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Theorem dalem1 29848
Description: Lemma for dath 29925. 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 29823 . 2  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
31dalem-clpjq 29826 . . . . . 6  |-  ( ph  ->  -.  C  .<_  ( P 
.\/  Q ) )
43adantr 451 . . . . 5  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  -.  C  .<_  ( P  .\/  Q
) )
51dalemkehl 29812 . . . . . . . . . 10  |-  ( ph  ->  K  e.  HL )
61dalempea 29815 . . . . . . . . . 10  |-  ( ph  ->  P  e.  A )
71dalemsea 29818 . . . . . . . . . 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 29564 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  P  .<_  ( P  .\/  S ) )
125, 6, 7, 11syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  P  .<_  ( P  .\/  S ) )
1312adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  P  .<_  ( P  .\/  S ) )
141dalemqea 29816 . . . . . . . . . . 11  |-  ( ph  ->  Q  e.  A )
151dalemtea 29819 . . . . . . . . . . 11  |-  ( ph  ->  T  e.  A )
168, 9, 10hlatlej1 29564 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  Q  .<_  ( Q  .\/  T ) )
175, 14, 15, 16syl3anc 1182 . . . . . . . . . 10  |-  ( ph  ->  Q  .<_  ( Q  .\/  T ) )
1817adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  Q  .<_  ( Q  .\/  T ) )
19 simpr 447 . . . . . . . . 9  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  ( P  .\/  S )  =  ( Q  .\/  T ) )
2018, 19breqtrrd 4049 . . . . . . . 8  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  Q  .<_  ( P  .\/  S ) )
211dalemkelat 29813 . . . . . . . . . 10  |-  ( ph  ->  K  e.  Lat )
221, 10dalempeb 29828 . . . . . . . . . 10  |-  ( ph  ->  P  e.  ( Base `  K ) )
231, 10dalemqeb 29829 . . . . . . . . . 10  |-  ( ph  ->  Q  e.  ( Base `  K ) )
24 eqid 2283 . . . . . . . . . . . 12  |-  ( Base `  K )  =  (
Base `  K )
2524, 9, 10hlatjcl 29556 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
265, 6, 7, 25syl3anc 1182 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
2724, 8, 9latjle12 14168 . . . . . . . . . 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 1184 . . . . . . . . 9  |-  ( ph  ->  ( ( P  .<_  ( P  .\/  S )  /\  Q  .<_  ( P 
.\/  S ) )  <-> 
( P  .\/  Q
)  .<_  ( P  .\/  S ) ) )
2928adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  ( ( P  .<_  ( P  .\/  S )  /\  Q  .<_  ( P  .\/  S ) )  <->  ( P  .\/  Q )  .<_  ( P  .\/  S ) ) )
3013, 20, 29mpbi2and 887 . . . . . . 7  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  ( P  .\/  Q )  .<_  ( P 
.\/  S ) )
311dalemrea 29817 . . . . . . . . . 10  |-  ( ph  ->  R  e.  A )
321dalemyeo 29821 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  O )
33 dalem1.o . . . . . . . . . . 11  |-  O  =  ( LPlanes `  K )
34 dalem1.y . . . . . . . . . . 11  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
359, 10, 33, 34lplnri1 29742 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  Y  e.  O )  ->  P  =/=  Q )
365, 6, 14, 31, 32, 35syl131anc 1195 . . . . . . . . 9  |-  ( ph  ->  P  =/=  Q )
378, 9, 10ps-1 29666 . . . . . . . . 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 1200 . . . . . . . 8  |-  ( ph  ->  ( ( P  .\/  Q )  .<_  ( P  .\/  S )  <->  ( P  .\/  Q )  =  ( P  .\/  S ) ) )
3938adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  ( ( P  .\/  Q )  .<_  ( P  .\/  S )  <-> 
( P  .\/  Q
)  =  ( P 
.\/  S ) ) )
4030, 39mpbid 201 . . . . . 6  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  ( P  .\/  Q )  =  ( P  .\/  S ) )
4140breq2d 4035 . . . . 5  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  ( C  .<_  ( P  .\/  Q
)  <->  C  .<_  ( P 
.\/  S ) ) )
424, 41mtbid 291 . . . 4  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  -.  C  .<_  ( P  .\/  S
) )
4342ex 423 . . 3  |-  ( ph  ->  ( ( P  .\/  S )  =  ( Q 
.\/  T )  ->  -.  C  .<_  ( P 
.\/  S ) ) )
4443necon2ad 2494 . 2  |-  ( ph  ->  ( C  .<_  ( P 
.\/  S )  -> 
( P  .\/  S
)  =/=  ( Q 
.\/  T ) ) )
452, 44mpd 14 1  |-  ( ph  ->  ( P  .\/  S
)  =/=  ( Q 
.\/  T ) )
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   Latclat 14151   Atomscatm 29453   HLchlt 29540   LPlanesclpl 29681
This theorem is referenced by:  dalemcea  29849  dalem2  29850
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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