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Theorem dalem25 32696
Description: Lemma for dath 32734. 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 32648 . . 3  |-  ( ph  ->  C  =/=  S )
653ad2ant1 1018 . 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 32686 . . . . . . . . . 10  |-  ( ps 
->  C  .<_  ( c 
.\/  d ) )
983ad2ant3 1020 . . . . . . . . 9  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  C  .<_  ( c  .\/  d ) )
109adantr 463 . . . . . . . 8  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  C  .<_  ( c  .\/  d ) )
11 simpr 459 . . . . . . . . . 10  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  c  =  G )
12 dalem23.g . . . . . . . . . . . . 13  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
131dalemkelat 32622 . . . . . . . . . . . . . . 15  |-  ( ph  ->  K  e.  Lat )
14133ad2ant1 1018 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
151dalemkehl 32621 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  K  e.  HL )
16153ad2ant1 1018 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
177dalemccea 32681 . . . . . . . . . . . . . . . 16  |-  ( ps 
->  c  e.  A
)
18173ad2ant3 1020 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
191dalempea 32624 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  P  e.  A )
20193ad2ant1 1018 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  e.  A )
21 eqid 2402 . . . . . . . . . . . . . . . 16  |-  ( Base `  K )  =  (
Base `  K )
2221, 3, 4hlatjcl 32365 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  c  e.  A  /\  P  e.  A )  ->  ( c  .\/  P
)  e.  ( Base `  K ) )
2316, 18, 20, 22syl3anc 1230 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  P
)  e.  ( Base `  K ) )
247dalemddea 32682 . . . . . . . . . . . . . . . 16  |-  ( ps 
->  d  e.  A
)
25243ad2ant3 1020 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  e.  A )
261dalemsea 32627 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  S  e.  A )
27263ad2ant1 1018 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  e.  A )
2821, 3, 4hlatjcl 32365 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  d  e.  A  /\  S  e.  A )  ->  ( d  .\/  S
)  e.  ( Base `  K ) )
2916, 25, 27, 28syl3anc 1230 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .\/  S
)  e.  ( Base `  K ) )
30 dalem23.m . . . . . . . . . . . . . . 15  |-  ./\  =  ( meet `  K )
3121, 2, 30latmle2 15923 . . . . . . . . . . . . . 14  |-  ( ( K  e.  Lat  /\  ( c  .\/  P
)  e.  ( Base `  K )  /\  (
d  .\/  S )  e.  ( Base `  K
) )  ->  (
( c  .\/  P
)  ./\  ( d  .\/  S ) )  .<_  ( d  .\/  S
) )
3214, 23, 29, 31syl3anc 1230 . . . . . . . . . . . . 13  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  P )  ./\  ( d  .\/  S ) )  .<_  ( d  .\/  S
) )
3312, 32syl5eqbr 4427 . . . . . . . . . . . 12  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  .<_  ( d  .\/  S ) )
343, 4hlatjcom 32366 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  d  e.  A  /\  S  e.  A )  ->  ( d  .\/  S
)  =  ( S 
.\/  d ) )
3516, 25, 27, 34syl3anc 1230 . . . . . . . . . . . 12  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .\/  S
)  =  ( S 
.\/  d ) )
3633, 35breqtrd 4418 . . . . . . . . . . 11  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  .<_  ( S  .\/  d ) )
3736adantr 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  G  .<_  ( S  .\/  d ) )
3811, 37eqbrtrd 4414 . . . . . . . . 9  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  c  .<_  ( S  .\/  d ) )
392, 3, 4hlatlej2 32374 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  S  e.  A  /\  d  e.  A )  ->  d  .<_  ( S  .\/  d ) )
4016, 27, 25, 39syl3anc 1230 . . . . . . . . . 10  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  .<_  ( S  .\/  d ) )
4140adantr 463 . . . . . . . . 9  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  d  .<_  ( S  .\/  d ) )
427, 4dalemcceb 32687 . . . . . . . . . . . 12  |-  ( ps 
->  c  e.  ( Base `  K ) )
43423ad2ant3 1020 . . . . . . . . . . 11  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  ( Base `  K ) )
4421, 4atbase 32288 . . . . . . . . . . . . 13  |-  ( d  e.  A  ->  d  e.  ( Base `  K
) )
4524, 44syl 17 . . . . . . . . . . . 12  |-  ( ps 
->  d  e.  ( Base `  K ) )
46453ad2ant3 1020 . . . . . . . . . . 11  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  e.  ( Base `  K ) )
4721, 3, 4hlatjcl 32365 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  S  e.  A  /\  d  e.  A )  ->  ( S  .\/  d
)  e.  ( Base `  K ) )
4816, 27, 25, 47syl3anc 1230 . . . . . . . . . . 11  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( S  .\/  d
)  e.  ( Base `  K ) )
4921, 2, 3latjle12 15908 . . . . . . . . . . 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 1232 . . . . . . . . . 10  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .<_  ( S  .\/  d )  /\  d  .<_  ( S 
.\/  d ) )  <-> 
( c  .\/  d
)  .<_  ( S  .\/  d ) ) )
5150adantr 463 . . . . . . . . 9  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  ( (
