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Theorem dalem1 34672
Description: Lemma for dath 34749. Show the lines  P S and  Q T are different. (Contributed by NM, 9-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 )
dalem1.o  |-  O  =  ( LPlanes `  K )
dalem1.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
Assertion
Ref Expression
dalem1  |-  ( ph  ->  ( P  .\/  S
)  =/=  ( Q 
.\/  T ) )

Proof of Theorem dalem1
StepHypRef Expression
1 dalema.ph . . 3  |-  ( 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 ) ) ) ) )
21dalemclpjs 34647 . 2  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
31dalem-clpjq 34650 . . . . . 6  |-  ( ph  ->  -.  C  .<_  ( P 
.\/  Q ) )
43adantr 465 . . . . 5  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  -.  C  .<_  ( P  .\/  Q
) )
51dalemkehl 34636 . . . . . . . . . 10  |-  ( ph  ->  K  e.  HL )
61dalempea 34639 . . . . . . . . . 10  |-  ( ph  ->  P  e.  A )
71dalemsea 34642 . . . . . . . . . 10  |-  ( ph  ->  S  e.  A )
8 dalemc.l . . . . . . . . . . 11  |-  .<_  =  ( le `  K )
9 dalemc.j . . . . . . . . . . 11  |-  .\/  =  ( join `  K )
10 dalemc.a . . . . . . . . . . 11  |-  A  =  ( Atoms `  K )
118, 9, 10hlatlej1 34388 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  P  .<_  ( P  .\/  S ) )
125, 6, 7, 11syl3anc 1228 . . . . . . . . 9  |-  ( ph  ->  P  .<_  ( P  .\/  S ) )
1312adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  P  .<_  ( P  .\/  S ) )
141dalemqea 34640 . . . . . . . . . . 11  |-  ( ph  ->  Q  e.  A )
151dalemtea 34643 . . . . . . . . . . 11  |-  ( ph  ->  T  e.  A )
168, 9, 10hlatlej1 34388 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  Q  .<_  ( Q  .\/  T ) )
175, 14, 15, 16syl3anc 1228 . . . . . . . . . 10  |-  ( ph  ->  Q  .<_  ( Q  .\/  T ) )
1817adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  Q  .<_  ( Q  .\/  T ) )
19 simpr 461 . . . . . . . . 9  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  ( P  .\/  S )  =  ( Q  .\/  T ) )
2018, 19breqtrrd 4473 . . . . . . . 8  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  Q  .<_  ( P  .\/  S ) )
211dalemkelat 34637 . . . . . . . . . 10  |-  ( ph  ->  K  e.  Lat )
221, 10dalempeb 34652 . . . . . . . . . 10  |-  ( ph  ->  P  e.  ( Base `  K ) )
231, 10dalemqeb 34653 . . . . . . . . . 10  |-  ( ph  ->  Q  e.  ( Base `  K ) )
24 eqid 2467 . . . . . . . . . . . 12  |-  ( Base `  K )  =  (
Base `  K )
2524, 9, 10hlatjcl 34380 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
265, 6, 7, 25syl3anc 1228 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
2724, 8, 9latjle12 15552 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
) ) )  -> 
( ( P  .<_  ( P  .\/  S )  /\  Q  .<_  ( P 
.\/  S ) )  <-> 
( P  .\/  Q
)  .<_  ( P  .\/  S ) ) )
2821, 22, 23, 26, 27syl13anc 1230 . . . . . . . . 9  |-  ( ph  ->  ( ( P  .<_  ( P  .\/  S )  /\  Q  .<_  ( P 
.\/  S ) )  <-> 
( P  .\/  Q
)  .<_  ( P  .\/  S ) ) )
2928adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  ( ( P  .<_  ( P  .\/  S )  /\  Q  .<_  ( P  .\/  S ) )  <->  ( P  .\/  Q )  .<_  ( P  .\/  S ) ) )
3013, 20, 29mpbi2and 919 . . . . . . 7  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  ( P  .\/  Q )  .<_  ( P 
.\/  S ) )
311dalemrea 34641 . . . . . . . . . 10  |-  ( ph  ->  R  e.  A )
321dalemyeo 34645 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  O )
33 dalem1.o . . . . . . . . . . 11  |-  O  =  ( LPlanes `  K )
34 dalem1.y . . . . . . . . . . 11  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
359, 10, 33, 34lplnri1 34566 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  Y  e.  O )  ->  P  =/=  Q )
365, 6, 14, 31, 32, 35syl131anc 1241 . . . . . . . . 9  |-  ( ph  ->  P  =/=  Q )
378, 9, 10ps-1 34490 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( P  e.  A  /\  S  e.  A ) )  -> 
( ( P  .\/  Q )  .<_  ( P  .\/  S )  <->  ( P  .\/  Q )  =  ( P  .\/  S ) ) )
385, 6, 14, 36, 6, 7, 37syl132anc 1246 . . . . . . . 8  |-  ( ph  ->  ( ( P  .\/  Q )  .<_  ( P  .\/  S )  <->  ( P  .\/  Q )  =  ( P  .\/  S ) ) )
3938adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  ( ( P  .\/  Q )  .<_  ( P  .\/  S )  <-> 
( P  .\/  Q
)  =  ( P 
.\/  S ) ) )
4030, 39mpbid 210 . . . . . 6  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  ( P  .\/  Q )  =  ( P  .\/  S ) )
4140breq2d 4459 . . . . 5  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  ( C  .<_  ( P  .\/  Q
)  <->  C  .<_  ( P 
.\/  S ) ) )
424, 41mtbid 300 . . . 4  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  -.  C  .<_  ( P  .\/  S
) )
4342ex 434 . . 3  |-  ( ph  ->  ( ( P  .\/  S )  =  ( Q 
.\/  T )  ->  -.  C  .<_  ( P 
.\/  S ) ) )
4443necon2ad 2680 . 2  |-  ( ph  ->  ( C  .<_  ( P 
.\/  S )  -> 
( P  .\/  S
)  =/=  ( Q 
.\/  T ) ) )
452, 44mpd 15 1  |-  ( ph  ->  ( P  .\/  S
)  =/=  ( Q 
.\/  T ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   Basecbs 14493   lecple 14565   joincjn 15434   Latclat 15535   Atomscatm 34277   HLchlt 34364   LPlanesclpl 34505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-poset 15436  df-plt 15448  df-lub 15464  df-glb 15465  df-join 15466  df-meet 15467  df-p0 15529  df-lat 15536  df-clat 15598  df-oposet 34190  df-ol 34192  df-oml 34193  df-covers 34280  df-ats 34281  df-atl 34312  df-cvlat 34336  df-hlat 34365  df-llines 34511  df-lplanes 34512
This theorem is referenced by:  dalemcea  34673  dalem2  34674
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