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Theorem dia2dimlem3 29945
Description: Lemma for dia2dim 29956. Define a translation  D whose trace is atom  V. Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)
Hypotheses
Ref Expression
dia2dimlem3.l  |-  .<_  =  ( le `  K )
dia2dimlem3.j  |-  .\/  =  ( join `  K )
dia2dimlem3.m  |-  ./\  =  ( meet `  K )
dia2dimlem3.a  |-  A  =  ( Atoms `  K )
dia2dimlem3.h  |-  H  =  ( LHyp `  K
)
dia2dimlem3.t  |-  T  =  ( ( LTrn `  K
) `  W )
dia2dimlem3.r  |-  R  =  ( ( trL `  K
) `  W )
dia2dimlem3.q  |-  Q  =  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )
dia2dimlem3.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
dia2dimlem3.u  |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )
dia2dimlem3.v  |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )
dia2dimlem3.p  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
dia2dimlem3.f  |-  ( ph  ->  ( F  e.  T  /\  ( F `  P
)  =/=  P ) )
dia2dimlem3.rf  |-  ( ph  ->  ( R `  F
)  .<_  ( U  .\/  V ) )
dia2dimlem3.uv  |-  ( ph  ->  U  =/=  V )
dia2dimlem3.ru  |-  ( ph  ->  ( R `  F
)  =/=  U )
dia2dimlem3.rv  |-  ( ph  ->  ( R `  F
)  =/=  V )
dia2dimlem3.d  |-  ( ph  ->  D  e.  T )
dia2dimlem3.dv  |-  ( ph  ->  ( D `  Q
)  =  ( F `
 P ) )
Assertion
Ref Expression
dia2dimlem3  |-  ( ph  ->  ( R `  D
)  =  V )

Proof of Theorem dia2dimlem3
StepHypRef Expression
1 dia2dimlem3.k . . . . . . 7  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
21simpld 447 . . . . . 6  |-  ( ph  ->  K  e.  HL )
3 dia2dimlem3.f . . . . . . . . 9  |-  ( ph  ->  ( F  e.  T  /\  ( F `  P
)  =/=  P ) )
43simpld 447 . . . . . . . 8  |-  ( ph  ->  F  e.  T )
5 dia2dimlem3.p . . . . . . . 8  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
6 dia2dimlem3.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
7 dia2dimlem3.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
8 dia2dimlem3.h . . . . . . . . 9  |-  H  =  ( LHyp `  K
)
9 dia2dimlem3.t . . . . . . . . 9  |-  T  =  ( ( LTrn `  K
) `  W )
106, 7, 8, 9ltrnel 29017 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
111, 4, 5, 10syl3anc 1187 . . . . . . 7  |-  ( ph  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
1211simpld 447 . . . . . 6  |-  ( ph  ->  ( F `  P
)  e.  A )
13 dia2dimlem3.v . . . . . . 7  |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )
1413simpld 447 . . . . . 6  |-  ( ph  ->  V  e.  A )
15 dia2dimlem3.j . . . . . . 7  |-  .\/  =  ( join `  K )
166, 15, 7hlatlej2 28254 . . . . . 6  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  V  e.  A )  ->  V  .<_  ( ( F `  P )  .\/  V
) )
172, 12, 14, 16syl3anc 1187 . . . . 5  |-  ( ph  ->  V  .<_  ( ( F `  P )  .\/  V ) )
18 hllat 28242 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
192, 18syl 17 . . . . . 6  |-  ( ph  ->  K  e.  Lat )
20 eqid 2253 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2120, 7atbase 28168 . . . . . . 7  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
2214, 21syl 17 . . . . . 6  |-  ( ph  ->  V  e.  ( Base `  K ) )
2320, 15, 7hlatjcl 28245 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  V  e.  A )  ->  (
