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Theorem dalem51 30694
Description: Lemma for dath 30707. Construct the condition  ph with  c,  G H I, and 
Y in place of  C,  Y, and  Z respectively. This lets us reuse the special case of Desargues' Theorem where  Y  =/=  Z, to eventually prove the case where  Y  =  Z. (Contributed by NM, 16-Aug-2012.)
Hypotheses
Ref Expression
dalem.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 ) ) ) ) )
dalem.l  |-  .<_  =  ( le `  K )
dalem.j  |-  .\/  =  ( join `  K )
dalem.a  |-  A  =  ( Atoms `  K )
dalem.ps  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
dalem44.m  |-  ./\  =  ( meet `  K )
dalem44.o  |-  O  =  ( LPlanes `  K )
dalem44.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem44.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem44.g  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
dalem44.h  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
dalem44.i  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
Assertion
Ref Expression
dalem51  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( ( K  e.  HL  /\  c  e.  A )  /\  ( G  e.  A  /\  H  e.  A  /\  I  e.  A
)  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  /\  ( ( ( G 
.\/  H )  .\/  I )  e.  O  /\  Y  e.  O
)  /\  ( ( -.  c  .<_  ( G 
.\/  H )  /\  -.  c  .<_  ( H 
.\/  I )  /\  -.  c  .<_  ( I 
.\/  G ) )  /\  ( -.  c  .<_  ( P  .\/  Q
)  /\  -.  c  .<_  ( Q  .\/  R
)  /\  -.  c  .<_  ( R  .\/  P
) )  /\  (
c  .<_  ( G  .\/  P )  /\  c  .<_  ( H  .\/  Q )  /\  c  .<_  ( I 
.\/  R ) ) ) )  /\  (
( G  .\/  H
)  .\/  I )  =/=  Y ) )

Proof of Theorem dalem51
StepHypRef Expression
1 dalem.ph . . . . . . 7  |-  ( 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 30594 . . . . . 6  |-  ( ph  ->  K  e.  HL )
323ad2ant1 979 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
4 dalem.ps . . . . . . 7  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
54dalemccea 30654 . . . . . 6  |-  ( ps 
->  c  e.  A
)
653ad2ant3 981 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
73, 6jca 520 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( K  e.  HL  /\  c  e.  A ) )
8 dalem.l . . . . . 6  |-  .<_  =  ( le `  K )
9 dalem.j . . . . . 6  |-  .\/  =  ( join `  K )
10 dalem.a . . . . . 6  |-  A  =  ( Atoms `  K )
11 dalem44.m . . . . . 6  |-  ./\  =  ( meet `  K )
12 dalem44.o . . . . . 6  |-  O  =  ( LPlanes `  K )
13 dalem44.y . . . . . 6  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
14 dalem44.z . . . . . 6  |-  Z  =  ( ( S  .\/  T )  .\/  U )
15 dalem44.g . . . . . 6  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
161, 8, 9, 10, 4, 11, 12, 13, 14, 15dalem23 30667 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
17 dalem44.h . . . . . 6  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
181, 8, 9, 10, 4, 11, 12, 13, 14, 17dalem29 30672 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  A )
19 dalem44.i . . . . . 6  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
201, 8, 9, 10, 4, 11, 12, 13, 14, 19dalem34 30677 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  A )
2116, 18, 203jca 1135 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  e.  A  /\  H  e.  A  /\  I  e.  A
) )
221dalempea 30597 . . . . . 6  |-  ( ph  ->  P  e.  A )
231dalemqea 30598 . . . . . 6  |-  ( ph  ->  Q  e.  A )
241dalemrea 30599 . . . . . 6  |-  ( ph  ->  R  e.  A )
2522, 23, 243jca 1135 . . . . 5  |-  ( ph  ->  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )
26253ad2ant1 979 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )
277, 21, 263jca 1135 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( K  e.  HL  /\  c  e.  A )  /\  ( G  e.  A  /\  H  e.  A  /\  I  e.  A )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) ) )
281, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem42 30685 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  e.  O )
291dalemyeo 30603 . . . . 5  |-  ( ph  ->  Y  e.  O )
30293ad2ant1 979 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  e.  O )
3128, 30jca 520 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( G 
.\/  H )  .\/  I )  e.  O  /\  Y  e.  O
) )
321, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem45 30688 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  ( G 
