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Theorem dia2dimlem3 35863
Description: Lemma for dia2dim 35874. 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 459 . . . . . 6  |-  ( ph  ->  K  e.  HL )
3 dia2dimlem3.f . . . . . . . . 9  |-  ( ph  ->  ( F  e.  T  /\  ( F `  P
)  =/=  P ) )
43simpld 459 . . . . . . . 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 34935 . . . . . . . 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 1228 . . . . . . 7  |-  ( ph  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
1211simpld 459 . . . . . 6  |-  ( ph  ->  ( F `  P
)  e.  A )
13 dia2dimlem3.v . . . . . . 7  |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )
1413simpld 459 . . . . . 6  |-  ( ph  ->  V  e.  A )
15 dia2dimlem3.j . . . . . . 7  |-  .\/  =  ( join `  K )
166, 15, 7hlatlej2 34172 . . . . . 6  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  V  e.  A )  ->  V  .<_  ( ( F `  P )  .\/  V
) )
172, 12, 14, 16syl3anc 1228 . . . . 5  |-  ( ph  ->  V  .<_  ( ( F `  P )  .\/  V ) )
18 hllat 34160 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
192, 18syl 16 . . . . . 6  |-  ( ph  ->  K  e.  Lat )
20 eqid 2467 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2120, 7atbase 34086 . . . . . . 7  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
2214, 21syl 16 . . . . . 6  |-  ( ph  ->  V  e.  ( Base `  K ) )
2320, 15, 7hlatjcl 34163 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  V  e.  A )  ->  (
( F `  P
)  .\/  V )  e.  ( Base `  K
) )
242, 12, 14, 23syl3anc 1228 . . . . . 6  |-  ( ph  ->  ( ( F `  P )  .\/  V
)  e.  ( Base `  K ) )
25 dia2dimlem3.r . . . . . . . . 9  |-  R  =  ( ( trL `  K
) `  W )
266, 7, 8, 9, 25trlat 34965 . . . . . . . 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 1228 . . . . . . 7  |-  ( ph  ->  ( R `  F
)  e.  A )
28 dia2dimlem3.u . . . . . . . 8  |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )
2928simpld 459 . . . . . . 7  |-  ( ph  ->  U  e.  A )
3020, 15, 7hlatjcl 34163 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( R `  F )  e.  A  /\  U  e.  A )  ->  (
( R `  F
)  .\/  U )  e.  ( Base `  K
) )
312, 27, 29, 30syl3anc 1228 . . . . . 6  |-  ( ph  ->  ( ( R `  F )  .\/  U
)  e.  ( Base `  K ) )
32 dia2dimlem3.m . . . . . . 7  |-  ./\  =  ( meet `  K )
3320, 6, 32latmlem2 15562 . . . . . 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 1230 . . . . 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 34164 . . . . . . . 8  |-  ( ( K  e.  HL  /\  U  e.  A  /\  V  e.  A )  ->  ( U  .\/  V
)  =  ( V 
.\/  U ) )
382, 29, 14, 37syl3anc 1228 . . . . . . 7  |-  ( ph  ->  ( U  .\/  V
)  =  ( V 
.\/  U ) )
3936, 38breqtrd 4471 . . . . . 6  |-  ( ph  ->  ( R `  F
)  .<_  ( V  .\/  U ) )
40 dia2dimlem3.ru . . . . . . 7  |-  ( ph  ->  ( R `  F
)  =/=  U )
416, 15, 7hlatexch2 34192 . . . . . . 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 1241 . . . . . 6  |-  ( ph  ->  ( ( R `  F )  .<_  ( V 
.\/  U )  ->  V  .<_  ( ( R `
 F )  .\/  U ) ) )
4339, 42mpd 15 . . . . 5  |-  ( ph  ->  V  .<_  ( ( R `  F )  .\/  U ) )
4420, 6, 32latleeqm2 15560 . . . . . 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 1228 . . . . 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 35861 . . . . . 6  |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
516, 15, 32, 7, 8, 9, 25trlval2 34959 . . . . . 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 1228 . . . . 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 6300 . . . . . . . 8  |-  ( ph  ->  ( Q  .\/  ( D `  Q )
)  =  ( ( ( P  .\/  U
)  ./\  ( ( F `  P )  .\/  V ) )  .\/  ( F `  P ) ) )
565simpld 459 . . . . . . . . . 10  |-  ( ph  ->  P  e.  A )
5720, 15, 7hlatjcl 34163 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
582, 56, 29, 57syl3anc 1228 . . . . . . . . 9  |-  ( ph  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
