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Theorem dalem59 30542
Description: Lemma for dath 30547. Analog of dalem57 30540 for  F. (Contributed by NM, 10-Aug-2012.)
Hypotheses
Ref Expression
dalem.ph  |-  ( ph  <->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) ) )
dalem.l  |-  .<_  =  ( le `  K )
dalem.j  |-  .\/  =  ( join `  K )
dalem.a  |-  A  =  ( Atoms `  K )
dalem.ps  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
dalem59.m  |-  ./\  =  ( meet `  K )
dalem59.o  |-  O  =  ( LPlanes `  K )
dalem59.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem59.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem59.f  |-  F  =  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )
dalem59.g  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
dalem59.h  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
dalem59.i  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
dalem59.b1  |-  B  =  ( ( ( G 
.\/  H )  .\/  I )  ./\  Y
)
Assertion
Ref Expression
dalem59  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  F  .<_  B )

Proof of Theorem dalem59
StepHypRef Expression
1 dalem.ph . . . . 5  |-  ( ph  <->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) ) )
2 dalem.l . . . . 5  |-  .<_  =  ( le `  K )
3 dalem.j . . . . 5  |-  .\/  =  ( join `  K )
4 dalem.a . . . . 5  |-  A  =  ( Atoms `  K )
5 dalem59.y . . . . 5  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
6 dalem59.z . . . . 5  |-  Z  =  ( ( S  .\/  T )  .\/  U )
71, 2, 3, 4, 5, 6dalemrot 30468 . . . 4  |-  ( ph  ->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( Q  e.  A  /\  R  e.  A  /\  P  e.  A )  /\  ( T  e.  A  /\  U  e.  A  /\  S  e.  A
) )  /\  (
( ( Q  .\/  R )  .\/  P )  e.  O  /\  (
( T  .\/  U
)  .\/  S )  e.  O )  /\  (
( -.  C  .<_  ( Q  .\/  R )  /\  -.  C  .<_  ( R  .\/  P )  /\  -.  C  .<_  ( P  .\/  Q ) )  /\  ( -.  C  .<_  ( T  .\/  U )  /\  -.  C  .<_  ( U  .\/  S )  /\  -.  C  .<_  ( S  .\/  T
) )  /\  ( C  .<_  ( Q  .\/  T )  /\  C  .<_  ( R  .\/  U )  /\  C  .<_  ( P 
.\/  S ) ) ) ) )
873ad2ant1 976 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( Q  e.  A  /\  R  e.  A  /\  P  e.  A )  /\  ( T  e.  A  /\  U  e.  A  /\  S  e.  A
) )  /\  (
( ( Q  .\/  R )  .\/  P )  e.  O  /\  (
( T  .\/  U
)  .\/  S )  e.  O )  /\  (
( -.  C  .<_  ( Q  .\/  R )  /\  -.  C  .<_  ( R  .\/  P )  /\  -.  C  .<_  ( P  .\/  Q ) )  /\  ( -.  C  .<_  ( T  .\/  U )  /\  -.  C  .<_  ( U  .\/  S )  /\  -.  C  .<_  ( S  .\/  T
) )  /\  ( C  .<_  ( Q  .\/  T )  /\  C  .<_  ( R  .\/  U )  /\  C  .<_  ( P 
.\/  S ) ) ) ) )
91, 2, 3, 4, 5, 6dalemrotyz 30469 . . . 4  |-  ( (
ph  /\  Y  =  Z )  ->  (
( Q  .\/  R
)  .\/  P )  =  ( ( T 
.\/  U )  .\/  S ) )
1093adant3 975 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( Q  .\/  R )  .\/  P )  =  ( ( T 
.\/  U )  .\/  S ) )
11 dalem.ps . . . . 5  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
121, 2, 3, 4, 11, 5dalemrotps 30502 . . . 4  |-  ( (
ph  /\  ps )  ->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  ( ( Q 
.\/  R )  .\/  P )  /\  ( d  =/=  c  /\  -.  d  .<_  ( ( Q 
.\/  R )  .\/  P )  /\  C  .<_  ( c  .\/  d ) ) ) )
13123adant2 974 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  ( ( Q 
.\/  R )  .\/  P )  /\  ( d  =/=  c  /\  -.  d  .<_  ( ( Q 
.\/  R )  .\/  P )  /\  C  .<_  ( c  .\/  d ) ) ) )
14 biid 227 . . . 4  |-  ( ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( Q  e.  A  /\  R  e.  A  /\  P  e.  A )  /\  ( T  e.  A  /\  U  e.  A  /\  S  e.  A
) )  /\  (
( ( Q  .\/  R )  .\/  P )  e.  O  /\  (
( T  .\/  U
)  .\/  S )  e.  O )  /\  (
( -.  C  .<_  ( Q  .\/  R )  /\  -.  C  .<_  ( R  .\/  P )  /\  -.  C  .<_  ( P  .\/  Q ) )  /\  ( -.  C  .<_  ( T  .\/  U )  /\  -.  C  .<_  ( U  .\/  S )  /\  -.  C  .<_  ( S  .\/  T
) )  /\  ( C  .<_  ( Q  .\/  T )  /\  C  .<_  ( R  .\/  U )  /\  C  .<_  ( P 
.\/  S ) ) ) )  <->  ( (
( K  e.  HL  /\  C  e.  ( Base `  K ) )  /\  ( Q  e.  A  /\  R  e.  A  /\  P  e.  A
)  /\  ( T  e.  A  /\  U  e.  A  /\  S  e.  A ) )  /\  ( ( ( Q 
.\/  R )  .\/  P )  e.  O  /\  ( ( T  .\/  U )  .\/  S )  e.  O )  /\  ( ( -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