.\/  H ) )
331, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem46 30689 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  ( H 
.\/  I ) )
341, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem47 30690 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  ( I 
.\/  G ) )
3532, 33, 343jca 1135 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( -.  c  .<_  ( G  .\/  H )  /\  -.  c  .<_  ( H  .\/  I )  /\  -.  c  .<_  ( I  .\/  G ) ) )
361, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem48 30691 . . . . . 6  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  ( P 
.\/  Q ) )
371, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem49 30692 . . . . . 6  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  ( Q 
.\/  R ) )
381, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem50 30693 . . . . . 6  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  ( R 
.\/  P ) )
3936, 37, 383jca 1135 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( -.  c  .<_  ( P  .\/  Q )  /\  -.  c  .<_  ( Q  .\/  R )  /\  -.  c  .<_  ( R  .\/  P ) ) )
40393adant2 977 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( -.  c  .<_  ( P  .\/  Q )  /\  -.  c  .<_  ( Q  .\/  R )  /\  -.  c  .<_  ( R  .\/  P ) ) )
411, 8, 9, 10, 4, 11, 12, 13, 14, 15dalem27 30670 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  .<_  ( G  .\/  P ) )
421, 8, 9, 10, 4, 11, 12, 13, 14, 17dalem32 30675 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  .<_  ( H  .\/  Q ) )
431, 8, 9, 10, 4, 11, 12, 13, 14, 19dalem36 30679 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  .<_  ( I  .\/  R ) )
4441, 42, 433jca 1135 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .<_  ( G 
.\/  P )  /\  c  .<_  ( H  .\/  Q )  /\  c  .<_  ( I  .\/  R ) ) )
4535, 40, 443jca 1135 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( -.  c  .<_  ( G  .\/  H
)  /\  -.  c  .<_  ( H  .\/  I
)  /\  -.  c  .<_  ( I  .\/  G
) )  /\  ( -.  c  .<_  ( P 
.\/  Q )  /\  -.  c  .<_  ( Q 
.\/  R )  /\  -.  c  .<_  ( R 
.\/  P ) )  /\  ( c  .<_  ( G  .\/  P )  /\  c  .<_  ( H 
.\/  Q )  /\  c  .<_  ( I  .\/  R ) ) ) )
4627, 31, 453jca 1135 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( K  e.  HL  /\  c  e.  A )  /\  ( G  e.  A  /\  H  e.  A  /\  I  e.  A )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  (
( ( G  .\/  H )  .\/  I )  e.  O  /\  Y  e.  O )  /\  (
( -.  c  .<_  ( G  .\/  H )  /\  -.  c  .<_  ( H  .\/  I )  /\  -.  c  .<_  ( I  .\/  G ) )  /\  ( -.  c  .<_  ( P  .\/  Q )  /\  -.  c  .<_  ( Q  .\/  R )  /\  -.  c  .<_  ( R  .\/  P
) )  /\  (
c  .<_  ( G  .\/  P )  /\  c  .<_  ( H  .\/  Q )  /\  c  .<_  ( I 
.\/  R ) ) ) ) )
471, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem43 30686 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  =/=  Y )
4846, 47jca 520 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( ( K  e.  HL  /\  c  e.  A )  /\  ( G  e.  A  /\  H  e.  A  /\  I  e.  A
)  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  /\  ( ( ( G 
.\/  H )  .\/  I )  e.  O  /\  Y  e.  O
)  /\  ( ( -.  c  .<_  ( G 
.\/  H )  /\  -.  c  .<_  ( H 
.\/  I )  /\  -.  c  .<_  ( I 
.\/  G ) )  /\  ( -.  c  .<_  ( P  .\/  Q
)  /\  -.  c  .<_  ( Q  .\/  R
)  /\  -.  c  .<_  ( R  .\/  P
) )  /\  (
c  .<_  ( G  .\/  P )  /\  c  .<_  ( H  .\/  Q )  /\  c  .<_  ( I 
.\/  R ) ) ) )  /\  (
( G  .\/  H
)  .\/  I )  =/=  Y ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1654    e. wcel 1728    =/= wne 2606   class class class wbr 4243   ` cfv 5489  (class class class)co 6117   Basecbs 13507   lecple 13574   joincjn 14439   meetcmee 14440   Atomscatm 30235   HLchlt 30322   LPlanesclpl 30463
This theorem is referenced by:  dalem53  30696  dalem54  30697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-1st 6385  df-2nd 6386  df-undef 6579  df-riota 6585  df-poset 14441  df-plt 14453  df-lub 14469  df-glb 14470  df-join 14471  df-meet 14472  df-p0 14506  df-lat 14513  df-clat 14575  df-oposet 30148  df-ol 30150  df-oml 30151  df-covers 30238  df-ats 30239  df-atl 30270  df-cvlat 30294  df-hlat 30323  df-llines 30469  df-lplanes 30470  df-lvols 30471
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