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Theorem dalem51 29983
Description: Lemma for dath 29996. 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 29883 . . . . . 6  |-  ( ph  ->  K  e.  HL )
323ad2ant1 977 . . . . 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 29943 . . . . . 6  |-  ( ps 
->  c  e.  A
)
653ad2ant3 979 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
73, 6jca 518 . . . 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 29956 . . . . 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 29961 . . . . 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 29966 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  A )
2116, 18, 203jca 1133 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  e.  A  /\  H  e.  A  /\  I  e.  A
) )
221dalempea 29886 . . . . . 6  |-  ( ph  ->  P  e.  A )
231dalemqea 29887 . . . . . 6  |-  ( ph  ->  Q  e.  A )
241dalemrea 29888 . . . . . 6  |-  ( ph  ->  R  e.  A )
2522, 23, 243jca 1133 . . . . 5  |-  ( ph  ->  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )
26253ad2ant1 977 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )
277, 21, 263jca 1133 . . 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 29974 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  e.  O )
291dalemyeo 29892 . . . . 5  |-  ( ph  ->  Y  e.  O )
30293ad2ant1 977 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  e.  O )
3128, 30jca 518 . . 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 29977 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  ( G 
.\/  H ) )
331, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem46 29978 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  ( H 
.\/  I ) )
341, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem47 29979 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  ( I 
.\/  G ) )
3532, 33, 343jca 1133 . . . 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 29980 . . . . . 6  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  ( P 
.\/  Q ) )
371, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem49 29981 . . . . . 6  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  ( Q 
.\/  R ) )
381, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem50 29982 . . . . . 6  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  ( R 
.\/  P ) )
3936, 37, 383jca 1133 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( -.  c  .<_  ( P  .\/  Q )  /\  -.  c  .<_  ( Q  .\/  R )  /\  -.  c  .<_  ( R  .\/  P ) ) )
40393adant2 975 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( -.  c  .<_  ( P  .\/  Q )  /\  -.  c  .<_  ( Q  .\/  R )  /\  -.  c  .<_  ( R  .\/  P ) ) )
411, 8, 9, 10, 4, 11, 12, 13, 14, 15dalem27 29959 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  .<_  ( G  .\/  P ) )
421, 8, 9, 10, 4, 11, 12, 13, 14, 17dalem32 29964 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  .<_  ( H  .\/  Q ) )
431, 8, 9, 10, 4, 11, 12, 13, 14, 19dalem36 29968 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  .<_  ( I  .\/  R ) )
4441, 42, 433jca 1133 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .<_  ( G 
.\/  P )  /\  c  .<_  ( H  .\/  Q )  /\  c  .<_  ( I  .\/  R ) ) )
4535, 40, 443jca 1133 . . 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 1133 . 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 29975 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  =/=  Y )
4846, 47jca 518 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 176    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   Basecbs 13356   lecple 13423   joincjn 14288   meetcmee 14289   Atomscatm 29524   HLchlt 29611   LPlanesclpl 29752
This theorem is referenced by:  dalem53  29985  dalem54  29986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-lat 14362  df-clat 14424  df-oposet 29437  df-ol 29439  df-oml 29440  df-covers 29527  df-ats 29528  df-atl 29559  df-cvlat 29583  df-hlat 29612  df-llines 29758  df-lplanes 29759  df-lvols 29760
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