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Theorem dalem1 33661
Description: Lemma for dath 33738. 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 33636 . 2  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
31dalem-clpjq 33639 . . . . . 6  |-  ( ph  ->  -.  C  .<_  ( P 
.\/  Q ) )
43adantr 465 . . . . 5  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  -.  C  .<_  ( P  .\/  Q
) )
51dalemkehl 33625 . . . . . . . . . 10  |-  ( ph  ->  K  e.  HL )
61dalempea 33628 . . . . . . . . . 10  |-  ( ph  ->  P  e.  A )
71dalemsea 33631 . . . . . . . . . 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 33377 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  P  .<_  ( P  .\/  S ) )
125, 6, 7, 11syl3anc 1219 . . . . . . . . 9  |-  ( ph  ->  P  .<_  ( P  .\/  S ) )
1312adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  P  .<_  ( P  .\/  S ) )
141dalemqea 33629 . . . . . . . . . . 11  |-  ( ph  ->  Q  e.  A )
151dalemtea 33632 . . . . . . . . . . 11  |-  ( ph  ->  T  e.  A )
168, 9, 10hlatlej1 33377 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  Q  .<_  ( Q  .\/  T ) )
175, 14, 15, 16syl3anc 1219 . . . . . . . . . 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 4429 . . . . . . . 8  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  Q  .<_  ( P  .\/  S ) )
211dalemkelat 33626 . . . . . . . . . 10  |-  ( ph  ->  K  e.  Lat )
221, 10dalempeb 33641 . . . . . . . . . 10  |-  ( ph  ->  P  e.  ( Base `  K ) )
231, 10dalemqeb 33642 . . . . . . . . . 10  |-  ( ph  ->  Q  e.  ( Base `  K ) )
24 eqid 2454 . . . . . . . . . . . 12  |-  ( Base `  K )  =  (
Base `  K )
2524, 9, 10hlatjcl 33369 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
265, 6, 7, 25syl3anc 1219 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
2724, 8, 9latjle12 15354 . . . . . . . . . 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 1221 . . . . . . . . 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 912 . . . . . . 7  |-  ( (
ph  /\  ( P  .\/  S )  =  ( Q  .\/  T ) )  ->  ( P  .\/  Q )  .<_  ( P 
.\/  S ) )
311dalemrea 33630 . . . . . . . . . 10  |-  ( ph  ->  R  e.  A )
321dalemyeo 33634 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  O )
33 dalem1.o . . . . . . . . . . 11  |-  O  =  ( LPlanes `  K )
34 dalem1.y . . . . . . . . . . 11  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
359, 10, 33, 34lplnri1 33555 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  Y  e.  O )  ->  P  =/=  Q )
365, 6, 14, 31, 32, 35syl131anc 1232 . . . . . . . . 9  |-  ( ph  ->  P  =/=  Q )
378, 9, 10ps-1 33479 . . . . . . . . 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 1237 . . . . . . . 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 4415 . . . . 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 2665 . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14295   lecple 14367   joincjn 15236   Latclat 15337   Atomscatm 33266   HLchlt 33353   LPlanesclpl 33494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-poset 15238  df-plt 15250  df-lub 15266  df-glb 15267  df-join 15268  df-meet 15269  df-p0 15331  df-lat 15338  df-clat 15400  df-oposet 33179  df-ol 33181  df-oml 33182  df-covers 33269  df-ats 33270  df-atl 33301  df-cvlat 33325  df-hlat 33354  df-llines 33500  df-lplanes 33501
This theorem is referenced by:  dalemcea  33662  dalem2  33663
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