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Theorem dalem3 29853
Description: Lemma for dalemdnee 29855. (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
dalem3  |-  ( (
ph  /\  D  =/=  Q )  ->  D  =/=  E )

Proof of Theorem dalem3
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 ) ) ) ) )
21dalemkehl 29812 . . . 4  |-  ( ph  ->  K  e.  HL )
31dalempea 29815 . . . 4  |-  ( ph  ->  P  e.  A )
41dalemqea 29816 . . . 4  |-  ( ph  ->  Q  e.  A )
51dalemrea 29817 . . . 4  |-  ( ph  ->  R  e.  A )
61dalemyeo 29821 . . . 4  |-  ( ph  ->  Y  e.  O )
7 dalemc.l . . . . 5  |-  .<_  =  ( le `  K )
8 dalemc.j . . . . 5  |-  .\/  =  ( join `  K )
9 dalemc.a . . . . 5  |-  A  =  ( Atoms `  K )
10 dalem3.o . . . . 5  |-  O  =  ( LPlanes `  K )
11 dalem3.y . . . . 5  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
127, 8, 9, 10, 11lplnric 29741 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  Y  e.  O )  ->  -.  R  .<_  ( P  .\/  Q ) )
132, 3, 4, 5, 6, 12syl131anc 1195 . . 3  |-  ( ph  ->  -.  R  .<_  ( P 
.\/  Q ) )
1413adantr 451 . 2  |-  ( (
ph  /\  D  =/=  Q )  ->  -.  R  .<_  ( P  .\/  Q
) )
15 dalem3.e . . . . . . 7  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
161dalemkelat 29813 . . . . . . . 8  |-  ( ph  ->  K  e.  Lat )
17 eqid 2283 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
1817, 8, 9hlatjcl 29556 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
192, 4, 5, 18syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
201, 8, 9dalemtjueb 29836 . . . . . . . 8  |-  ( ph  ->  ( T  .\/  U
)  e.  ( Base `  K ) )
21 dalem3.m . . . . . . . . 9  |-  ./\  =  ( meet `  K )
2217, 7, 21latmle1 14182 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  ( T  .\/  U )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  .<_  ( Q  .\/  R ) )
2316, 19, 20, 22syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  .<_  ( Q  .\/  R ) )
2415, 23syl5eqbr 4056 . . . . . 6  |-  ( ph  ->  E  .<_  ( Q  .\/  R ) )
25 breq1 4026 . . . . . 6  |-  ( D  =  E  ->  ( D  .<_  ( Q  .\/  R )  <->  E  .<_  ( Q 
.\/  R ) ) )
2624, 25syl5ibrcom 213 . . . . 5  |-  ( ph  ->  ( D  =  E  ->  D  .<_  ( Q 
.\/  R ) ) )
2726adantr 451 . . . 4  |-  ( (
ph  /\  D  =/=  Q )  ->  ( D  =  E  ->  D  .<_  ( Q  .\/  R ) ) )
282adantr 451 . . . . 5  |-  ( (
ph  /\  D  =/=  Q )  ->  K  e.  HL )
29 dalem3.z . . . . . . 7  |-  Z  =  ( ( S  .\/  T )  .\/  U )
30 dalem3.d . . . . . . 7  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
311, 7, 8, 9, 21, 10, 11, 29, 30dalemdea 29851 . . . . . 6  |-  ( ph  ->  D  e.  A )
3231adantr 451 . . . . 5  |-  ( (
ph  /\  D  =/=  Q )  ->  D  e.  A )
335adantr 451 . . . . 5  |-  ( (
ph  /\  D  =/=  Q )  ->  R  e.  A )
344adantr 451 . . . . 5  |-  ( (
ph  /\  D  =/=  Q )  ->  Q  e.  A )
35 simpr 447 . . . . 5  |-  ( (
ph  /\  D  =/=  Q )  ->  D  =/=  Q )
367, 8, 9hlatexch1 29584 . . . . 5  |-  ( ( K  e.  HL  /\  ( D  e.  A  /\  R  e.  A  /\  Q  e.  A
)  /\  D  =/=  Q )  ->  ( D  .<_  ( Q  .\/  R
)  ->  R  .<_  ( Q  .\/  D ) ) )