( F `  P
)  .\/  V )  e.  ( Base `  K
) )
242, 12, 14, 23syl3anc 1187 . . . . . 6  |-  ( ph  ->  ( ( F `  P )  .\/  V
)  e.  ( Base `  K ) )
25 dia2dimlem3.r . . . . . . . . 9  |-  R  =  ( ( trL `  K
) `  W )
266, 7, 8, 9, 25trlat 29047 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( R `  F )  e.  A
)
271, 5, 3, 26syl3anc 1187 . . . . . . 7  |-  ( ph  ->  ( R `  F
)  e.  A )
28 dia2dimlem3.u . . . . . . . 8  |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )
2928simpld 447 . . . . . . 7  |-  ( ph  ->  U  e.  A )
3020, 15, 7hlatjcl 28245 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( R `  F )  e.  A  /\  U  e.  A )  ->  (
( R `  F
)  .\/  U )  e.  ( Base `  K
) )
312, 27, 29, 30syl3anc 1187 . . . . . 6  |-  ( ph  ->  ( ( R `  F )  .\/  U
)  e.  ( Base `  K ) )
32 dia2dimlem3.m . . . . . . 7  |-  ./\  =  ( meet `  K )
3320, 6, 32latmlem2 14032 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( V  e.  ( Base `  K )  /\  ( ( F `  P )  .\/  V
)  e.  ( Base `  K )  /\  (
( R `  F
)  .\/  U )  e.  ( Base `  K
) ) )  -> 
( V  .<_  ( ( F `  P ) 
.\/  V )  -> 
( ( ( R `
 F )  .\/  U )  ./\  V )  .<_  ( ( ( R `
 F )  .\/  U )  ./\  ( ( F `  P )  .\/  V ) ) ) )
3419, 22, 24, 31, 33syl13anc 1189 . . . . 5  |-  ( ph  ->  ( V  .<_  ( ( F `  P ) 
.\/  V )  -> 
( ( ( R `
 F )  .\/  U )  ./\  V )  .<_  ( ( ( R `
 F )  .\/  U )  ./\  ( ( F `  P )  .\/  V ) ) ) )
3517, 34mpd 16 . . . 4  |-  ( ph  ->  ( ( ( R `
 F )  .\/  U )  ./\  V )  .<_  ( ( ( R `
 F )  .\/  U )  ./\  ( ( F `  P )  .\/  V ) ) )
36 dia2dimlem3.rf . . . . . . 7  |-  ( ph  ->  ( R `  F
)  .<_  ( U  .\/  V ) )
3715, 7hlatjcom 28246 . . . . . . . 8  |-  ( ( K  e.  HL  /\  U  e.  A  /\  V  e.  A )  ->  ( U  .\/  V
)  =  ( V 
.\/  U ) )
382, 29, 14, 37syl3anc 1187 . . . . . . 7  |-  ( ph  ->  ( U  .\/  V
)  =  ( V 
.\/  U ) )
3936, 38breqtrd 3944 . . . . . 6  |-  ( ph  ->  ( R `  F
)  .<_  ( V  .\/  U ) )
40 dia2dimlem3.ru . . . . . . 7  |-  ( ph  ->  ( R `  F
)  =/=  U )
416, 15, 7hlatexch2 28274 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( ( R `  F )  e.  A  /\  V  e.  A  /\  U  e.  A
)  /\  ( R `  F )  =/=  U
)  ->  ( ( R `  F )  .<_  ( V  .\/  U
)  ->  V  .<_  ( ( R `  F
)  .\/  U )
) )
422, 27, 14, 29, 40, 41syl131anc 1200 . . . . . 6  |-  ( ph  ->  ( ( R `  F )  .<_  ( V 
.\/  U )  ->  V  .<_  ( ( R `
 F )  .\/  U ) ) )
4339, 42mpd 16 . . . . 5  |-  ( ph  ->  V  .<_  ( ( R `  F )  .\/  U ) )
4420, 6, 32latleeqm2 14030 . . . . . 6  |-  ( ( K  e.  Lat  /\  V  e.  ( Base `  K )  /\  (
( R `  F
)  .\/  U )  e.  ( Base `  K
) )  ->  ( V  .<_  ( ( R `
 F )  .\/  U )  <->  ( ( ( R `  F ) 
.\/  U )  ./\  V )  =  V ) )
4519, 22, 31, 44syl3anc 1187 . . . . 5  |-  ( ph  ->  ( V  .<_  ( ( R `  F ) 
.\/  U )  <->  ( (
( R `  F
)  .\/  U )  ./\  V )  =  V ) )
4643, 45mpbid 203 . . . 4  |-  ( ph  ->  ( ( ( R `
 F )  .\/  U )  ./\  V )  =  V )
47 dia2dimlem3.d . . . . . 6  |-  ( ph  ->  D  e.  T )
48 dia2dimlem3.q . . . . . . 7  |-  Q  =  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )
49 dia2dimlem3.uv . . . . . . 7  |-  ( ph  ->  U  =/=  V )
