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Theorem dalem4 30159
Description: Lemma for dalemdnee 30160. (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 30150 . . . 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 452 . . 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 30118 . . . . . . 7  |-  ( ph  ->  K  e.  Lat )
91, 3, 4dalempjqeb 30139 . . . . . . 7  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
101, 3, 4dalemsjteb 30140 . . . . . . 7  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
11 eqid 2412 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
12 dalem3.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
1311, 12latmcom 14467 . . . . . . 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 1184 . . . . . 6  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  =  ( ( S  .\/  T )  ./\  ( P  .\/  Q ) ) )
157, 14syl5eq 2456 . . . . 5  |-  ( ph  ->  D  =  ( ( S  .\/  T ) 
./\  ( P  .\/  Q ) ) )
1615neeq1d 2588 . . . 4  |-  ( ph  ->  ( D  =/=  T  <->  ( ( S  .\/  T
)  ./\  ( P  .\/  Q ) )  =/= 
T ) )
1716biimpa 471 . . 3  |-  ( (
ph  /\  D  =/=  T )  ->  ( ( S  .\/  T )  ./\  ( P  .\/  Q ) )  =/=  T )
18 biid 228 . . . 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 2412 . . . 4  |-  ( ( S  .\/  T ) 
./\  ( P  .\/  Q ) )  =  ( ( S  .\/  T
)  ./\  ( P  .\/  Q ) )
23 eqid 2412 . . . 4  |-  ( ( T  .\/  U ) 
./\  ( Q  .\/  R ) )  =  ( ( T  .\/  U
)  ./\  ( Q  .\/  R ) )
2418, 2, 3, 4, 12, 19, 20, 21, 22, 23dalem3 30158 . . 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 643 . 2  |-  ( (
ph  /\  D  =/=  T )  ->  ( ( S  .\/  T )  ./\  ( P  .\/  Q ) )  =/=  ( ( T  .\/  U ) 
./\  ( Q  .\/  R ) ) )
2615adantr 452 . 2  |-  ( (
ph  /\  D  =/=  T )  ->  D  =  ( ( S  .\/  T )  ./\  ( P  .\/  Q ) ) )
27 dalem3.e . . . 4  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
281dalemkehl 30117 . . . . . 6  |-  ( ph  ->  K  e.  HL )
291dalemqea 30121 . . . . . 6  |-  ( ph  ->  Q  e.  A )
301dalemrea 30122 . . . . . 6  |-  ( ph  ->  R  e.  A )
3111, 3, 4hlatjcl 29861 . . . . . 6  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
3228, 29, 30, 31syl3anc 1184 . . . . 5  |-  ( ph  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
331, 3, 4dalemtjueb 30141 . . . . 5  |-  ( ph  ->  ( T  .\/  U
)  e.  ( Base `  K ) )
3411, 12latmcom 14467 . . . . 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 1184 . . . 4  |-  ( ph  ->  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  =  ( ( T  .\/  U )  ./\  ( Q  .\/  R ) ) )
3627, 35syl5eq 2456 . . 3  |-  ( ph  ->  E  =  ( ( T  .\/  U ) 
./\  ( Q  .\/  R ) ) )
3736adantr 452 . 2  |-  ( (
ph  /\  D  =/=  T )  ->  E  =  ( ( T  .\/  U )  ./\  ( Q  .\/  R ) ) )
3825, 26, 373netr4d 2602 1  |-  ( (
ph  /\  D  =/=  T )  ->  D  =/=  E )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   Basecbs 13432   lecple 13499   joincjn 14364   meetcmee 14365   Latclat 14437   Atomscatm 29758   HLchlt 29845   LPlanesclpl 29986
This theorem is referenced by:  dalemdnee  30160
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-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
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