596, 15, 7hlatlej1 34171 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  V  e.  A )  ->  ( F `  P )  .<_  ( ( F `  P )  .\/  V
) )
602, 12, 14, 59syl3anc 1228 . . . . . . . . 9  |-  ( ph  ->  ( F `  P
)  .<_  ( ( F `
 P )  .\/  V ) )
6120, 6, 15, 32, 7atmod4i1 34662 . . . . . . . . 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 1241 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
.\/  U )  ./\  ( ( F `  P )  .\/  V
) )  .\/  ( F `  P )
)  =  ( ( ( P  .\/  U
)  .\/  ( F `  P ) )  ./\  ( ( F `  P )  .\/  V
) ) )
6315, 7hlatj32 34168 . . . . . . . . . 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 1230 . . . . . . . . 9  |-  ( ph  ->  ( ( P  .\/  U )  .\/  ( F `
 P ) )  =  ( ( P 
.\/  ( F `  P ) )  .\/  U ) )
6564oveq1d 6297 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
.\/  U )  .\/  ( F `  P ) )  ./\  ( ( F `  P )  .\/  V ) )  =  ( ( ( P 
.\/  ( F `  P ) )  .\/  U )  ./\  ( ( F `  P )  .\/  V ) ) )
6655, 62, 653eqtrd 2512 . . . . . . 7  |-  ( ph  ->  ( Q  .\/  ( D `  Q )
)  =  ( ( ( P  .\/  ( F `  P )
)  .\/  U )  ./\  ( ( F `  P )  .\/  V
) ) )
6766oveq1d 6297 . . . . . 6  |-  ( ph  ->  ( ( Q  .\/  ( D `  Q ) )  ./\  W )  =  ( ( ( ( P  .\/  ( F `  P )
)  .\/  U )  ./\  ( ( F `  P )  .\/  V
) )  ./\  W
) )
68 hlol 34158 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OL )
692, 68syl 16 . . . . . . 7  |-  ( ph  ->  K  e.  OL )
7020, 15, 7hlatjcl 34163 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  P )  e.  A )  -> 
( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
712, 56, 12, 70syl3anc 1228 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
7220, 7atbase 34086 . . . . . . . . 9  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
7329, 72syl 16 . . . . . . . 8  |-  ( ph  ->  U  e.  ( Base `  K ) )
7420, 15latjcl 15531 . . . . . . . 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 1228 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  ( F `  P ) )  .\/  U )  e.  ( Base `  K
) )
761simprd 463 . . . . . . . 8  |-  ( ph  ->  W  e.  H )
7720, 8lhpbase 34794 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
7876, 77syl 16 . . . . . . 7  |-  ( ph  ->  W  e.  ( Base `  K ) )
7920, 32latm32 34028 . . . . . . 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 1230 . . . . . 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 34959 . . . . . . . . . 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 1228 . . . . . . . . 9  |-  ( ph  ->  ( R `  F
)  =  ( ( P  .\/  ( F `
 P ) ) 
./\  W ) )
8382oveq1d 6297 . . . . . . . 8  |-  ( ph  ->  ( ( R `  F )  .\/  U
)  =  ( ( ( P  .\/  ( F `  P )
)  ./\  W )  .\/  U ) )
8428simprd 463 . . . . . . . . 9  |-  ( ph  ->  U  .<_  W )
8520, 6, 15, 32, 7atmod4i1 34662 . . . . . . . . 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 1241 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
.\/  ( F `  P ) )  ./\  W )  .\/  U )  =  ( ( ( P  .\/  ( F `
 P ) ) 
.\/  U )  ./\  W ) )
8783, 86eqtr2d 2509 . . . . . . 7  |-  ( ph  ->  ( ( ( P 
.\/  ( F `  P ) )  .\/  U )  ./\  W )  =  ( ( R `
 F )  .\/  U ) )
8887oveq1d 6297 . . . . . 6  |-  ( ph  ->  ( ( ( ( P  .\/  ( F `
 P ) ) 
.\/  U )  ./\  W )  ./\  ( ( F `  P )  .\/  V ) )  =  ( ( ( R `
 F )  .\/  U )  ./\  ( ( F `  P )  .\/  V ) ) )
8967, 80, 883eqtrd 2512 . . . . 5  |-  ( ph  ->  ( ( Q  .\/  ( D `  Q ) )  ./\  W )  =  ( ( ( R `  F ) 
.\/  U )  ./\  ( ( F `  P )  .\/  V
) ) )
9052, 89eqtr2d 2509 . . . 4  |-  ( ph  ->  ( ( ( R `
 F )  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )  =  ( R `  D
) )
9135, 46, 903brtr3d 4476 . . 3  |-  ( ph  ->  V  .<_  ( R `  D ) )
92 hlatl 34157 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
932, 92syl 16 . . . 4  |-  ( ph  ->  K  e.  AtLat )
94 hlop 34159 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OP )
952, 94syl 16 . . . . . . . . 9  |-  ( ph  ->  K  e.  OP )
96 eqid 2467 . . . . . . . . . 10  |-  ( 0.
