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Theorem dia2dimlem3 34067
Description: Lemma for dia2dim 34078. 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 457 . . . . . 6  |-  ( ph  ->  K  e.  HL )
3 dia2dimlem3.f . . . . . . . . 9  |-  ( ph  ->  ( F  e.  T  /\  ( F `  P
)  =/=  P ) )
43simpld 457 . . . . . . . 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 33137 . . . . . . . 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 1230 . . . . . . 7  |-  ( ph  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
1211simpld 457 . . . . . 6  |-  ( ph  ->  ( F `  P
)  e.  A )
13 dia2dimlem3.v . . . . . . 7  |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )
1413simpld 457 . . . . . 6  |-  ( ph  ->  V  e.  A )
15 dia2dimlem3.j . . . . . . 7  |-  .\/  =  ( join `  K )
166, 15, 7hlatlej2 32374 . . . . . 6  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  V  e.  A )  ->  V  .<_  ( ( F `  P )  .\/  V
) )
172, 12, 14, 16syl3anc 1230 . . . . 5  |-  ( ph  ->  V  .<_  ( ( F `  P )  .\/  V ) )
18 hllat 32362 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
192, 18syl 17 . . . . . 6  |-  ( ph  ->  K  e.  Lat )
20 eqid 2402 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2120, 7atbase 32288 . . . . . . 7  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
2214, 21syl 17 . . . . . 6  |-  ( ph  ->  V  e.  ( Base `  K ) )
2320, 15, 7hlatjcl 32365 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  V  e.  A )  ->  (
( F `  P
)  .\/  V )  e.  ( Base `  K
) )
242, 12, 14, 23syl3anc 1230 . . . . . 6  |-  ( ph  ->  ( ( F `  P )  .\/  V
)  e.  ( Base `  K ) )
25 dia2dimlem3.r . . . . . . . . 9  |-  R  =  ( ( trL `  K
) `  W )
266, 7, 8, 9, 25trlat 33168 . . . . . . . 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 1230 . . . . . . 7  |-  ( ph  ->  ( R `  F
)  e.  A )
28 dia2dimlem3.u . . . . . . . 8  |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )
2928simpld 457 . . . . . . 7  |-  ( ph  ->  U  e.  A )
3020, 15, 7hlatjcl 32365 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( R `  F )  e.  A  /\  U  e.  A )  ->  (
( R `  F
)  .\/  U )  e.  ( Base `  K
) )
312, 27, 29, 30syl3anc 1230 . . . . . 6  |-  ( ph  ->  ( ( R `  F )  .\/  U
)  e.  ( Base `  K ) )
32 dia2dimlem3.m . . . . . . 7  |-  ./\  =  ( meet `  K )
3320, 6, 32latmlem2 15928 . . . . . 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 1232 . . . . 5  |-  ( ph  ->  ( V  .<_  ( ( F `  P ) 
.\/  V )  -> 
( ( ( R `
 F )  .\/  U )  ./\  V )  .<_  ( ( ( R `
 F )  .\/  U )  ./\  ( ( F `  P )  .\/  V ) ) ) )
3517, 34mpd 15 . . . 4  |-  ( ph  ->  ( ( ( R `
 F )  .\/  U )  ./\  V )  .<_  ( ( ( R `
 F )  .\/  U )  ./\  ( ( F `  P )  .\/  V ) ) )
36 dia2dimlem3.rf . . . . . . 7  |-  ( ph  ->  ( R `  F
)  .<_  ( U  .\/  V ) )
3715, 7hlatjcom 32366 . . . . . . . 8  |-  ( ( K  e.  HL  /\  U  e.  A  /\  V  e.  A )  ->  ( U  .\/  V
)  =  ( V 
.\/  U ) )
382, 29, 14, 37syl3anc 1230 . . . . . . 7  |-  ( ph  ->  ( U  .\/  V
)  =  ( V 
.\/  U ) )
3936, 38breqtrd 4418 . . . . . 6  |-  ( ph  ->  ( R `  F
)  .<_  ( V  .\/  U ) )
40 dia2dimlem3.ru . . . . . . 7  |-  ( ph  ->  ( R `  F
)  =/=  U )
