . . . . . . 7  |-  ( ( ( K  e.  HL  /\ 
-.  ( ( P 
.\/  Q )  .\/  R )  e.  O  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  Q  e.  A )
4 simp23 1031 . . . . . . 7  |-  ( ( ( K  e.  HL  /\ 
-.  ( ( P 
.\/  Q )  .\/  R )  e.  O  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  R  e.  A )
5 simp21 1029 . . . . . . 7  |-  ( ( ( K  e.  HL  /\ 
-.  ( ( P 
.\/  Q )  .\/  R )  e.  O  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  P  e.  A )
6 dalawlem.l . . . . . . . 8  |-  .<_  =  ( le `  K )
7 dalawlem.j . . . . . . . 8  |-  .\/  =  ( join `  K )
8 dalawlem.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
9 dalawlem2.o . . . . . . . 8  |-  O  =  ( LPlanes `  K )
106, 7, 8, 9islpln2a 34750 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  P  e.  A
) )  ->  (
( ( Q  .\/  R )  .\/  P )  e.  O  <->  ( Q  =/=  R  /\  -.  P  .<_  ( Q  .\/  R
) ) ) )
111, 3, 4, 5, 10syl13anc 1230 . . . . . 6  |-  ( ( ( K  e.  HL  /\ 
-.  ( ( P 
.\/  Q )  .\/  R )  e.  O  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( (
( Q  .\/  R
)  .\/  P )  e.  O  <->  ( Q  =/= 
R  /\  -.  P  .<_  ( Q  .\/  R
) ) ) )
12 df-ne 2664 . . . . . . . 8  |-  ( Q  =/=  R  <->  -.  Q  =  R )
1312anbi1i 695 . . . . . . 7  |-  ( ( Q  =/=  R  /\  -.  P  .<_  ( Q 
.\/  R ) )  <-> 
( -.  Q  =  R  /\  -.  P  .<_  ( Q  .\/  R
) ) )
14 pm4.56 495 . . . . . . 7  |-  ( ( -.  Q  =  R  /\  -.  P  .<_  ( Q  .\/  R ) )  <->  -.  ( Q  =  R  \/  P  .<_  ( Q  .\/  R
) ) )
1513, 14bitri 249 . . . . . 6  |-  ( ( Q  =/=  R  /\  -.  P  .<_  ( Q 
.\/  R ) )  <->  -.  ( Q  =  R  \/  P  .<_  ( Q 
.\/  R ) ) )
1611, 15syl6rbb 262 . . . . 5  |-  ( ( ( K  e.  HL  /\ 
-.  ( ( P 
.\/  Q )  .\/  R )  e.  O  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( -.  ( Q  =  R  \/  P  .<_  ( Q 
.\/  R ) )  <-> 
( ( Q  .\/  R )  .\/  P )  e.  O ) )
177, 8hlatjrot 34575 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  P  e.  A
) )  ->  (
( Q  .\/  R
)  .\/  P )  =  ( ( P 
.\/  Q )  .\/  R ) )
181, 3, 4, 5, 17syl13anc 1230 . . . . . 6  |-  ( ( ( K  e.  HL  /\ 
-.  ( ( P 
.\/  Q )  .\/  R )  e.  O  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( Q  .\/  R )  .\/  P )  =  ( ( P  .\/  Q ) 
.\/  R ) )
1918eleq1d 2536 . . . . 5  |-  ( ( ( K  e.  HL  /\ 
-.  ( ( P 
.\/  Q )  .\/  R )  e.  O  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( (
( Q  .\/  R
)  .\/  P )  e.  O  <->  ( ( P 
.\/  Q )  .\/  R )  e.  O ) )
2016, 19bitrd 253 . . . 4  |-  ( ( ( K  e.  HL  /\ 
-.  ( ( P 
.\/  Q )  .\/  R )  e.  O  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( -.  ( Q  =  R  \/  P  .<_  ( Q 
.\/  R ) )  <-> 
( ( P  .\/  Q )  .\/  R )  e.  O ) )
2120con1bid 330 . . 3  |-  ( ( ( K  e.  HL  /\ 
-.  ( ( P 
.\/  Q )  .\/  R )  e.  O  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( -.  ( ( P  .\/  Q )  .\/  R )  e.  O  <->  ( Q  =  R  \/  P  .<_  ( Q  .\/  R
) ) ) )
222, 21mpbid 210 . 2  |-  ( ( ( K  e.  HL  /\ 
-.  ( ( P 
.\/  Q )  .\/  R )  e.  O  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( Q  =  R  \/  P  .<_  ( Q  .\/  R
) ) )
23 simp13 1028 . 2  |-  ( ( ( K  e.  HL  /\ 
-.  ( ( P 
.\/  Q )  .\/  R )  e.  O  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )
24 simp2 997 . 2  |-  ( ( ( K  e.  HL  /\ 
-.  ( ( P 
