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Theorem dalem25 30557
Description: Lemma for dath 30595. Show that the dummy center of perspectivity  c is different from auxiliary atom  G. (Contributed by NM, 3-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
) ) ) )
dalem23.m  |-  ./\  =  ( meet `  K )
dalem23.o  |-  O  =  ( LPlanes `  K )
dalem23.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem23.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem23.g  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
Assertion
Ref Expression
dalem25  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  =/=  G )

Proof of Theorem dalem25
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 )
51, 2, 3, 4dalemcnes 30509 . . 3  |-  ( ph  ->  C  =/=  S )
653ad2ant1 979 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  C  =/=  S )
7 dalem.ps . . . . . . . . . . 11  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
87dalemclccjdd 30547 . . . . . . . . . 10  |-  ( ps 
->  C  .<_  ( c 
.\/  d ) )
983ad2ant3 981 . . . . . . . . 9  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  C  .<_  ( c  .\/  d ) )
109adantr 453 . . . . . . . 8  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  C  .<_  ( c  .\/  d ) )
11 simpr 449 . . . . . . . . . 10  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  c  =  G )
12 dalem23.g . . . . . . . . . . . . 13  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
131dalemkelat 30483 . . . . . . . . . . . . . . 15  |-  ( ph  ->  K  e.  Lat )
14133ad2ant1 979 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
151dalemkehl 30482 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  K  e.  HL )
16153ad2ant1 979 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
177dalemccea 30542 . . . . . . . . . . . . . . . 16  |-  ( ps 
->  c  e.  A
)
18173ad2ant3 981 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
191dalempea 30485 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  P  e.  A )
20193ad2ant1 979 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  e.  A )
21 eqid 2438 . . . . . . . . . . . . . . . 16  |-  ( Base `  K )  =  (
Base `  K )
2221, 3, 4hlatjcl 30226 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  c  e.  A  /\  P  e.  A )  ->  ( c  .\/  P
)  e.  ( Base `  K ) )
2316, 18, 20, 22syl3anc 1185 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  P
)  e.  ( Base `  K ) )
247dalemddea 30543 . . . . . . . . . . . . . . . 16  |-  ( ps 
->  d  e.  A
)
25243ad2ant3 981 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  e.  A )
261dalemsea 30488 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  S  e.  A )
27263ad2ant1 979 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  e.  A )
2821, 3, 4hlatjcl 30226 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  d  e.  A  /\  S  e.  A )  ->  ( d  .\/  S
)  e.  ( Base `  K ) )
2916, 25, 27, 28syl3anc 1185 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .\/  S
)  e.  ( Base `  K ) )
30 dalem23.m . . . . . . . . . . . . . . 15  |-  ./\  =  ( meet `  K )
3121, 2, 30latmle2 14508 . . . . . . . . . . . . . 14  |-  ( ( K  e.  Lat  /\  ( c  .\/  P
)  e.  ( Base `  K )  /\  (
d  .\/  S )  e.  ( Base `  K
) )  ->  (
( c  .\/  P
)  ./\  ( d  .\/  S ) )  .<_  ( d  .\/  S
) )
3214, 23, 29, 31syl3anc 1185 . . . . . . . . . . . . 13  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  P )  ./\  ( d  .\/  S ) )  .<_  ( d  .\/  S
) )
3312, 32syl5eqbr 4247 . . . . . . . . . . . 12  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  .<_  ( d  .\/  S ) )
343, 4hlatjcom 30227 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  d  e.  A  /\  S  e.  A )  ->  ( d  .\/  S
)  =  ( S 
.\/  d ) )
3516, 25, 27, 34syl3anc 1185 . . . . . . . . . . . 12  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .\/  S
)  =  ( S 
