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Theorem dalem1 32936
Description: Lemma for dath 33013. 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 32911 . 2  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
31dalem-clpjq 32914 . . . . . 6  |-  ( ph  ->  -.  C  .<_  ( P 
.\/  Q ) )
43adantr 466 . . . . 5  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  -.  C  .<_  ( P  .\/  Q
) )
51dalemkehl 32900 . . . . . . . . . 10  |-  ( ph  ->  K  e.  HL )
61dalempea 32903 . . . . . . . . . 10  |-  ( ph  ->  P  e.  A )
71dalemsea 32906 . . . . . . . . . 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 32652 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  P  .<_  ( P  .\/  S ) )
125, 6, 7, 11syl3anc 1264 . . . . . . . . 9  |-  ( ph  ->  P  .<_  ( P  .\/  S ) )
1312adantr 466 . . . . . . . 8  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  P  .<_  ( P  .\/  S ) )
141dalemqea 32904 . . . . . . . . . . 11  |-  ( ph  ->  Q  e.  A )
151dalemtea 32907 . . . . . . . . . . 11  |-  ( ph  ->  T  e.  A )
168, 9, 10hlatlej1 32652 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  Q  .<_  ( Q  .\/  T ) )
175, 14, 15, 16syl3anc 1264 . . . . . . . . . 10  |-  ( ph  ->  Q  .<_  ( Q  .\/  T ) )
1817adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  Q  .<_  ( Q  .\/  T ) )
19 simpr 462 . . . . . . . . 9  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  ( P  .\/  S )  =  ( Q  .\/  T ) )
2018, 19breqtrrd 4452 . . . . . . . 8  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  Q  .<_  ( P  .\/  S ) )
211dalemkelat 32901 . . . . . . . . . 10  |-  ( ph  ->  K  e.  Lat )
221, 10dalempeb 32916 . . . . . . . . . 10  |-  ( ph  ->  P  e.  ( Base `  K ) )
231, 10dalemqeb 32917 . . . . . . . . . 10  |-  ( ph  ->  Q  e.  ( Base `  K ) )
24 eqid 2429 . . . . . . . . . . . 12  |-  ( Base `  K )  =  (
Base `  K )
2524, 9, 10hlatjcl 32644 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
265, 6, 7, 25syl3anc 1264 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
2724, 8, 9latjle12 16259 . . . . . . . . . 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 1266 . . . . . . . . 9  |-  ( ph  ->  ( ( P  .<_  ( P  .\/  S )  /\  Q  .<_  ( P 
.\/  S ) )  <-> 
( P  .\/  Q
)  .<_  ( P  .\/  S ) ) )
2928adantr 466 . . . . . . . 8  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  ( ( P  .<_  ( P  .\/  S )  /\  Q  .<_  ( P  .\/  S ) )  <->  ( P  .\/  Q )  .<_  ( P  .\/  S ) ) )
3013, 20, 29mpbi2and 929 . . . . . . 7  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  ( P  .\/  Q )  .<_  ( P 
.\/  S ) )
311dalemrea 32905 . . . . . . . . . 10  |-  ( ph  ->  R  e.  A )
321dalemyeo 32909 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  O )
33 dalem1.o . . . . . . . . . . 11  |-  O  =  ( LPlanes `  K )
34 dalem1.y . . . . . . . . . . 11  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
359, 10, 33, 34lplnri1 32830 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  Y  e.  O )  ->  P  =/=  Q )
365, 6, 14, 31, 32, 35syl131anc 1277 . . . . . . . . 9  |-  ( ph  ->  P  =/=  Q )
378, 9, 10ps-1 32754 . . . . . . . . 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 1282 . . . . . . . 8  |-  ( ph  ->  ( ( P  .\/  Q )  .<_  ( P  .\/  S )  <->  ( P  .\/  Q )  =  ( P  .\/  S ) ) )
3938adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  ( ( P  .\/  Q )  .<_  ( P  .\/  S )  <-> 
( P  .\/  Q
)  =  ( P 
.\/  S ) ) )
4030, 39mpbid 213 . . . . . 6  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  ( P  .\/  Q )  =  ( P  .\/  S ) )
4140breq2d 4438 . . . . 5  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  ( C  .<_  ( P  .\/  Q
)  <->  C  .<_  ( P 
.\/  S ) ) )
424, 41mtbid 301 . . . 4  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  -.  C  .<_  ( P  .\/  S
) )
4342ex 435 . . 3  |-  ( ph  ->  ( ( P  .\/  S )  =  ( Q 
.\/  T )  ->  -.  C  .<_  ( P 
.\/  S ) ) )
4443necon2ad 2644 . 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   lecple 15159   joincjn 16140   Latclat 16242   Atomscatm 32541   HLchlt 32628   LPlanesclpl 32769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-lat 16243  df-clat 16305  df-oposet 32454  df-ol 32456  df-oml 32457  df-covers 32544  df-ats 32545  df-atl 32576  df-cvlat 32600  df-hlat 32629  df-llines 32775  df-lplanes 32776
This theorem is referenced by:  dalemcea  32937  dalem2  32938
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