506, 15, 32, 7, 8, 9, 25, 48, 1, 28, 13, 5, 3, 36, 49, 40dia2dimlem1 29943 . . . . . 6  |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
516, 15, 32, 7, 8, 9, 25trlval2 29041 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  D  e.  T  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( R `  D )  =  ( ( Q  .\/  ( D `  Q )
)  ./\  W )
)
521, 47, 50, 51syl3anc 1187 . . . . 5  |-  ( ph  ->  ( R `  D
)  =  ( ( Q  .\/  ( D `
 Q ) ) 
./\  W ) )
5348a1i 12 . . . . . . . . 9  |-  ( ph  ->  Q  =  ( ( P  .\/  U ) 
./\  ( ( F `
 P )  .\/  V ) ) )
54 dia2dimlem3.dv . . . . . . . . 9  |-  ( ph  ->  ( D `  Q
)  =  ( F `
 P ) )
5553, 54oveq12d 5728 . . . . . . . 8  |-  ( ph  ->  ( Q  .\/  ( D `  Q )
)  =  ( ( ( P  .\/  U
)  ./\  ( ( F `  P )  .\/  V ) )  .\/  ( F `  P ) ) )
565simpld 447 . . . . . . . . . 10  |-  ( ph  ->  P  e.  A )
5720, 15, 7hlatjcl 28245 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
582, 56, 29, 57syl3anc 1187 . . . . . . . . 9  |-  ( ph  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
596, 15, 7hlatlej1 28253 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  V  e.  A )  ->  ( F `  P )  .<_  ( ( F `  P )  .\/  V
) )
602, 12, 14, 59syl3anc 1187 . . . . . . . . 9  |-  ( ph  ->  ( F `  P
)  .<_  ( ( F `
 P )  .\/  V ) )
6120, 6, 15, 32, 7atmod4i1 28744 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( ( F `  P )  e.  A  /\  ( P  .\/  U
)  e.  ( Base `  K )  /\  (
( F `  P
)  .\/  V )  e.  ( Base `  K
) )  /\  ( F `  P )  .<_  ( ( F `  P )  .\/  V
) )  ->  (
( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )  .\/  ( F `  P ) )  =  ( ( ( P  .\/  U
)  .\/  ( F `  P ) )  ./\  ( ( F `  P )  .\/  V
) ) )
622, 12, 58, 24, 60, 61syl131anc 1200 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
.\/  U )  ./\  ( ( F `  P )  .\/  V
) )  .\/  ( F `  P )
)  =  ( ( ( P  .\/  U
)  .\/  ( F `  P ) )  ./\  ( ( F `  P )  .\/  V
) ) )
6315, 7hlatj32 28250 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  U  e.  A  /\  ( F `  P
)  e.  A ) )  ->  ( ( P  .\/  U )  .\/  ( F `  P ) )  =  ( ( P  .\/  ( F `
 P ) ) 
.\/  U ) )
642, 56, 29, 12, 63syl13anc 1189 . . . . . . . . 9  |-  ( ph  ->  ( ( P  .\/  U )  .\/  ( F `
 P ) )  =  ( ( P 
.\/  ( F `  P ) )  .\/  U ) )
6564oveq1d 5725 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
.\/  U )  .\/  ( F `  P ) )  ./\  ( ( F `  P )  .\/  V ) )  =  ( ( ( P 
.\/  ( F `  P ) )  .\/  U )  ./\  ( ( F `  P )  .\/  V ) ) )
6655, 62, 653eqtrd 2289 . . . . . . 7  |-  ( ph  ->  ( Q  .\/  ( D `  Q )
)  =  ( ( ( P  .\/  ( F `  P )
)  .\/  U )  ./\  ( ( F `  P )  .\/  V
) ) )
6766oveq1d 5725 . . . . . 6  |-  ( ph  ->  ( ( Q  .\/  ( D `  Q ) )  ./\  W )  =  ( ( ( ( P  .\/  ( F `  P )
)  .\/  U )  ./\  ( ( F `  P )  .\/  V
) )  ./\  W
) )
68 hlol 28240 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OL )
692, 68syl 17 . . . . . . 7  |-  ( ph  ->  K  e.  OL )
7020, 15, 7hlatjcl 28245 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  P )  e.  A )  -> 
( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
712, 56, 12, 70syl3anc 1187 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
