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Theorem dalem57 29970
Description: Lemma for dath 29977. Axis of perspectivity point  D is on the auxiliary line  B. (Contributed by NM, 9-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
) ) ) )
dalem57.m  |-  ./\  =  ( meet `  K )
dalem57.o  |-  O  =  ( LPlanes `  K )
dalem57.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem57.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem57.d  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
dalem57.g  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
dalem57.h  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
dalem57.i  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
dalem57.b1  |-  B  =  ( ( ( G 
.\/  H )  .\/  I )  ./\  Y
)
Assertion
Ref Expression
dalem57  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  D  .<_  B )

Proof of Theorem dalem57
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 ) ) ) ) )
2 dalem.l . . . . . . 7  |-  .<_  =  ( le `  K )
3 dalem.j . . . . . . 7  |-  .\/  =  ( join `  K )
4 dalem.a . . . . . . 7  |-  A  =  ( Atoms `  K )
5 dalem.ps . . . . . . 7  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
6 dalem57.m . . . . . . 7  |-  ./\  =  ( meet `  K )
7 dalem57.o . . . . . . 7  |-  O  =  ( LPlanes `  K )
8 dalem57.y . . . . . . 7  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
9 dalem57.z . . . . . . 7  |-  Z  =  ( ( S  .\/  T )  .\/  U )
10 dalem57.g . . . . . . 7  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
11 dalem57.h . . . . . . 7  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
12 dalem57.i . . . . . . 7  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
13 dalem57.b1 . . . . . . 7  |-  B  =  ( ( ( G 
.\/  H )  .\/  I )  ./\  Y
)
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dalem55 29968 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  =  ( ( G  .\/  H )  ./\  B )
)
151dalemkelat 29865 . . . . . . . 8  |-  ( ph  ->  K  e.  Lat )
16153ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
171dalemkehl 29864 . . . . . . . . 9  |-  ( ph  ->  K  e.  HL )
18173ad2ant1 976 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
191, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem23 29937 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
201, 2, 3, 4, 5, 6, 7, 8, 9, 11dalem29 29942 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  A )
21 eqid 2358 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2221, 3, 4hlatjcl 29608 . . . . . . . 8  |-  ( ( K  e.  HL  /\  G  e.  A  /\  H  e.  A )  ->  ( G  .\/  H
)  e.  ( Base `  K ) )
2318, 19, 20, 22syl3anc 1182 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  e.  ( Base `  K ) )
241, 3, 4dalempjqeb 29886 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
25243ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
2621, 2, 6latmle2 14276 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
)  ->  ( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( P  .\/  Q ) )
2716, 23, 25, 26syl3anc 1182 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( P  .\/  Q ) )
2814, 27eqbrtrrd 4124 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  ( P  .\/  Q
) )
291, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dalem56 29969 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( S  .\/  T ) )  =  ( ( G  .\/  H )  ./\  B )
)
301, 3, 4dalemsjteb 29887 . . . . . . . 8  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
31303ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( S  .\/  T
)  e.  ( Base `  K ) )
3221, 2, 6latmle2 14276 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
)  ->  ( ( G  .\/  H )  ./\  ( S  .\/  T ) )  .<_  ( S  .\/  T ) )
3316, 23, 31, 32syl3anc 1182 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( S  .\/  T ) )  .<_  ( S  .\/  T ) )
3429, 33eqbrtrrd 4124 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  ( S  .\/  T
) )
351, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dalem54 29967 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  e.  A )
3621, 4atbase 29531 . . . . . . 7  |-  ( ( ( G  .\/  H
)  ./\  B )  e.  A  ->  ( ( G  .\/  H ) 
./\  B )  e.  ( Base `  K
) )
3735, 36syl 15 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  e.  ( Base `  K
) )
3821, 2, 6latlem12 14277 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( ( ( G 
.\/  H )  ./\  B )  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
) )  ->  (
( ( ( G 
.\/  H )  ./\  B )  .<_  ( P  .\/  Q )  /\  (
( G  .\/  H
)  ./\  B )  .<_  ( S  .\/  T
) )  <->  ( ( G  .\/  H )  ./\  B )  .<_  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) ) ) )
3916, 37, 25, 31, 38syl13anc 1184 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( ( G  .\/  H ) 
./\  B )  .<_  ( P  .\/  Q )  /\  ( ( G 
.\/  H )  ./\  B )  .<_  ( S  .\/  T ) )  <->  ( ( G  .\/  H )  ./\  B )  .<_  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) ) ) )
4028, 34, 39mpbi2and 887 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) ) )
41 dalem57.d . . . 4  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
4240, 41syl6breqr 4142 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  D )
43 hlatl 29602 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
4418, 43syl 15 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  AtLat )
451, 2, 3, 4, 6, 7, 8, 9, 41dalemdea 29903 . . . . 5  |-  ( ph  ->  D  e.  A )
46453ad2ant1 976 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  D  e.  A )
472, 4atcmp 29553 . . . 4  |-  ( ( K  e.  AtLat  /\  (
( G  .\/  H
)  ./\  B )  e.  A  /\  D  e.  A )  ->  (
( ( G  .\/  H )  ./\  B )  .<_  D  <->  ( ( G 
.\/  H )  ./\  B )  =  D ) )
4844, 35, 46, 47syl3anc 1182 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( G 
.\/  H )  ./\  B )  .<_  D  <->  ( ( G  .\/  H )  ./\  B )  =  D ) )
4942, 48mpbid 201 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  =  D )
50 eqid 2358 . . . . 5  |-  ( LLines `  K )  =  (
LLines `  K )
511, 2, 3, 4, 5, 6, 50, 7, 8, 9, 10, 11, 12, 13dalem53 29966 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( LLines `  K ) )
5221, 50llnbase 29750 . . . 4  |-  ( B  e.  ( LLines `  K
)  ->  B  e.  ( Base `  K )
)
5351, 52syl 15 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( Base `  K ) )
5421, 2, 6latmle2 14276 . . 3  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  B  e.  ( Base `  K )
)  ->  ( ( G  .\/  H )  ./\  B )  .<_  B )
5516, 23, 53, 54syl3anc 1182 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  B )
5649, 55eqbrtrrd 4124 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  D  .<_  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   class class class wbr 4102   ` cfv 5334  (class class class)co 5942   Basecbs 13239   lecple 13306   joincjn 14171   meetcmee 14172   Latclat 14244   Atomscatm 29505   AtLatcal 29506   HLchlt 29592   LLinesclln 29732   LPlanesclpl 29733
This theorem is referenced by:  dalem58  29971  dalem60  29973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-undef 6382  df-riota 6388  df-poset 14173  df-plt 14185  df-lub 14201  df-glb 14202  df-join 14203  df-meet 14204  df-p0 14238  df-lat 14245  df-clat 14307  df-oposet 29418  df-ol 29420  df-oml 29421  df-covers 29508  df-ats 29509  df-atl 29540  df-cvlat 29564  df-hlat 29593  df-llines 29739  df-lplanes 29740  df-lvols 29741
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