.\/  Q )  .\/  R )  e.  O  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )
25 simp3 998 . 2  |-  ( ( ( K  e.  HL  /\ 
-.  ( ( P 
.\/  Q )  .\/  R )  e.  O  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )
26 dalawlem.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
276, 7, 26, 8dalawlem12 35084 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  Q  =  R  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( (
( Q  .\/  R
)  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )
28273expib 1199 . . . . . 6  |-  ( ( K  e.  HL  /\  Q  =  R  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  ->  ( (
( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  -> 
( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) ) )
29283exp 1195 . . . . 5  |-  ( K  e.  HL  ->  ( Q  =  R  ->  ( ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U )  ->  ( ( ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  ->  (
( P  .\/  Q
)  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) ) ) ) )
306, 7, 26, 8dalawlem11 35083 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( (
( Q  .\/  R
)  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )
31303expib 1199 . . . . . 6  |-  ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  U ) )  ->  ( ( ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  ->  (
( P  .\/  Q
)  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) ) )
32313exp 1195 . . . . 5  |-  ( K  e.  HL  ->  ( P  .<_  ( Q  .\/  R )  ->  ( (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U )  ->  ( ( ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  ->  (
( P  .\/  Q
)  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) ) ) ) )
3329, 32jaod 380 . . . 4  |-  ( K  e.  HL  ->  (
( Q  =  R  \/  P  .<_  ( Q 
.\/  R ) )  ->  ( ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  U )  -> 
( ( ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( (
( Q  .\/  R
)  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) ) ) ) )
34333imp 1190 . . 3  |-  ( ( K  e.  HL  /\  ( Q  =  R  \/  P  .<_  ( Q 
.\/  R ) )  /\  ( ( P 
.\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  -> 
( ( ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( (
( Q  .\/  R
)  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) ) )
35343impib 1194 . 2  |-  ( ( ( K  e.  HL  /\  ( Q  =  R  \/  P  .<_  ( Q 
.\/  R ) )  /\  ( ( P 
.\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  -> 
( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) )
361, 22, 23, 24, 25, 35syl311anc 1242 1  |-  ( ( ( K  e.  HL  /\ 
-.  ( ( P 
.\/  Q )  .\/  R )  e.  O  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( (
( Q  .\/  R
)  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   lecple 14578   joincjn 15447   meetcmee 15448   Atomscatm 34466   HLchlt 34553   LPlanesclpl 34694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-poset 15449  df-plt 15461  df-lub 15477  df-glb 15478  df-join 15479  df-meet 15480  df-p0 15542  df-lat 15549  df-clat 15611  df-oposet 34379  df-ol 34381  df-oml 34382  df-covers 34469  df-ats 34470  df-atl 34501  df-cvlat 34525  df-hlat 34554  df-llines 34700  df-lplanes 34701  df-psubsp 34705  df-pmap 34706  df-padd 34998
This theorem is referenced by:  dalawlem14  35086
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