`  K )  =  ( 0. `  K
)
97 eqid 2467 . . . . . . . . . 10  |-  ( lt
`  K )  =  ( lt `  K
)
9896, 97, 70ltat 34088 . . . . . . . . 9  |-  ( ( K  e.  OP  /\  V  e.  A )  ->  ( 0. `  K
) ( lt `  K ) V )
9995, 14, 98syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( 0. `  K
) ( lt `  K ) V )
100 hlpos 34162 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  Poset )
1012, 100syl 16 . . . . . . . . 9  |-  ( ph  ->  K  e.  Poset )
10220, 96op0cl 33981 . . . . . . . . . 10  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  ( Base `  K
) )
10395, 102syl 16 . . . . . . . . 9  |-  ( ph  ->  ( 0. `  K
)  e.  ( Base `  K ) )
10420, 8, 9, 25trlcl 34960 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  D  e.  T
)  ->  ( R `  D )  e.  (
Base `  K )
)
1051, 47, 104syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( R `  D
)  e.  ( Base `  K ) )
10620, 6, 97pltletr 15451 . . . . . . . . 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 1230 . . . . . . . 8  |-  ( ph  ->  ( ( ( 0.
`  K ) ( lt `  K ) V  /\  V  .<_  ( R `  D ) )  ->  ( 0. `  K ) ( lt
`  K ) ( R `  D ) ) )
10899, 91, 107mp2and 679 . . . . . . 7  |-  ( ph  ->  ( 0. `  K
) ( lt `  K ) ( R `
 D ) )
10920, 97, 96opltn0 33987 . . . . . . . 8  |-  ( ( K  e.  OP  /\  ( R `  D )  e.  ( Base `  K
) )  ->  (
( 0. `  K
) ( lt `  K ) ( R `
 D )  <->  ( R `  D )  =/=  ( 0. `  K ) ) )
11095, 105, 109syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( ( 0. `  K ) ( lt
`  K ) ( R `  D )  <-> 
( R `  D
)  =/=  ( 0.
`  K ) ) )
111108, 110mpbid 210 . . . . . 6  |-  ( ph  ->  ( R `  D
)  =/=  ( 0.
`  K ) )
112111neneqd 2669 . . . . 5  |-  ( ph  ->  -.  ( R `  D )  =  ( 0. `  K ) )
11396, 7, 8, 9, 25trlator0 34967 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  D  e.  T
)  ->  ( ( R `  D )  e.  A  \/  ( R `  D )  =  ( 0. `  K ) ) )
1141, 47, 113syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( ( R `  D )  e.  A  \/  ( R `  D
)  =  ( 0.
`  K ) ) )
115114orcomd 388 . . . . . 6  |-  ( ph  ->  ( ( R `  D )  =  ( 0. `  K )  \/  ( R `  D )  e.  A
) )
116115ord 377 . . . . 5  |-  ( ph  ->  ( -.  ( R `
 D )  =  ( 0. `  K
)  ->  ( R `  D )  e.  A
) )
117112, 116mpd 15 . . . 4  |-  ( ph  ->  ( R `  D
)  e.  A )
1186, 7atcmp 34108 . . . 4  |-  ( ( K  e.  AtLat  /\  V  e.  A  /\  ( R `  D )  e.  A )  ->  ( V  .<_  ( R `  D )  <->  V  =  ( R `  D ) ) )
11993, 14, 117, 118syl3anc 1228 . . 3  |-  ( ph  ->  ( V  .<_  ( R `
 D )  <->  V  =  ( R `  D ) ) )
12091, 119mpbid 210 . 2  |-  ( ph  ->  V  =  ( R `
 D ) )
121120eqcomd 2475 1  |-  ( ph  ->  ( R `  D
)  =  V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14483   lecple 14555   Posetcpo 15420   ltcplt 15421   joincjn 15424   meetcmee 15425   0.cp0 15517   Latclat 15525   OPcops 33969   OLcol 33971   Atomscatm 34060   AtLatcal 34061   HLchlt 34147   LHypclh 34780   LTrncltrn 34897   trLctrl 34954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-map 7419  df-poset 15426  df-plt 15438  df-lub 15454  df-glb 15455  df-join 15456  df-meet 15457  df-p0 15519  df-p1 15520  df-lat 15526  df-clat 15588  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294  df-psubsp 34299  df-pmap 34300  df-padd 34592  df-lhyp 34784  df-laut 34785  df-ldil 34900  df-ltrn 34901  df-trl 34955
This theorem is referenced by:  dia2dimlem5  35865
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