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Theorem dalem3 34335
Description: Lemma for dalemdnee 34337. (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 34294 . . . 4  |-  ( ph  ->  K  e.  HL )
31dalempea 34297 . . . 4  |-  ( ph  ->  P  e.  A )
41dalemqea 34298 . . . 4  |-  ( ph  ->  Q  e.  A )
51dalemrea 34299 . . . 4  |-  ( ph  ->  R  e.  A )
61dalemyeo 34303 . . . 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 34223 . . . 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 1236 . . 3  |-  ( ph  ->  -.  R  .<_  ( P 
.\/  Q ) )
1413adantr 465 . 2  |-  ( (
ph  /\  D  =/=  Q )  ->  -.  R  .<_  ( P  .\/  Q
) )
15 dalem3.e . . . . . . 7  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
161dalemkelat 34295 . . . . . . . 8  |-  ( ph  ->  K  e.  Lat )
17 eqid 2460 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
1817, 8, 9hlatjcl 34038 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
192, 4, 5, 18syl3anc 1223 . . . . . . . 8  |-  ( ph  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
201, 8, 9dalemtjueb 34318 . . . . . . . 8  |-  ( ph  ->  ( T  .\/  U
)  e.  ( Base `  K ) )
21 dalem3.m . . . . . . . . 9  |-  ./\  =  ( meet `  K )
2217, 7, 21latmle1 15552 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  ( T  .\/  U )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  .<_  ( Q  .\/  R ) )
2316, 19, 20, 22syl3anc 1223 . . . . . . 7  |-  ( ph  ->  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  .<_  ( Q  .\/  R ) )
2415, 23syl5eqbr 4473 . . . . . 6  |-  ( ph  ->  E  .<_  ( Q  .\/  R ) )
25 breq1 4443 . . . . . 6  |-  ( D  =  E  ->  ( D  .<_  ( Q  .\/  R )  <->  E  .<_  ( Q 
.\/  R ) ) )
2624, 25syl5ibrcom 222 . . . . 5  |-  ( ph  ->  ( D  =  E  ->  D  .<_  ( Q 
.\/  R ) ) )
2726adantr 465 . . . 4  |-  ( (
ph  /\  D  =/=  Q )  ->  ( D  =  E  ->  D  .<_  ( Q  .\/  R ) ) )
282adantr 465 . . . . 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 34333 . . . . . 6  |-  ( ph  ->  D  e.  A )
3231adantr 465 . . . . 5  |-  ( (
ph  /\  D  =/=  Q )  ->  D  e.  A )
335adantr 465 . . . . 5  |-  ( (
ph  /\  D  =/=  Q )  ->  R  e.  A )
344adantr 465 . . . . 5  |-  ( (
ph  /\  D  =/=  Q )  ->  Q  e.  A )
35 simpr 461 . . . . 5  |-  ( (
ph  /\  D  =/=  Q )  ->  D  =/=  Q )
367, 8, 9hlatexch1 34066 . . . . 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 1236 . . . 4  |-  ( (
ph  /\  D  =/=  Q )  ->  ( D  .<_  ( Q  .\/  R
)  ->  R  .<_  ( Q  .\/  D ) ) )
387, 8, 9hlatlej2 34047 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  Q  .<_  ( P  .\/  Q ) )
392, 3, 4, 38syl3anc 1223 . . . . . . 7  |-  ( ph  ->  Q  .<_  ( P  .\/  Q ) )
401, 8, 9dalempjqeb 34316 . . . . . . . . 9  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
411, 8, 9dalemsjteb 34317 . . . . . . . . 9  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
4217, 7, 21latmle1 15552 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( P  .\/  Q ) )
4316, 40, 41, 42syl3anc 1223 . . . . . . . 8  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( P  .\/  Q ) )
4430, 43syl5eqbr 4473 . . . . . . 7  |-  ( ph  ->  D  .<_  ( P  .\/  Q ) )
451, 9dalemqeb 34311 . . . . . . . 8  |-  ( ph  ->  Q  e.  ( Base `  K ) )
4617, 9atbase 33961 . . . . . . . . 9  |-  ( D  e.  A  ->  D  e.  ( Base `  K
) )
4731, 46syl 16 . . . . . . . 8  |-  ( ph  ->  D  e.  ( Base `  K ) )
4817, 7, 8latjle12 15538 . . . . . . . 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 1225 . . . . . . 7  |-  ( ph  ->  ( ( Q  .<_  ( P  .\/  Q )  /\  D  .<_  ( P 
.\/  Q ) )  <-> 
( Q  .\/  D
)  .<_  ( P  .\/  Q ) ) )
5039, 44, 49mpbi2and 914 . . . . . 6  |-  ( ph  ->  ( Q  .\/  D
)  .<_  ( P  .\/  Q ) )
511, 9dalemreb 34312 . . . . . . 7  |-  ( ph  ->  R  e.  ( Base `  K ) )
5217, 8, 9hlatjcl 34038 . . . . . . . 8  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  D  e.  A )  ->  ( Q  .\/  D
)  e.  ( Base `  K ) )
532, 4, 31, 52syl3anc 1223 . . . . . . 7  |-  ( ph  ->  ( Q  .\/  D
)  e.  ( Base `  K ) )
5417, 7lattr 15532 . . . . . . 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 1225 . . . . . 6  |-  ( ph  ->  ( ( R  .<_  ( Q  .\/  D )  /\  ( Q  .\/  D )  .<_  ( P  .\/  Q ) )  ->  R  .<_  ( P  .\/  Q ) ) )
5650, 55mpan2d 674 . . . . 5  |-  ( ph  ->  ( R  .<_  ( Q 
.\/  D )  ->  R  .<_  ( P  .\/  Q ) ) )
5756adantr 465 . . . 4  |-  ( (
ph  /\  D  =/=  Q )  ->  ( R  .<_  ( Q  .\/  D
)  ->  R  .<_  ( P  .\/  Q ) ) )
5827, 37, 573syld 55 . . 3  |-  ( (
ph  /\  D  =/=  Q )  ->  ( D  =  E  ->  R  .<_  ( P  .\/  Q ) ) )
5958necon3bd 2672 . 2  |-  ( (
ph  /\  D  =/=  Q )  ->  ( -.  R  .<_  ( P  .\/  Q )  ->  D  =/=  E ) )
6014, 59mpd 15 1  |-  ( (
ph  /\  D  =/=  Q )  ->  D  =/=  E )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   lecple 14551   joincjn 15420   meetcmee 15421   Latclat 15521   Atomscatm 33935   HLchlt 34022   LPlanesclpl 34163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-llines 34169  df-lplanes 34170
This theorem is referenced by:  dalem4  34336  dalemdnee  34337
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