7220, 7atbase 28168 . . . . . . . . 9  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
7329, 72syl 17 . . . . . . . 8  |-  ( ph  ->  U  e.  ( Base `  K ) )
7420, 15latjcl 14000 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  ( F `
 P ) )  e.  ( Base `  K
)  /\  U  e.  ( Base `  K )
)  ->  ( ( P  .\/  ( F `  P ) )  .\/  U )  e.  ( Base `  K ) )
7519, 71, 73, 74syl3anc 1187 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  ( F `  P ) )  .\/  U )  e.  ( Base `  K
) )
761simprd 451 . . . . . . . 8  |-  ( ph  ->  W  e.  H )
7720, 8lhpbase 28876 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
7876, 77syl 17 . . . . . . 7  |-  ( ph  ->  W  e.  ( Base `  K ) )
7920, 32latm32 28110 . . . . . . 7  |-  ( ( K  e.  OL  /\  ( ( ( P 
.\/  ( F `  P ) )  .\/  U )  e.  ( Base `  K )  /\  (
( F `  P
)  .\/  V )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
) )  ->  (
( ( ( P 
.\/  ( F `  P ) )  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )  ./\  W )  =  ( ( ( ( P  .\/  ( F `  P ) )  .\/  U ) 
./\  W )  ./\  ( ( F `  P )  .\/  V
) ) )
8069, 75, 24, 78, 79syl13anc 1189 . . . . . 6  |-  ( ph  ->  ( ( ( ( P  .\/  ( F `
 P ) ) 
.\/  U )  ./\  ( ( F `  P )  .\/  V
) )  ./\  W
)  =  ( ( ( ( P  .\/  ( F `  P ) )  .\/  U ) 
./\  W )  ./\  ( ( F `  P )  .\/  V
) ) )
816, 15, 32, 7, 8, 9, 25trlval2 29041 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  F )  =  ( ( P  .\/  ( F `  P )
)  ./\  W )
)
821, 4, 5, 81syl3anc 1187 . . . . . . . . 9  |-  ( ph  ->  ( R `  F
)  =  ( ( P  .\/  ( F `
 P ) ) 
./\  W ) )
8382oveq1d 5725 . . . . . . . 8  |-  ( ph  ->  ( ( R `  F )  .\/  U
)  =  ( ( ( P  .\/  ( F `  P )
)  ./\  W )  .\/  U ) )
8428simprd 451 . . . . . . . . 9  |-  ( ph  ->  U  .<_  W )
8520, 6, 15, 32, 7atmod4i1 28744 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( U  e.  A  /\  ( P  .\/  ( F `  P )
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  U  .<_  W )  ->  (
( ( P  .\/  ( F `  P ) )  ./\  W )  .\/  U )  =  ( ( ( P  .\/  ( F `  P ) )  .\/  U ) 
./\  W ) )
862, 29, 71, 78, 84, 85syl131anc 1200 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
.\/  ( F `  P ) )  ./\  W )  .\/  U )  =  ( ( ( P  .\/  ( F `
 P ) ) 
.\/  U )  ./\  W ) )
8783, 86eqtr2d 2286 . . . . . . 7  |-  ( ph  ->  ( ( ( P 
.\/  ( F `  P ) )  .\/  U )  ./\  W )  =  ( ( R `
 F )  .\/  U ) )
8887oveq1d 5725 . . . . . 6  |-  ( ph  ->  ( ( ( ( P  .\/  ( F `
 P ) ) 
.\/  U )  ./\  W )  ./\  ( ( F `  P )  .\/  V ) )  =  ( ( ( R `
 F )  .\/  U )  ./\  ( ( F `  P )  .\/  V ) ) )
8967, 80, 883eqtrd 2289 . . . . 5  |-  ( ph  ->  ( ( Q  .\/  ( D `  Q ) )  ./\  W )  =  ( ( ( R `  F ) 
.\/  U )  ./\  ( ( F `  P )  .\/  V
) ) )
9052, 89eqtr2d 2286 . . . 4  |-  ( ph  ->  ( ( ( R `
 F )  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )  =  ( R `  D
) )
9135, 46, 903brtr3d 3949 . . 3  |-  ( ph  ->  V  .<_  ( R `  D ) )
92 hlatl 28239 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
932, 92syl 17 . . . 4  |-  ( ph  ->  K  e.  AtLat )
94 hlop 28241 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OP )
952, 94syl 17 . . . . . . . . 9  |-  ( ph  ->  K  e.  OP )
96 eqid 2253 . . . . . . . . . 10  |-  ( 0.
