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Theorem dalem60 35578
Description: Lemma for dath 35582. 
B is an axis of perspectivity (almost). (Contributed by NM, 11-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
) ) ) )
dalem60.m  |-  ./\  =  ( meet `  K )
dalem60.o  |-  O  =  ( LPlanes `  K )
dalem60.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem60.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem60.d  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
dalem60.e  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
dalem60.g  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
dalem60.h  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
dalem60.i  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
dalem60.b1  |-  B  =  ( ( ( G 
.\/  H )  .\/  I )  ./\  Y
)
Assertion
Ref Expression
dalem60  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( D  .\/  E
)  =  B )

Proof of Theorem dalem60
StepHypRef Expression
1 dalem.ph . . . 4  |-  ( 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 . . . 4  |-  .<_  =  ( le `  K )
3 dalem.j . . . 4  |-  .\/  =  ( join `  K )
4 dalem.a . . . 4  |-  A  =  ( Atoms `  K )
5 dalem.ps . . . 4  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
6 dalem60.m . . . 4  |-  ./\  =  ( meet `  K )
7 dalem60.o . . . 4  |-  O  =  ( LPlanes `  K )
8 dalem60.y . . . 4  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
9 dalem60.z . . . 4  |-  Z  =  ( ( S  .\/  T )  .\/  U )
10 dalem60.d . . . 4  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
11 dalem60.g . . . 4  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
12 dalem60.h . . . 4  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
13 dalem60.i . . . 4  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
14 dalem60.b1 . . . 4  |-  B  =  ( ( ( G 
.\/  H )  .\/  I )  ./\  Y
)
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14dalem57 35575 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  D  .<_  B )
16 dalem60.e . . . 4  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
171, 2, 3, 4, 5, 6, 7, 8, 9, 16, 11, 12, 13, 14dalem58 35576 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  E  .<_  B )
181dalemkelat 35470 . . . . 5  |-  ( ph  ->  K  e.  Lat )
19183ad2ant1 1017 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
201, 2, 3, 4, 6, 7, 8, 9, 10dalemdea 35508 . . . . . 6  |-  ( ph  ->  D  e.  A )
21 eqid 2457 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2221, 4atbase 35136 . . . . . 6  |-  ( D  e.  A  ->  D  e.  ( Base `  K
) )
2320, 22syl 16 . . . . 5  |-  ( ph  ->  D  e.  ( Base `  K ) )
24233ad2ant1 1017 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  D  e.  ( Base `  K ) )
251, 2, 3, 4, 6, 7, 8, 9, 16dalemeea 35509 . . . . . 6  |-  ( ph  ->  E  e.  A )
2621, 4atbase 35136 . . . . . 6  |-  ( E  e.  A  ->  E  e.  ( Base `  K
) )
2725, 26syl 16 . . . . 5  |-  ( ph  ->  E  e.  ( Base `  K ) )
28273ad2ant1 1017 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  E  e.  ( Base `  K ) )
29 eqid 2457 . . . . . 6  |-  ( LLines `  K )  =  (
LLines `  K )
301, 2, 3, 4, 5, 6, 29, 7, 8, 9, 11, 12, 13, 14dalem53 35571 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( LLines `  K ) )
3121, 29llnbase 35355 . . . . 5  |-  ( B  e.  ( LLines `  K
)  ->  B  e.  ( Base `  K )
)
3230, 31syl 16 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( Base `  K ) )
3321, 2, 3latjle12 15819 . . . 4  |-  ( ( K  e.  Lat  /\  ( D  e.  ( Base `  K )  /\  E  e.  ( Base `  K )  /\  B  e.  ( Base `  K
) ) )  -> 
( ( D  .<_  B  /\  E  .<_  B )  <-> 
( D  .\/  E
)  .<_  B ) )
3419, 24, 28, 32, 33syl13anc 1230 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( D  .<_  B  /\  E  .<_  B )  <-> 
( D  .\/  E
)  .<_  B ) )
3515, 17, 34mpbi2and 921 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( D  .\/  E
)  .<_  B )
361dalemkehl 35469 . . . 4  |-  ( ph  ->  K  e.  HL )
37363ad2ant1 1017 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
381, 2, 3, 4, 6, 7, 8, 9, 10, 16dalemdnee 35512 . . . . 5  |-  ( ph  ->  D  =/=  E )
393, 4, 29llni2 35358 . . . . 5  |-  ( ( ( K  e.  HL  /\  D  e.  A  /\  E  e.  A )  /\  D  =/=  E
)  ->  ( D  .\/  E )  e.  (
LLines `  K ) )
4036, 20, 25, 38, 39syl31anc 1231 . . . 4  |-  ( ph  ->  ( D  .\/  E
)  e.  ( LLines `  K ) )
41403ad2ant1 1017 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( D  .\/  E
)  e.  ( LLines `  K ) )
422, 29llncmp 35368 . . 3  |-  ( ( K  e.  HL  /\  ( D  .\/  E )  e.  ( LLines `  K
)  /\  B  e.  ( LLines `  K )
)  ->  ( ( D  .\/  E )  .<_  B 
<->  ( D  .\/  E
)  =  B ) )
4337, 41, 30, 42syl3anc 1228 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( D  .\/  E )  .<_  B  <->  ( D  .\/  E )  =  B ) )
4435, 43mpbid 210 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( D  .\/  E
)  =  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Basecbs 14644   lecple 14719   joincjn 15700   meetcmee 15701   Latclat 15802   Atomscatm 35110   HLchlt 35197   LLinesclln 35337   LPlanesclpl 35338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-preset 15684  df-poset 15702  df-plt 15715  df-lub 15731  df-glb 15732  df-join 15733  df-meet 15734  df-p0 15796  df-lat 15803  df-clat 15865  df-oposet 35023  df-ol 35025  df-oml 35026  df-covers 35113  df-ats 35114  df-atl 35145  df-cvlat 35169  df-hlat 35198  df-llines 35344  df-lplanes 35345  df-lvols 35346
This theorem is referenced by:  dalem61  35579
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