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Theorem dalem60 29921
Description: Lemma for dath 29925. 
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 29918 . . 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 29919 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  E  .<_  B )
181dalemkelat 29813 . . . . 5  |-  ( ph  ->  K  e.  Lat )
19183ad2ant1 976 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
201, 2, 3, 4, 6, 7, 8, 9, 10dalemdea 29851 . . . . . 6  |-  ( ph  ->  D  e.  A )
21 eqid 2283 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2221, 4atbase 29479 . . . . . 6  |-  ( D  e.  A  ->  D  e.  ( Base `  K
) )
2320, 22syl 15 . . . . 5  |-  ( ph  ->  D  e.  ( Base `  K ) )
24233ad2ant1 976 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  D  e.  ( Base `  K ) )
251, 2, 3, 4, 6, 7, 8, 9, 16dalemeea 29852 . . . . . 6  |-  ( ph  ->  E  e.  A )
2621, 4atbase 29479 . . . . . 6  |-  ( E  e.  A  ->  E  e.  ( Base `  K
) )
2725, 26syl 15 . . . . 5  |-  ( ph  ->  E  e.  ( Base `  K ) )
28273ad2ant1 976 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  E  e.  ( Base `  K ) )
29 eqid 2283 . . . . . 6  |-  ( LLines `  K )  =  (
LLines `  K )
301, 2, 3, 4, 5, 6, 29, 7, 8, 9, 11, 12, 13, 14dalem53 29914 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( LLines `  K ) )
3121, 29llnbase 29698 . . . . 5  |-  ( B  e.  ( LLines `  K
)  ->  B  e.  ( Base `  K )
)
3230, 31syl 15 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( Base `  K ) )
3321, 2, 3latjle12 14168 . . . 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 1184 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( D  .<_  B  /\  E  .<_  B )  <-> 
( D  .\/  E
)  .<_  B ) )
3515, 17, 34mpbi2and 887 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( D  .\/  E
)  .<_  B )
361dalemkehl 29812 . . . 4  |-  ( ph  ->  K  e.  HL )
37363ad2ant1 976 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
381, 2, 3, 4, 6, 7, 8, 9, 10, 16dalemdnee 29855 . . . . 5  |-  ( ph  ->  D  =/=  E )
393, 4, 29llni2 29701 . . . . 5  |-  ( ( ( K  e.  HL  /\  D  e.  A  /\  E  e.  A )  /\  D  =/=  E
)  ->  ( D  .\/  E )  e.  (
LLines `  K ) )
4036, 20, 25, 38, 39syl31anc 1185 . . . 4  |-  ( ph  ->  ( D  .\/  E
)  e.  ( LLines `  K ) )
41403ad2ant1 976 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( D  .\/  E
)  e.  ( LLines `  K ) )
422, 29llncmp 29711 . . 3  |-  ( ( K  e.  HL  /\  ( D  .\/  E )  e.  ( LLines `  K
)  /\  B  e.  ( LLines `  K )
)  ->  ( ( D  .\/  E )  .<_  B 
<->  ( D  .\/  E
)  =  B ) )
4337, 41, 30, 42syl3anc 1182 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( D  .\/  E )  .<_  B  <->  ( D  .\/  E )  =  B ) )
4435, 43mpbid 201 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( D  .\/  E
)  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Latclat 14151   Atomscatm 29453   HLchlt 29540   LLinesclln 29680   LPlanesclpl 29681
This theorem is referenced by:  dalem61  29922
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689
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