416, 15, 7hlatexch2 32394 . . . . . . 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 1243 . . . . . 6  |-  ( ph  ->  ( ( R `  F )  .<_  ( V 
.\/  U )  ->  V  .<_  ( ( R `
 F )  .\/  U ) ) )
4339, 42mpd 15 . . . . 5  |-  ( ph  ->  V  .<_  ( ( R `  F )  .\/  U ) )
4420, 6, 32latleeqm2 15926 . . . . . 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 1230 . . . . 5  |-  ( ph  ->  ( V  .<_  ( ( R `  F ) 
.\/  U )  <->  ( (
( R `  F
)  .\/  U )  ./\  V )  =  V ) )
4643, 45mpbid 210 . . . 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 34065 . . . . . 6  |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
516, 15, 32, 7, 8, 9, 25trlval2 33162 . . . . . 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 1230 . . . . 5  |-  ( ph  ->  ( R `  D
)  =  ( ( Q  .\/  ( D `
 Q ) ) 
./\  W ) )
5348a1i 11 . . . . . . . . 9  |-  ( ph  ->  Q  =  ( ( P  .\/  U ) 
./\  ( ( F `
 P )  .\/  V ) ) )
54 dia2dimlem3.dv . . . . . . . . 9  |-  ( ph  ->  ( D `  Q
)  =  ( F `
 P ) )
5553, 54oveq12d 6252 . . . . . . . 8  |-  ( ph  ->  ( Q  .\/  ( D `  Q )
)  =  ( ( ( P  .\/  U
)  ./\  ( ( F `  P )  .\/  V ) )  .\/  ( F `  P ) ) )
565simpld 457 . . . . . . . . . 10  |-  ( ph  ->  P  e.  A )
5720, 15, 7hlatjcl 32365 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
582, 56, 29, 57syl3anc 1230 . . . . . . . . 9  |-  ( ph  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
596, 15, 7hlatlej1 32373 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  V  e.  A )  ->  ( F `  P )  .<_  ( ( F `  P )  .\/  V
) )
602, 12, 14, 59syl3anc 1230 . . . . . . . . 9  |-  ( ph  ->  ( F `  P
)  .<_  ( ( F `
 P )  .\/  V ) )
6120, 6, 15, 32, 7atmod4i1 32864 . . . . . . . . 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 1243 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
.\/  U )  ./\  ( ( F `  P )  .\/  V
) )  .\/  ( F `  P )
)  =  ( ( ( P  .\/  U
)  .\/  ( F `  P ) )  ./\  ( ( F `  P )  .\/  V
) ) )
6315, 7hlatj32 32370 . . . . . . . . . 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 1232 . . . . . . . . 9  |-  ( ph  ->  ( ( P  .\/  U )  .\/  ( F `
 P ) )  =  ( ( P 
.\/  ( F `  P ) )  .\/  U ) )
6564oveq1d 6249 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
.\/  U )  .\/  ( F `  P ) )  ./\  ( ( F `  P )  .\/  V ) )  =  ( ( ( P 
.\/  ( F `  P ) )  .\/  U )  ./\  ( ( F `  P )  .\/  V ) ) )
6655, 62, 653eqtrd 2447 . . . . . . 7  |-  ( ph  ->  ( Q  .\/  ( D `  Q )
)  =  ( ( ( P  .\/  ( F `  P )
)  .\/  U )  ./\  ( ( F `  P )  .\/  V
) ) )
6766oveq1d 6249 . . . . . 6  |-  ( ph  ->  ( ( Q  .\/  ( D `  Q ) )  ./\  W )  =  ( ( ( ( P  .\/  ( F `  P )
)  .\/  U )  ./\  ( ( F `  P )  .\/  V
) )  ./\  W
) )
68 hlol 32360 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OL )
692, 68syl 17 . . . . . . 7  |-  ( ph  ->  K  e.  OL )
7020, 15, 7hlatjcl 32365 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  P )  e.  A )  -> 
( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
712, 56, 12, 70syl3anc 1230 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
7220, 7atbase 32288 . . . . . . . . 9  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