3728, 32, 33, 34, 35, 36syl131anc 1195 . . . 4  |-  ( (
ph  /\  D  =/=  Q )  ->  ( D  .<_  ( Q  .\/  R
)  ->  R  .<_  ( Q  .\/  D ) ) )
387, 8, 9hlatlej2 29565 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  Q  .<_  ( P  .\/  Q ) )
392, 3, 4, 38syl3anc 1182 . . . . . . 7  |-  ( ph  ->  Q  .<_  ( P  .\/  Q ) )
401, 8, 9dalempjqeb 29834 . . . . . . . . 9  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
411, 8, 9dalemsjteb 29835 . . . . . . . . 9  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
4217, 7, 21latmle1 14182 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( P  .\/  Q ) )
4316, 40, 41, 42syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( P  .\/  Q ) )
4430, 43syl5eqbr 4056 . . . . . . 7  |-  ( ph  ->  D  .<_  ( P  .\/  Q ) )
451, 9dalemqeb 29829 . . . . . . . 8  |-  ( ph  ->  Q  e.  ( Base `  K ) )
4617, 9atbase 29479 . . . . . . . . 9  |-  ( D  e.  A  ->  D  e.  ( Base `  K
) )
4731, 46syl 15 . . . . . . . 8  |-  ( ph  ->  D  e.  ( Base `  K ) )
4817, 7, 8latjle12 14168 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  D  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
) ) )  -> 
( ( Q  .<_  ( P  .\/  Q )  /\  D  .<_  ( P 
.\/  Q ) )  <-> 
( Q  .\/  D
)  .<_  ( P  .\/  Q ) ) )
4916, 45, 47, 40, 48syl13anc 1184 . . . . . . 7  |-  ( ph  ->  ( ( Q  .<_  ( P  .\/  Q )  /\  D  .<_  ( P 
.\/  Q ) )  <-> 
( Q  .\/  D
)  .<_  ( P  .\/  Q ) ) )
5039, 44, 49mpbi2and 887 . . . . . 6  |-  ( ph  ->  ( Q  .\/  D
)  .<_  ( P  .\/  Q ) )
511, 9dalemreb 29830 . . . . . . 7  |-  ( ph  ->  R  e.  ( Base `  K ) )
5217, 8, 9hlatjcl 29556 . . . . . . . 8  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  D  e.  A )  ->  ( Q  .\/  D
)  e.  ( Base `  K ) )
532, 4, 31, 52syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( Q  .\/  D
)  e.  ( Base `  K ) )
5417, 7lattr 14162 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( R  e.  ( Base `  K )  /\  ( Q  .\/  D )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
) )  ->  (
( R  .<_  ( Q 
.\/  D )  /\  ( Q  .\/  D ) 
.<_  ( P  .\/  Q
) )  ->  R  .<_  ( P  .\/  Q
) ) )
5516, 51, 53, 40, 54syl13anc 1184 . . . . . 6  |-  ( ph  ->  ( ( R  .<_  ( Q  .\/  D )  /\  ( Q  .\/  D )  .<_  ( P  .\/  Q ) )  ->  R  .<_  ( P  .\/  Q ) ) )
5650, 55mpan2d 655 . . . . 5  |-  ( ph  ->  ( R  .<_  ( Q 
.\/  D )  ->  R  .<_  ( P  .\/  Q ) ) )
5756adantr 451 . . . 4  |-  ( (
ph  /\  D  =/=  Q )  ->  ( R  .<_  ( Q  .\/  D
)  ->  R  .<_  ( P  .\/  Q ) ) )
5827, 37, 573syld 51 . . 3  |-  ( (
ph  /\  D  =/=  Q )  ->  ( D  =  E  ->  R  .<_  ( P  .\/  Q ) ) )
5958necon3bd 2483 . 2  |-  ( (
ph  /\  D  =/=  Q )  ->  ( -.  R  .<_  ( P  .\/  Q )  ->  D  =/=  E ) )
6014, 59mpd 14 1  |-  ( (
ph  /\  D  =/=  Q )  ->  D  =/=  E )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Latclat 14151   Atomscatm 29453   HLchlt 29540   LPlanesclpl 29681
This theorem is referenced by:  dalem4  29854  dalemdnee  29855
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688
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