.\/  d ) )
3633, 35breqtrd 4238 . . . . . . . . . . 11  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  .<_  ( S  .\/  d ) )
3736adantr 453 . . . . . . . . . 10  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  G  .<_  ( S  .\/  d ) )
3811, 37eqbrtrd 4234 . . . . . . . . 9  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  c  .<_  ( S  .\/  d ) )
392, 3, 4hlatlej2 30235 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  S  e.  A  /\  d  e.  A )  ->  d  .<_  ( S  .\/  d ) )
4016, 27, 25, 39syl3anc 1185 . . . . . . . . . 10  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  .<_  ( S  .\/  d ) )
4140adantr 453 . . . . . . . . 9  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  d  .<_  ( S  .\/  d ) )
427, 4dalemcceb 30548 . . . . . . . . . . . 12  |-  ( ps 
->  c  e.  ( Base `  K ) )
43423ad2ant3 981 . . . . . . . . . . 11  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  ( Base `  K ) )
4421, 4atbase 30149 . . . . . . . . . . . . 13  |-  ( d  e.  A  ->  d  e.  ( Base `  K
) )
4524, 44syl 16 . . . . . . . . . . . 12  |-  ( ps 
->  d  e.  ( Base `  K ) )
46453ad2ant3 981 . . . . . . . . . . 11  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  e.  ( Base `  K ) )
4721, 3, 4hlatjcl 30226 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  S  e.  A  /\  d  e.  A )  ->  ( S  .\/  d
)  e.  ( Base `  K ) )
4816, 27, 25, 47syl3anc 1185 . . . . . . . . . . 11  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( S  .\/  d
)  e.  ( Base `  K ) )
4921, 2, 3latjle12 14493 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( c  e.  (
Base `  K )  /\  d  e.  ( Base `  K )  /\  ( S  .\/  d )  e.  ( Base `  K
) ) )  -> 
( ( c  .<_  ( S  .\/  d )  /\  d  .<_  ( S 
.\/  d ) )  <-> 
( c  .\/  d
)  .<_  ( S  .\/  d ) ) )
5014, 43, 46, 48, 49syl13anc 1187 . . . . . . . . . 10  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .<_  ( S  .\/  d )  /\  d  .<_  ( S 
.\/  d ) )  <-> 
( c  .\/  d
)  .<_  ( S  .\/  d ) ) )
5150adantr 453 . . . . . . . . 9  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  ( (
c  .<_  ( S  .\/  d )  /\  d  .<_  ( S  .\/  d
) )  <->  ( c  .\/  d )  .<_  ( S 
.\/  d ) ) )
5238, 41, 51mpbi2and 889 . . . . . . . 8  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  ( c  .\/  d )  .<_  ( S 
.\/  d ) )
531, 4dalemceb 30497 . . . . . . . . . . 11  |-  ( ph  ->  C  e.  ( Base `  K ) )
54533ad2ant1 979 . . . . . . . . . 10  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  C  e.  ( Base `  K ) )
5521, 3, 4hlatjcl 30226 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  c  e.  A  /\  d  e.  A )  ->  ( c  .\/  d
)  e.  ( Base `  K ) )
5616, 18, 25, 55syl3anc 1185 . . . . . . . . . 10  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  d
)  e.  ( Base `  K ) )
5721, 2lattr 14487 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( C  e.  ( Base `  K )  /\  ( c  .\/  d
)  e.  ( Base `  K )  /\  ( S  .\/  d )  e.  ( Base `  K
) ) )  -> 
( ( C  .<_  ( c  .\/  d )  /\  ( c  .\/  d )  .<_  ( S 
.\/  d ) )  ->  C  .<_  ( S 
.\/  d ) ) )
5814, 54, 56, 48, 57syl13anc 1187 . . . . . . . . 9  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( C  .<_  ( c  .\/  d )  /\  ( c  .\/  d )  .<_  ( S 
.\/  d ) )  ->  C  .<_  ( S 
.\/  d ) ) )
5958adantr 453 . . . . . . . 8  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  ( ( C  .<_  ( c  .\/  d )  /\  (
c  .\/  d )  .<_  ( S  .\/  d
) )  ->  C  .<_  ( S  .\/  d
) ) )
6010, 52, 59mp2and 662 . . . . . . 7  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  C  .<_  ( S  .\/  d ) )
61 dalem23.o . . . . . . . . . . 11  |-  O  =  ( LPlanes `  K )
621, 61dalemyeb 30508 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  ( Base `  K ) )
63623ad2ant1 979 . . . . . . . . 9  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  e.  ( Base `  K ) )