7329, 72syl 17 . . . . . . . 8  |-  ( ph  ->  U  e.  ( Base `  K ) )
7420, 15latjcl 15897 . . . . . . . 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 1230 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  ( F `  P ) )  .\/  U )  e.  ( Base `  K
) )
761simprd 461 . . . . . . . 8  |-  ( ph  ->  W  e.  H )
7720, 8lhpbase 32996 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
7876, 77syl 17 . . . . . . 7  |-  ( ph  ->  W  e.  ( Base `  K ) )
7920, 32latm32 32230 . . . . . . 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 1232 . . . . . 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 33162 . . . . . . . . . 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 1230 . . . . . . . . 9  |-  ( ph  ->  ( R `  F
)  =  ( ( P  .\/  ( F `
 P ) ) 
./\  W ) )
8382oveq1d 6249 . . . . . . . 8  |-  ( ph  ->  ( ( R `  F )  .\/  U
)  =  ( ( ( P  .\/  ( F `  P )
)  ./\  W )  .\/  U ) )
8428simprd 461 . . . . . . . . 9  |-  ( ph  ->  U  .<_  W )
8520, 6, 15, 32, 7atmod4i1 32864 . . . . . . . . 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 1243 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
.\/  ( F `  P ) )  ./\  W )  .\/  U )  =  ( ( ( P  .\/  ( F `
 P ) ) 
.\/  U )  ./\  W ) )
8783, 86eqtr2d 2444 . . . . . . 7  |-  ( ph  ->  ( ( ( P 
.\/  ( F `  P ) )  .\/  U )  ./\  W )  =  ( ( R `
 F )  .\/  U ) )
8887oveq1d 6249 . . . . . 6  |-  ( ph  ->  ( ( ( ( P  .\/  ( F `
 P ) ) 
.\/  U )  ./\  W )  ./\  ( ( F `  P )  .\/  V ) )  =  ( ( ( R `
 F )  .\/  U )  ./\  ( ( F `  P )  .\/  V ) ) )
8967, 80, 883eqtrd 2447 . . . . 5  |-  ( ph  ->  ( ( Q  .\/  ( D `  Q ) )  ./\  W )  =  ( ( ( R `  F ) 
.\/  U )  ./\  ( ( F `  P )  .\/  V
) ) )
9052, 89eqtr2d 2444 . . . 4  |-  ( ph  ->  ( ( ( R `
 F )  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )  =  ( R `  D
) )
9135, 46, 903brtr3d 4423 . . 3  |-  ( ph  ->  V  .<_  ( R `  D ) )
92 hlatl 32359 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
932, 92syl 17 . . . 4  |-  ( ph  ->  K  e.  AtLat )
94 hlop 32361 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OP )
952, 94syl 17 . . . . . . . . 9  |-  ( ph  ->  K  e.  OP )
96 eqid 2402 . . . . . . . . . 10  |-  ( 0.
`  K )  =  ( 0. `  K
)
97 eqid 2402 . . . . . . . . . 10  |-  ( lt
`  K )  =  ( lt `  K
)
9896, 97, 70ltat 32290 . . . . . . . . 9  |-  ( ( K  e.  OP  /\  V  e.  A )  ->  ( 0. `  K
) ( lt `  K ) V )
9995, 14, 98syl2anc 659 . . . . . . . 8  |-  ( ph  ->  ( 0. `  K
) ( lt `  K ) V )
100 hlpos 32364 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  Poset )
1012, 100syl 17 . . . . . . . . 9  |-  ( ph  ->  K  e.  Poset )
10220, 96op0cl 32183 . . . . . . . . . 10  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  ( Base `  K
) )
10395, 102syl 17 . . . . . . . . 9  |-  ( ph  ->  ( 0. `  K
)  e.  ( Base `  K ) )
10420, 8, 9, 25trlcl 33163 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  D  e.  T
)  ->  ( R `  D )  e.  (
Base `  K )
)
1051, 47, 104syl2anc 659 . . . . . . . . 9  |-  ( ph  ->  ( R `  D
)  e.  ( Base `  K ) )
10620, 6, 97pltletr 15817 . . . . . . . . 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 1232 . . . . . . . 8  |-  ( ph  ->  ( ( ( 0.