)  /\  -.  C  .<_  ( P  .\/  Q
) )  /\  ( -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S )  /\  -.  C  .<_  ( S 
.\/  T ) )  /\  ( C  .<_  ( Q  .\/  T )  /\  C  .<_  ( R 
.\/  U )  /\  C  .<_  ( P  .\/  S ) ) ) ) )
15 biid 227 . . . 4  |-  ( ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  ( ( Q  .\/  R )  .\/  P )  /\  ( d  =/=  c  /\  -.  d  .<_  ( ( Q  .\/  R )  .\/  P )  /\  C  .<_  ( c 
.\/  d ) ) )  <->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  ( ( Q 
.\/  R )  .\/  P )  /\  ( d  =/=  c  /\  -.  d  .<_  ( ( Q 
.\/  R )  .\/  P )  /\  C  .<_  ( c  .\/  d ) ) ) )
16 dalem59.m . . . 4  |-  ./\  =  ( meet `  K )
17 dalem59.o . . . 4  |-  O  =  ( LPlanes `  K )
18 eqid 2296 . . . 4  |-  ( ( Q  .\/  R ) 
.\/  P )  =  ( ( Q  .\/  R )  .\/  P )
19 eqid 2296 . . . 4  |-  ( ( T  .\/  U ) 
.\/  S )  =  ( ( T  .\/  U )  .\/  S )
20 dalem59.f . . . 4  |-  F  =  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )
21 dalem59.h . . . 4  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
22 dalem59.i . . . 4  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
23 dalem59.g . . . 4  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
24 eqid 2296 . . . 4  |-  ( ( ( H  .\/  I
)  .\/  G )  ./\  ( ( Q  .\/  R )  .\/  P ) )  =  ( ( ( H  .\/  I
)  .\/  G )  ./\  ( ( Q  .\/  R )  .\/  P ) )
2514, 2, 3, 4, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24dalem58 30541 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( Q  e.  A  /\  R  e.  A  /\  P  e.  A )  /\  ( T  e.  A  /\  U  e.  A  /\  S  e.  A
) )  /\  (
( ( Q  .\/  R )  .\/  P )  e.  O  /\  (
( T  .\/  U
)  .\/  S )  e.  O )  /\  (
( -.  C  .<_  ( Q  .\/  R )  /\  -.  C  .<_  ( R  .\/  P )  /\  -.  C  .<_  ( P  .\/  Q ) )  /\  ( -.  C  .<_  ( T  .\/  U )  /\  -.  C  .<_  ( U  .\/  S )  /\  -.  C  .<_  ( S  .\/  T
) )  /\  ( C  .<_  ( Q  .\/  T )  /\  C  .<_  ( R  .\/  U )  /\  C  .<_  ( P 
.\/  S ) ) ) )  /\  (
( Q  .\/  R
)  .\/  P )  =  ( ( T 
.\/  U )  .\/  S )  /\  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  ( ( Q  .\/  R ) 
.\/  P )  /\  ( d  =/=  c  /\  -.  d  .<_  ( ( Q  .\/  R ) 
.\/  P )  /\  C  .<_  ( c  .\/  d ) ) ) )  ->  F  .<_  ( ( ( H  .\/  I )  .\/  G
)  ./\  ( ( Q  .\/  R )  .\/  P ) ) )
268, 10, 13, 25syl3anc 1182 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  F  .<_  ( ( ( H  .\/  I ) 
.\/  G )  ./\  ( ( Q  .\/  R )  .\/  P ) ) )
271dalemkehl 30434 . . . . . 6  |-  ( ph  ->  K  e.  HL )
28273ad2ant1 976 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
291, 2, 3, 4, 11, 16, 17, 5, 6, 21dalem29 30512 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  A )
301, 2, 3, 4, 11, 16, 17, 5, 6, 22dalem34 30517 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  A )
311, 2, 3, 4, 11, 16, 17, 5, 6, 23dalem23 30507 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
323, 4hlatjrot 30184 . . . . 5  |-  ( ( K  e.  HL  /\  ( H  e.  A  /\  I  e.  A  /\  G  e.  A
) )  ->  (
( H  .\/  I
)  .\/  G )  =  ( ( G 
.\/  H )  .\/  I ) )
3328, 29, 30, 31, 32syl13anc 1184 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( H  .\/  I )  .\/  G
)  =  ( ( G  .\/  H ) 
.\/  I ) )
341, 3, 4dalemqrprot 30459 . . . . . 6  |-  ( ph  ->  ( ( Q  .\/  R )  .\/  P )  =  ( ( P 
.\/  Q )  .\/  R ) )
3534, 5syl6eqr 2346 . . . . 5  |-  ( ph  ->  ( ( Q  .\/  R )  .\/  P )  =  Y )
36353ad2ant1 976 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( Q  .\/  R )  .\/  P )  =  Y )
3733, 36oveq12d 5892 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( H 
.\/  I )  .\/  G )  ./\  ( ( Q  .\/  R )  .\/  P ) )  =  ( ( ( G  .\/  H )  .\/  I ) 
./\  Y ) )
38 dalem59.b1 . . 3  |-  B  =  ( ( ( G 
.\/  H )  .\/  I )  ./\  Y
)
3937, 38syl6eqr 2346 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( H 
.\/  I )  .\/  G )  ./\  ( ( Q  .\/  R )  .\/  P ) )  =  B )
4026, 39breqtrd 4063 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  F  .<_  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Atomscatm 30075   HLchlt 30162   LPlanesclpl 30303
This theorem is referenced by:  dalem61  30544
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311
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