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Theorem dalem51 30209
Description: Lemma for dath 30222. 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 30109 . . . . . 6  |-  ( ph  ->  K  e.  HL )
323ad2ant1 978 . . . . 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 30169 . . . . . 6  |-  ( ps 
->  c  e.  A
)
653ad2ant3 980 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
73, 6jca 519 . . . 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 30182 . . . . 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 30187 . . . . 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 30192 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  A )
2116, 18, 203jca 1134 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  e.  A  /\  H  e.  A  /\  I  e.  A
) )
221dalempea 30112 . . . . . 6  |-  ( ph  ->  P  e.  A )
231dalemqea 30113 . . . . . 6  |-  ( ph  ->  Q  e.  A )
241dalemrea 30114 . . . . . 6  |-  ( ph  ->  R  e.  A )
2522, 23, 243jca 1134 . . . . 5  |-  ( ph  ->  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )
26253ad2ant1 978 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )
277, 21, 263jca 1134 . . 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 30200 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  e.  O )
291dalemyeo 30118 . . . . 5  |-  ( ph  ->  Y  e.  O )
30293ad2ant1 978 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  e.  O )
3128, 30jca 519 . . 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 30203 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  ( G 
.\/  H ) )
331, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem46 30204 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  ( H 
.\/  I ) )
341, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem47 30205 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  ( I 
.\/  G ) )
3532, 33, 343jca 1134 . . . 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 30206 . . . . . 6  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  ( P 
.\/  Q ) )
371, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem49 30207 . . . . . 6  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  ( Q 
.\/  R ) )
381, 8, 9, 10, 4, 11, 12, 13, 14, 15, 17, 19dalem50 30208 . . . . . 6  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  ( R 
.\/  P ) )
3936, 37, 383jca 1134 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( -.  c  .<_  ( P  .\/  Q )  /\  -.  c  .<_  ( Q  .\/  R )  /\  -.  c  .<_  ( R  .\/  P ) ) )
40393adant2 976 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( -.  c  .<_  ( P  .\/  Q )  /\  -.  c  .<_  ( Q  .\/  R )  /\  -.  c  .<_  ( R  .\/  P ) ) )
411, 8, 9, 10, 4, 11, 12, 13, 14, 15dalem27 30185 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  .<_  ( G  .\/  P ) )
421, 8, 9, 10, 4, 11, 12, 13, 14, 17dalem32 30190 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  .<_  ( H  .\/  Q ) )
431, 8, 9, 10, 4, 11, 12, 13, 14, 19dalem36 30194 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  .<_  ( I  .\/  R ) )
4441, 42, 433jca 1134 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .<_  ( G 
.\/  P )  /\  c  .<_  ( H  .\/  Q )  /\  c  .<_  ( I  .\/  R ) ) )
4535, 40, 443jca 1134 . . 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 1134 . 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 30201 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  =/=  Y )
4846, 47jca 519 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2571   class class class wbr 4176   ` cfv 5417  (class class class)co 6044   Basecbs 13428   lecple 13495   joincjn 14360   meetcmee 14361   Atomscatm 29750   HLchlt 29837   LPlanesclpl 29978
This theorem is referenced by:  dalem53  30211  dalem54  30212
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-lat 14434  df-clat 14496  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-llines 29984  df-lplanes 29985  df-lvols 29986
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