`  K )  =  ( 0. `  K
)
97 eqid 2253 . . . . . . . . . 10  |-  ( lt
`  K )  =  ( lt `  K
)
9896, 97, 70ltat 28170 . . . . . . . . 9  |-  ( ( K  e.  OP  /\  V  e.  A )  ->  ( 0. `  K
) ( lt `  K ) V )
9995, 14, 98syl2anc 645 . . . . . . . 8  |-  ( ph  ->  ( 0. `  K
) ( lt `  K ) V )
100 hlpos 28244 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  Poset )
1012, 100syl 17 . . . . . . . . 9  |-  ( ph  ->  K  e.  Poset )
10220, 96op0cl 28063 . . . . . . . . . 10  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  ( Base `  K
) )
10395, 102syl 17 . . . . . . . . 9  |-  ( ph  ->  ( 0. `  K
)  e.  ( Base `  K ) )
10420, 8, 9, 25trlcl 29042 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  D  e.  T
)  ->  ( R `  D )  e.  (
Base `  K )
)
1051, 47, 104syl2anc 645 . . . . . . . . 9  |-  ( ph  ->  ( R `  D
)  e.  ( Base `  K ) )
10620, 6, 97pltletr 13949 . . . . . . . . 9  |-  ( ( K  e.  Poset  /\  (
( 0. `  K
)  e.  ( Base `  K )  /\  V  e.  ( Base `  K
)  /\  ( R `  D )  e.  (
Base `  K )
) )  ->  (
( ( 0. `  K ) ( lt
`  K ) V  /\  V  .<_  ( R `
 D ) )  ->  ( 0. `  K ) ( lt
`  K ) ( R `  D ) ) )
107101, 103, 22, 105, 106syl13anc 1189 . . . . . . . 8  |-  ( ph  ->  ( ( ( 0.
`  K ) ( lt `  K ) V  /\  V  .<_  ( R `  D ) )  ->  ( 0. `  K ) ( lt
`  K ) ( R `  D ) ) )
10899, 91, 107mp2and 663 . . . . . . 7  |-  ( ph  ->  ( 0. `  K
) ( lt `  K ) ( R `
 D ) )
10920, 97, 96opltn0 28069 . . . . . . . 8  |-  ( ( K  e.  OP  /\  ( R `  D )  e.  ( Base `  K
) )  ->  (
( 0. `  K
) ( lt `  K ) ( R `
 D )  <->  ( R `  D )  =/=  ( 0. `  K ) ) )
11095, 105, 109syl2anc 645 . . . . . . 7  |-  ( ph  ->  ( ( 0. `  K ) ( lt
`  K ) ( R `  D )  <-> 
( R `  D
)  =/=  ( 0.
`  K ) ) )
111108, 110mpbid 203 . . . . . 6  |-  ( ph  ->  ( R `  D
)  =/=  ( 0.
`  K ) )
112111neneqd 2428 . . . . 5  |-  ( ph  ->  -.  ( R `  D )  =  ( 0. `  K ) )
11396, 7, 8, 9, 25trlator0 29049 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  D  e.  T
)  ->  ( ( R `  D )  e.  A  \/  ( R `  D )  =  ( 0. `  K ) ) )
1141, 47, 113syl2anc 645 . . . . . . 7  |-  ( ph  ->  ( ( R `  D )  e.  A  \/  ( R `  D
)  =  ( 0.
`  K ) ) )
115114orcomd 379 . . . . . 6  |-  ( ph  ->  ( ( R `  D )  =  ( 0. `  K )  \/  ( R `  D )  e.  A
) )
116115ord 368 . . . . 5  |-  ( ph  ->  ( -.  ( R `
 D )  =  ( 0. `  K
)  ->  ( R `  D )  e.  A
) )
117112, 116mpd 16 . . . 4  |-  ( ph  ->  ( R `  D
)  e.  A )
1186, 7atcmp 28190 . . . 4  |-  ( ( K  e.  AtLat  /\  V  e.  A  /\  ( R `  D )  e.  A )  ->  ( V  .<_  ( R `  D )  <->  V  =  ( R `  D ) ) )
11993, 14, 117, 118syl3anc 1187 . . 3  |-  ( ph  ->  ( V  .<_  ( R `
 D )  <->  V  =  ( R `  D ) ) )
12091, 119mpbid 203 . 2  |-  ( ph  ->  V  =  ( R `
 D ) )
121120eqcomd 2258 1  |-  ( ph  ->  ( R `  D
)  =  V )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2412   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   Basecbs 13022   lecple 13089   Posetcpo 13918   ltcplt 13919   joincjn 13922   meetcmee 13923   0.cp0 13987   Latclat 13995   OPcops 28051   OLcol 28053   Atomscatm 28142   AtLatcal 28143   HLchlt 28229   LHypclh 28862   LTrncltrn 28979   trLctrl 29036
This theorem is referenced by:  dia2dimlem5  29947
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-map 6660  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-llines 28376  df-psubsp 28381  df-pmap 28382  df-padd 28674  df-lhyp 28866  df-laut 28867  df-ldil 28982  df-ltrn 28983  df-trl 29037
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