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Theorem dalem4 30536
Description: Lemma for dalemdnee 30537. (Contributed by NM, 10-Aug-2012.)
Hypotheses
Ref Expression
dalema.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 ) ) ) ) )
dalemc.l  |-  .<_  =  ( le `  K )
dalemc.j  |-  .\/  =  ( join `  K )
dalemc.a  |-  A  =  ( Atoms `  K )
dalem3.m  |-  ./\  =  ( meet `  K )
dalem3.o  |-  O  =  ( LPlanes `  K )
dalem3.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem3.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem3.d  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
dalem3.e  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
Assertion
Ref Expression
dalem4  |-  ( (
ph  /\  D  =/=  T )  ->  D  =/=  E )

Proof of Theorem dalem4
StepHypRef Expression
1 dalema.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 dalemc.l . . . . 5  |-  .<_  =  ( le `  K )
3 dalemc.j . . . . 5  |-  .\/  =  ( join `  K )
4 dalemc.a . . . . 5  |-  A  =  ( Atoms `  K )
51, 2, 3, 4dalemswapyz 30527 . . . 4  |-  ( ph  ->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  ( Z  e.  O  /\  Y  e.  O )  /\  ( ( -.  C  .<_  ( S  .\/  T
)  /\  -.  C  .<_  ( T  .\/  U
)  /\  -.  C  .<_  ( U  .\/  S
) )  /\  ( -.  C  .<_  ( P 
.\/  Q )  /\  -.  C  .<_  ( Q 
.\/  R )  /\  -.  C  .<_  ( R 
.\/  P ) )  /\  ( C  .<_  ( S  .\/  P )  /\  C  .<_  ( T 
.\/  Q )  /\  C  .<_  ( U  .\/  R ) ) ) ) )
65adantr 453 . . 3  |-  ( (
ph  /\  D  =/=  T )  ->  ( (
( K  e.  HL  /\  C  e.  ( Base `  K ) )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
)  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  /\  ( Z  e.  O  /\  Y  e.  O
)  /\  ( ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( C  .<_  ( S  .\/  P )  /\  C  .<_  ( T  .\/  Q )  /\  C  .<_  ( U 
.\/  R ) ) ) ) )
7 dalem3.d . . . . . 6  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
81dalemkelat 30495 . . . . . . 7  |-  ( ph  ->  K  e.  Lat )
91, 3, 4dalempjqeb 30516 . . . . . . 7  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
101, 3, 4dalemsjteb 30517 . . . . . . 7  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
11 eqid 2438 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
12 dalem3.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
1311, 12latmcom 14509 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  =  ( ( S  .\/  T ) 
./\  ( P  .\/  Q ) ) )
148, 9, 10, 13syl3anc 1185 . . . . . 6  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  =  ( ( S  .\/  T )  ./\  ( P  .\/  Q ) ) )
157, 14syl5eq 2482 . . . . 5  |-  ( ph  ->  D  =  ( ( S  .\/  T ) 
./\  ( P  .\/  Q ) ) )
1615neeq1d 2616 . . . 4  |-  ( ph  ->  ( D  =/=  T  <->  ( ( S  .\/  T
)  ./\  ( P  .\/  Q ) )  =/= 
T ) )
1716biimpa 472 . . 3  |-  ( (
ph  /\  D  =/=  T )  ->  ( ( S  .\/  T )  ./\  ( P  .\/  Q ) )  =/=  T )
18 biid 229 . . . 4  |-  ( ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  ( Z  e.  O  /\  Y  e.  O )  /\  ( ( -.  C  .<_  ( S  .\/  T
)  /\  -.  C  .<_  ( T  .\/  U
)  /\  -.  C  .<_  ( U  .\/  S
) )  /\  ( -.  C  .<_  ( P 
.\/  Q )  /\  -.  C  .<_  ( Q 
.\/  R )  /\  -.  C  .<_  ( R 
