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Theorem dalem58 30601
Description: Lemma for dath 30607. Analog of dalem57 30600 for  E. (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
) ) ) )
dalem58.m  |-  ./\  =  ( meet `  K )
dalem58.o  |-  O  =  ( LPlanes `  K )
dalem58.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem58.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem58.e  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
dalem58.g  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
dalem58.h  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
dalem58.i  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
dalem58.b1  |-  B  =  ( ( ( G 
.\/  H )  .\/  I )  ./\  Y
)
Assertion
Ref Expression
dalem58  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  E  .<_  B )

Proof of Theorem dalem58
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 dalem58.y . . . . 5  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
6 dalem58.z . . . . 5  |-  Z  =  ( ( S  .\/  T )  .\/  U )
71, 2, 3, 4, 5, 6dalemrot 30528 . . . 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 979 . . 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 30529 . . . 4  |-  ( (
ph  /\  Y  =  Z )  ->  (
( Q  .\/  R
)  .\/  P )  =  ( ( T 
.\/  U )  .\/  S ) )
1093adant3 978 . . 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 30562 . . . 4  |-  ( (
ph  /\  ps )  ->  ( ( c  e.  A  /\  d  e.  A )  /\  -.  c  .<_  ( ( Q 
.\/  R )  .\/  P )  /\  ( d  =/=  c  /\  -.  d  .<_  ( ( Q 
.\/  R )  .\/  P )  /\  C  .<_  ( c  .\/  d ) ) ) )
13123adant2 977 . . 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 229 . . . 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 229 . . . 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 dalem58.m . . . 4  |-  ./\  =  ( meet `  K )
17 dalem58.o . . . 4  |-  O  =  ( LPlanes `  K )
18 eqid 2438 . . . 4  |-  ( ( Q  .\/  R ) 
.\/  P )  =  ( ( Q  .\/  R )  .\/  P )
19 eqid 2438 . . . 4  |-  ( ( T  .\/  U ) 
.\/  S )  =  ( ( T  .\/  U )  .\/  S )
20 dalem58.e . . . 4  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
21 dalem58.h . . . 4  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
22 dalem58.i . . . 4  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
23 dalem58.g . . . 4  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
24 eqid 2438 . . . 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, 24dalem57 30600 . . 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 ) ) ) )  ->  E  .<_  ( ( ( H  .\/  I )  .\/  G
)  ./\  ( ( Q  .\/  R )  .\/  P ) ) )
268, 10, 13, 25syl3anc 1185 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  E  .<_  ( ( ( H  .\/  I ) 
.\/  G )  ./\  ( ( Q  .\/  R )  .\/  P ) ) )
271dalemkehl 30494 . . . . . 6  |-  ( ph  ->  K  e.  HL )
28273ad2ant1 979 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
291, 2, 3, 4, 11, 16, 17, 5, 6, 21dalem29 30572 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  A )
301, 2, 3, 4, 11, 16, 17, 5, 6, 22dalem34 30577 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  A )
311, 2, 3, 4, 11, 16, 17, 5, 6, 23dalem23 30567 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
323, 4hlatjrot 30244 . . . . 5  |-  ( ( K  e.  HL  /\  ( H  e.  A  /\  I  e.  A  /\  G  e.  A
) )  ->  (
( H  .\/  I
)  .\/  G )  =  ( ( G 
.\/  H )  .\/  I ) )
3328, 29, 30, 31, 32syl13anc 1187 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( H  .\/  I )  .\/  G
)  =  ( ( G  .\/  H ) 
.\/  I ) )
341, 3, 4dalemqrprot 30519 . . . . . 6  |-  ( ph  ->  ( ( Q  .\/  R )  .\/  P )  =  ( ( P 
.\/  Q )  .\/  R ) )
3534, 5syl6eqr 2488 . . . . 5  |-  ( ph  ->  ( ( Q  .\/  R )  .\/  P )  =  Y )
36353ad2ant1 979 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( Q  .\/  R )  .\/  P )  =  Y )
3733, 36oveq12d 6102 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( H 
.\/  I )  .\/  G )  ./\  ( ( Q  .\/  R )  .\/  P ) )  =  ( ( ( G  .\/  H )  .\/  I ) 
./\  Y ) )
38 dalem58.b1 . . 3  |-  B  =  ( ( ( G 
.\/  H )  .\/  I )  ./\  Y
)
3937, 38syl6eqr 2488 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( H 
.\/  I )  .\/  G )  ./\  ( ( Q  .\/  R )  .\/  P ) )  =  B )
4026, 39breqtrd 4239 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  E  .<_  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   joincjn 14406   meetcmee 14407   Atomscatm 30135   HLchlt 30222   LPlanesclpl 30363
This theorem is referenced by:  dalem59  30602  dalem60  30603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-llines 30369  df-lplanes 30370  df-lvols 30371
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