`  K ) ( lt `  K ) V  /\  V  .<_  ( R `  D ) )  ->  ( 0. `  K ) ( lt
`  K ) ( R `  D ) ) )
10899, 91, 107mp2and 677 . . . . . . 7  |-  ( ph  ->  ( 0. `  K
) ( lt `  K ) ( R `
 D ) )
10920, 97, 96opltn0 32189 . . . . . . . 8  |-  ( ( K  e.  OP  /\  ( R `  D )  e.  ( Base `  K
) )  ->  (
( 0. `  K
) ( lt `  K ) ( R `
 D )  <->  ( R `  D )  =/=  ( 0. `  K ) ) )
11095, 105, 109syl2anc 659 . . . . . . 7  |-  ( ph  ->  ( ( 0. `  K ) ( lt
`  K ) ( R `  D )  <-> 
( R `  D
)  =/=  ( 0.
`  K ) ) )
111108, 110mpbid 210 . . . . . 6  |-  ( ph  ->  ( R `  D
)  =/=  ( 0.
`  K ) )
112111neneqd 2605 . . . . 5  |-  ( ph  ->  -.  ( R `  D )  =  ( 0. `  K ) )
11396, 7, 8, 9, 25trlator0 33170 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  D  e.  T
)  ->  ( ( R `  D )  e.  A  \/  ( R `  D )  =  ( 0. `  K ) ) )
1141, 47, 113syl2anc 659 . . . . . . 7  |-  ( ph  ->  ( ( R `  D )  e.  A  \/  ( R `  D
)  =  ( 0.
`  K ) ) )
115114orcomd 386 . . . . . 6  |-  ( ph  ->  ( ( R `  D )  =  ( 0. `  K )  \/  ( R `  D )  e.  A
) )
116115ord 375 . . . . 5  |-  ( ph  ->  ( -.  ( R `
 D )  =  ( 0. `  K
)  ->  ( R `  D )  e.  A
) )
117112, 116mpd 15 . . . 4  |-  ( ph  ->  ( R `  D
)  e.  A )
1186, 7atcmp 32310 . . . 4  |-  ( ( K  e.  AtLat  /\  V  e.  A  /\  ( R `  D )  e.  A )  ->  ( V  .<_  ( R `  D )  <->  V  =  ( R `  D ) ) )
11993, 14, 117, 118syl3anc 1230 . . 3  |-  ( ph  ->  ( V  .<_  ( R `
 D )  <->  V  =  ( R `  D ) ) )
12091, 119mpbid 210 . 2  |-  ( ph  ->  V  =  ( R `
 D ) )
121120eqcomd 2410 1  |-  ( ph  ->  ( R `  D
)  =  V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   class class class wbr 4394   ` cfv 5525  (class class class)co 6234   Basecbs 14733   lecple 14808   Posetcpo 15785   ltcplt 15786   joincjn 15789   meetcmee 15790   0.cp0 15883   Latclat 15891   OPcops 32171   OLcol 32173   Atomscatm 32262   AtLatcal 32263   HLchlt 32349   LHypclh 32982   LTrncltrn 33099   trLctrl 33157
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-iin 4273  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-mpt2 6239  df-1st 6738  df-2nd 6739  df-map 7379  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-p1 15886  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-psubsp 32501  df-pmap 32502  df-padd 32794  df-lhyp 32986  df-laut 32987  df-ldil 33102  df-ltrn 33103  df-trl 33158
This theorem is referenced by:  dia2dimlem5  34069
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