6421, 2, 30latmlem1 14512 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( C  e.  ( Base `  K )  /\  ( S  .\/  d )  e.  ( Base `  K
)  /\  Y  e.  ( Base `  K )
) )  ->  ( C  .<_  ( S  .\/  d )  ->  ( C  ./\  Y )  .<_  ( ( S  .\/  d )  ./\  Y
) ) )
6514, 54, 48, 63, 64syl13anc 1187 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( C  .<_  ( S 
.\/  d )  -> 
( C  ./\  Y
)  .<_  ( ( S 
.\/  d )  ./\  Y ) ) )
6665adantr 453 . . . . . . 7  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  ( C  .<_  ( S  .\/  d
)  ->  ( C  ./\ 
Y )  .<_  ( ( S  .\/  d ) 
./\  Y ) ) )
6760, 66mpd 15 . . . . . 6  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  ( C  ./\ 
Y )  .<_  ( ( S  .\/  d ) 
./\  Y ) )
68 dalem23.y . . . . . . . . . 10  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
69 dalem23.z . . . . . . . . . 10  |-  Z  =  ( ( S  .\/  T )  .\/  U )
701, 2, 3, 4, 61, 68, 69dalem17 30539 . . . . . . . . 9  |-  ( (
ph  /\  Y  =  Z )  ->  C  .<_  Y )
71703adant3 978 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  C  .<_  Y )
7221, 2, 30latleeqm1 14510 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  C  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  ( C  .<_  Y  <->  ( C  ./\ 
Y )  =  C ) )
7314, 54, 63, 72syl3anc 1185 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( C  .<_  Y  <->  ( C  ./\ 
Y )  =  C ) )
7471, 73mpbid 203 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( C  ./\  Y
)  =  C )
7574adantr 453 . . . . . 6  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  ( C  ./\ 
Y )  =  C )
761, 2, 3, 4, 69dalemsly 30514 . . . . . . . . 9  |-  ( (
ph  /\  Y  =  Z )  ->  S  .<_  Y )
77763adant3 978 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  .<_  Y )
787dalem-ddly 30545 . . . . . . . . 9  |-  ( ps 
->  -.  d  .<_  Y )
79783ad2ant3 981 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  d  .<_  Y )
8021, 2, 3, 30, 42atjm 30304 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  d  e.  A  /\  Y  e.  ( Base `  K ) )  /\  ( S  .<_  Y  /\  -.  d  .<_  Y ) )  -> 
( ( S  .\/  d )  ./\  Y
)  =  S )
8116, 27, 25, 63, 77, 79, 80syl132anc 1203 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( S  .\/  d )  ./\  Y
)  =  S )
8281adantr 453 . . . . . 6  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  ( ( S  .\/  d )  ./\  Y )  =  S )
8367, 75, 823brtr3d 4243 . . . . 5  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  C  .<_  S )
84 hlatl 30220 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  AtLat )
8515, 84syl 16 . . . . . . . 8  |-  ( ph  ->  K  e.  AtLat )
861, 2, 3, 4, 61, 68dalemcea 30519 . . . . . . . 8  |-  ( ph  ->  C  e.  A )
872, 4atcmp 30171 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  C  e.  A  /\  S  e.  A )  ->  ( C  .<_  S  <->  C  =  S ) )
8885, 86, 26, 87syl3anc 1185 . . . . . . 7  |-  ( ph  ->  ( C  .<_  S  <->  C  =  S ) )
89883ad2ant1 979 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( C  .<_  S  <->  C  =  S ) )
9089adantr 453 . . . . 5  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  ( C  .<_  S  <->  C  =  S
) )
9183, 90mpbid 203 . . . 4  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  C  =  S )
9291ex 425 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  =  G  ->  C  =  S ) )
9392necon3d 2641 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( C  =/=  S  ->  c  =/=  G ) )
946, 93mpd 15 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  =/=  G )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   meetcmee 14404   Latclat 14476   Atomscatm 30123   AtLatcal 30124   HLchlt 30210   LPlanesclpl 30351
This theorem is referenced by:  dalem28  30559  dalem31N  30562
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-llines 30357  df-lplanes 30358
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