c  .<_  ( S  .\/  d )  /\  d  .<_  ( S  .\/  d
) )  <->  ( c  .\/  d )  .<_  ( S 
.\/  d ) ) )
5238, 41, 51mpbi2and 922 . . . . . . . 8  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  ( c  .\/  d )  .<_  ( S 
.\/  d ) )
531, 4dalemceb 32636 . . . . . . . . . . 11  |-  ( ph  ->  C  e.  ( Base `  K ) )
54533ad2ant1 1018 . . . . . . . . . 10  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  C  e.  ( Base `  K ) )
5521, 3, 4hlatjcl 32365 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  c  e.  A  /\  d  e.  A )  ->  ( c  .\/  d
)  e.  ( Base `  K ) )
5616, 18, 25, 55syl3anc 1230 . . . . . . . . . 10  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  d
)  e.  ( Base `  K ) )
5721, 2lattr 15902 . . . . . . . . . 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 1232 . . . . . . . . 9  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( C  .<_  ( c  .\/  d )  /\  ( c  .\/  d )  .<_  ( S 
.\/  d ) )  ->  C  .<_  ( S 
.\/  d ) ) )
5958adantr 463 . . . . . . . 8  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  ( ( C  .<_  ( c  .\/  d )  /\  (
c  .\/  d )  .<_  ( S  .\/  d
) )  ->  C  .<_  ( S  .\/  d
) ) )
6010, 52, 59mp2and 677 . . . . . . 7  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  C  .<_  ( S  .\/  d ) )
61 dalem23.o . . . . . . . . . . 11  |-  O  =  ( LPlanes `  K )
621, 61dalemyeb 32647 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  ( Base `  K ) )
63623ad2ant1 1018 . . . . . . . . 9  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  e.  ( Base `  K ) )
6421, 2, 30latmlem1 15927 . . . . . . . . 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 1232 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( C  .<_  ( S 
.\/  d )  -> 
( C  ./\  Y
)  .<_  ( ( S 
.\/  d )  ./\  Y ) ) )
6665adantr 463 . . . . . . 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 32678 . . . . . . . . 9  |-  ( (
ph  /\  Y  =  Z )  ->  C  .<_  Y )
71703adant3 1017 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  C  .<_  Y )
7221, 2, 30latleeqm1 15925 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  C  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  ( C  .<_  Y  <->  ( C  ./\ 
Y )  =  C ) )
7314, 54, 63, 72syl3anc 1230 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( C  .<_  Y  <->  ( C  ./\ 
Y )  =  C ) )
7471, 73mpbid 210 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( C  ./\  Y
)  =  C )
7574adantr 463 . . . . . 6  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  ( C  ./\ 
Y )  =  C )
761, 2, 3, 4, 69dalemsly 32653 . . . . . . . . 9  |-  ( (
ph  /\  Y  =  Z )  ->  S  .<_  Y )
77763adant3 1017 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  .<_  Y )
787dalem-ddly 32684 . . . . . . . . 9  |-  ( ps 
->  -.  d  .<_  Y )
79783ad2ant3 1020 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  d  .<_  Y )
8021, 2, 3, 30, 42atjm 32443 . . . . . . . 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 1248 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( S  .\/  d )  ./\  Y
)  =  S )
8281adantr 463 . . . . . 6  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  ( ( S  .\/  d )  ./\  Y )  =  S )
8367, 75, 823brtr3d 4423 . . . . 5  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  C  .<_  S )
84 hlatl 32359 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  AtLat )
8515, 84syl 17 . . . . . . . 8  |-  ( ph  ->  K  e.  AtLat )
861, 2, 3, 4, 61, 68dalemcea 32658 . . . . . . . 8  |-  ( ph  ->  C  e.  A )
872, 4atcmp 32310 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  C  e.  A  /\  S  e.  A )  ->  ( C  .<_  S  <->  C  =  S ) )
8885, 86, 26, 87syl3anc 1230 . . . . . . 7  |-  ( ph  ->  ( C  .<_  S  <->  C  =  S ) )
89883ad2ant1 1018 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( C  .<_  S  <->  C  =  S ) )
9089adantr 463 . . . . 5  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  ( C  .<_  S  <->  C  =  S
) )
9183, 90mpbid 210 . . . 4  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  c  =  G
)  ->  C  =  S )
9291ex 432 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  =  G  ->  C  =  S ) )
9392necon3d 2627 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( C  =/=  S  ->  c  =/=  G ) )
946, 93mpd 15 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  =/=  G )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   class class class wbr 4394   ` cfv 5525  (class class class)co 6234   Basecbs 14733   lecple 14808   joincjn 15789   meetcmee 15790   Latclat 15891   Atomscatm 32262   AtLatcal 32263   HLchlt 32349   LPlanesclpl 32490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-preset 15773  df-poset 15791  df-plt 15804  df-lub 15820  df-glb 15821  df-join 15822  df-meet 15823  df-p0 15885  df-lat 15892  df-clat 15954  df-oposet 32175  df-ol 32177  df-oml 32178  df-covers 32265  df-ats 32266  df-atl 32297  df-cvlat 32321  df-hlat 32350  df-llines 32496  df-lplanes 32497
This theorem is referenced by:  dalem28  32698  dalem31N  32701
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