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Theorem dalawlem13 30377
Description: Lemma for dalaw 30380. Special case to eliminate the requirement  ( ( P  .\/  Q )  .\/  R )  e.  O in dalawlem1 30365. (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 987 . 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 988 . . 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 991 . . . . . . 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 992 . . . . . . 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 990 . . . . . . 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 30042 . . . . . . 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 1186 . . . . . 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 2577 . . . . . . . 8  |-  ( Q  =/=  R  <->  -.  Q  =  R )
1312anbi1i 677 . . . . . . 7  |-  ( ( Q  =/=  R  /\  -.  P  .<_  ( Q 
.\/  R ) )  <-> 
( -.  Q  =  R  /\  -.  P  .<_  ( Q  .\/  R
) ) )
14 pm4.56 482 . . . . . . 7  |-  ( ( -.  Q  =  R  /\  -.  P  .<_  ( Q  .\/  R ) )  <->  -.  ( Q  =  R  \/  P  .<_  ( Q  .\/  R
) ) )
1513, 14bitri 241 . . . . . 6  |-  ( ( Q  =/=  R  /\  -.  P  .<_  ( Q 
.\/  R ) )  <->  -.  ( Q  =  R  \/  P  .<_  ( Q 
.\/  R ) ) )
1611, 15syl6rbb 254 . . . . 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 29867 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  P  e.  A
) )  ->  (
( Q  .\/  R
)  .\/  P )  =  ( ( P 
.\/  Q )  .\/  R ) )
181, 3, 4, 5, 17syl13anc 1186 . . . . . 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 2478 . . . . 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 245 . . . 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 321 . . 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 202 . 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 989 . 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 958 . 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 959 . 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 30376 . . . . . . 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 1156 . . . . . 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 1152 . . . . 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 30375 . . . . . . 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 1156 . . . . . 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 1152 . . . . 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 370 . . . 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 1147 . . 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 1151 . 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 1198 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   lecple 13499   joincjn 14364   meetcmee 14365   Atomscatm 29758   HLchlt 29845   LPlanesclpl 29986
This theorem is referenced by:  dalawlem14  30378
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-lat 14438  df-clat 14500  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-llines 29992  df-lplanes 29993  df-psubsp 29997  df-pmap 29998  df-padd 30290
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