.\/  P ) )  /\  ( C  .<_  ( S  .\/  P )  /\  C  .<_  ( T 
.\/  Q )  /\  C  .<_  ( U  .\/  R ) ) ) )  <-> 
( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  ( Z  e.  O  /\  Y  e.  O )  /\  ( ( -.  C  .<_  ( S  .\/  T
)  /\  -.  C  .<_  ( T  .\/  U
)  /\  -.  C  .<_  ( U  .\/  S
) )  /\  ( -.  C  .<_  ( P 
.\/  Q )  /\  -.  C  .<_  ( Q 
.\/  R )  /\  -.  C  .<_  ( R 
.\/  P ) )  /\  ( C  .<_  ( S  .\/  P )  /\  C  .<_  ( T 
.\/  Q )  /\  C  .<_  ( U  .\/  R ) ) ) ) )
19 dalem3.o . . . 4  |-  O  =  ( LPlanes `  K )
20 dalem3.z . . . 4  |-  Z  =  ( ( S  .\/  T )  .\/  U )
21 dalem3.y . . . 4  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
22 eqid 2438 . . . 4  |-  ( ( S  .\/  T ) 
./\  ( P  .\/  Q ) )  =  ( ( S  .\/  T
)  ./\  ( P  .\/  Q ) )
23 eqid 2438 . . . 4  |-  ( ( T  .\/  U ) 
./\  ( Q  .\/  R ) )  =  ( ( T  .\/  U
)  ./\  ( Q  .\/  R ) )
2418, 2, 3, 4, 12, 19, 20, 21, 22, 23dalem3 30535 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  ( Z  e.  O  /\  Y  e.  O )  /\  ( ( -.  C  .<_  ( S  .\/  T
)  /\  -.  C  .<_  ( T  .\/  U
)  /\  -.  C  .<_  ( U  .\/  S
) )  /\  ( -.  C  .<_  ( P 
.\/  Q )  /\  -.  C  .<_  ( Q 
.\/  R )  /\  -.  C  .<_  ( R 
.\/  P ) )  /\  ( C  .<_  ( S  .\/  P )  /\  C  .<_  ( T 
.\/  Q )  /\  C  .<_  ( U  .\/  R ) ) ) )  /\  ( ( S 
.\/  T )  ./\  ( P  .\/  Q ) )  =/=  T )  ->  ( ( S 
.\/  T )  ./\  ( P  .\/  Q ) )  =/=  ( ( T  .\/  U ) 
./\  ( Q  .\/  R ) ) )
256, 17, 24syl2anc 644 . 2  |-  ( (
ph  /\  D  =/=  T )  ->  ( ( S  .\/  T )  ./\  ( P  .\/  Q ) )  =/=  ( ( T  .\/  U ) 
./\  ( Q  .\/  R ) ) )
2615adantr 453 . 2  |-  ( (
ph  /\  D  =/=  T )  ->  D  =  ( ( S  .\/  T )  ./\  ( P  .\/  Q ) ) )
27 dalem3.e . . . 4  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
281dalemkehl 30494 . . . . . 6  |-  ( ph  ->  K  e.  HL )
291dalemqea 30498 . . . . . 6  |-  ( ph  ->  Q  e.  A )
301dalemrea 30499 . . . . . 6  |-  ( ph  ->  R  e.  A )
3111, 3, 4hlatjcl 30238 . . . . . 6  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
3228, 29, 30, 31syl3anc 1185 . . . . 5  |-  ( ph  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
331, 3, 4dalemtjueb 30518 . . . . 5  |-  ( ph  ->  ( T  .\/  U
)  e.  ( Base `  K ) )
3411, 12latmcom 14509 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  ( T  .\/  U )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  =  ( ( T  .\/  U ) 
./\  ( Q  .\/  R ) ) )
358, 32, 33, 34syl3anc 1185 . . . 4  |-  ( ph  ->  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  =  ( ( T  .\/  U )  ./\  ( Q  .\/  R ) ) )
3627, 35syl5eq 2482 . . 3  |-  ( ph  ->  E  =  ( ( T  .\/  U ) 
./\  ( Q  .\/  R ) ) )
3736adantr 453 . 2  |-  ( (
ph  /\  D  =/=  T )  ->  E  =  ( ( T  .\/  U )  ./\  ( Q  .\/  R ) ) )
3825, 26, 373netr4d 2630 1  |-  ( (
ph  /\  D  =/=  T )  ->  D  =/=  E )
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   Latclat 14479   Atomscatm 30135   HLchlt 30222   LPlanesclpl 30363
This theorem is referenced by:  dalemdnee  30537
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-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
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