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Theorem dalawlem13 30124
Description: Lemma for dalaw 30127. Special case to eliminate the requirement  ( ( P  .\/  Q )  .\/  R )  e.  O in dalawlem1 30112. (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 985 . 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 986 . . 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 989 . . . . . . 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 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 )
)  ->  R  e.  A )
5 simp21 988 . . . . . . 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 29789 . . . . . . 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 1184 . . . . . 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 2523 . . . . . . . 8  |-  ( Q  =/=  R  <->  -.  Q  =  R )
1312anbi1i 676 . . . . . . 7  |-  ( ( Q  =/=  R  /\  -.  P  .<_  ( Q 
.\/  R ) )  <-> 
( -.  Q  =  R  /\  -.  P  .<_  ( Q  .\/  R
) ) )
14 pm4.56 481 . . . . . . 7  |-  ( ( -.  Q  =  R  /\  -.  P  .<_  ( Q  .\/  R ) )  <->  -.  ( Q  =  R  \/  P  .<_  ( Q  .\/  R
) ) )
1513, 14bitri 240 . . . . . 6  |-  ( ( Q  =/=  R  /\  -.  P  .<_  ( Q 
.\/  R ) )  <->  -.  ( Q  =  R  \/  P  .<_  ( Q 
.\/  R ) ) )
1611, 15syl6rbb 253 . . . . 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 29614 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  P  e.  A
) )  ->  (
( Q  .\/  R
)  .\/  P )  =  ( ( P 
.\/  Q )  .\/  R ) )
181, 3, 4, 5, 17syl13anc 1184 . . . . . 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 2424 . . . . 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 244 . . . 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 320 . . 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 201 . 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 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 )
)  ->  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )
24 simp2 956 . 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 957 . 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 30123 . . . . . . 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 1154 . . . . . 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 1150 . . . . 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 30122 . . . . . . 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 1154 . . . . . 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 1150 . . . . 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 369 . . . 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 1145 . . 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 1149 . 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 1196 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   class class class wbr 4102   ` cfv 5334  (class class class)co 5942   lecple 13306   joincjn 14171   meetcmee 14172   Atomscatm 29505   HLchlt 29592   LPlanesclpl 29733
This theorem is referenced by:  dalawlem14  30125
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-iin 3987  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-undef 6382  df-riota 6388  df-poset 14173  df-plt 14185  df-lub 14201  df-glb 14202  df-join 14203  df-meet 14204  df-p0 14238  df-lat 14245  df-clat 14307  df-oposet 29418  df-ol 29420  df-oml 29421  df-covers 29508  df-ats 29509  df-atl 29540  df-cvlat 29564  df-hlat 29593  df-llines 29739  df-lplanes 29740  df-psubsp 29744  df-pmap 29745  df-padd 30037
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