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Theorem dia2dimlem3 35030
Description: Lemma for dia2dim 35041. 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 34102 . . . . . . . 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 1219 . . . . . . 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 33339 . . . . . 6  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  V  e.  A )  ->  V  .<_  ( ( F `  P )  .\/  V
) )
172, 12, 14, 16syl3anc 1219 . . . . 5  |-  ( ph  ->  V  .<_  ( ( F `  P )  .\/  V ) )
18 hllat 33327 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
192, 18syl 16 . . . . . 6  |-  ( ph  ->  K  e.  Lat )
20 eqid 2452 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2120, 7atbase 33253 . . . . . . 7  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
2214, 21syl 16 . . . . . 6  |-  ( ph  ->  V  e.  ( Base `  K ) )
2320, 15, 7hlatjcl 33330 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  V  e.  A )  ->  (
( F `  P
)  .\/  V )  e.  ( Base `  K
) )
242, 12, 14, 23syl3anc 1219 . . . . . 6  |-  ( ph  ->  ( ( F `  P )  .\/  V
)  e.  ( Base `  K ) )
25 dia2dimlem3.r . . . . . . . . 9  |-  R  =  ( ( trL `  K
) `  W )
266, 7, 8, 9, 25trlat 34132 . . . . . . . 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 1219 . . . . . . 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 33330 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( R `  F )  e.  A  /\  U  e.  A )  ->  (
( R `  F
)  .\/  U )  e.  ( Base `  K
) )
312, 27, 29, 30syl3anc 1219 . . . . . 6  |-  ( ph  ->  ( ( R `  F )  .\/  U
)  e.  ( Base `  K ) )
32 dia2dimlem3.m . . . . . . 7  |-  ./\  =  ( meet `  K )
3320, 6, 32latmlem2 15366 . . . . . 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 1221 . . . . 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 33331 . . . . . . . 8  |-  ( ( K  e.  HL  /\  U  e.  A  /\  V  e.  A )  ->  ( U  .\/  V
)  =  ( V 
.\/  U ) )
382, 29, 14, 37syl3anc 1219 . . . . . . 7  |-  ( ph  ->  ( U  .\/  V
)  =  ( V 
.\/  U ) )
3936, 38breqtrd 4419 . . . . . 6  |-  ( ph  ->  ( R `  F
)  .<_  ( V  .\/  U ) )
40 dia2dimlem3.ru . . . . . . 7  |-  ( ph  ->  ( R `  F
)  =/=  U )
416, 15, 7hlatexch2 33359 . . . . . . 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 1232 . . . . . 6  |-  ( ph  ->  ( ( R `  F )  .<_  ( V 
.\/  U )  ->  V  .<_  ( ( R `
 F )  .\/  U ) ) )
4339, 42mpd 15 . . . . 5  |-  ( ph  ->  V  .<_  ( ( R `  F )  .\/  U ) )
4420, 6, 32latleeqm2 15364 . . . . . 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 1219 . . . . 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 35028 . . . . . 6  |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
516, 15, 32, 7, 8, 9, 25trlval2 34126 . . . . . 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 1219 . . . . 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 6213 . . . . . . . 8  |-  ( ph  ->  ( Q  .\/  ( D `  Q )
)  =  ( ( ( P  .\/  U
)  ./\  ( ( F `  P )  .\/  V ) )  .\/  ( F `  P ) ) )
565simpld 459 . . . . . . . . . 10  |-  ( ph  ->  P  e.  A )
5720, 15, 7hlatjcl 33330 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
582, 56, 29, 57syl3anc 1219 . . . . . . . . 9  |-  ( ph  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
596, 15, 7hlatlej1 33338 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  V  e.  A )  ->  ( F `  P )  .<_  ( ( F `  P )  .\/  V
) )
602, 12, 14, 59syl3anc 1219 . . . . . . . . 9  |-  ( ph  ->  ( F `  P
)  .<_  ( ( F `
 P )  .\/  V ) )
6120, 6, 15, 32, 7atmod4i1 33829 . . . . . . . . 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 1232 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
.\/  U )  ./\  ( ( F `  P )  .\/  V
) )  .\/  ( F `  P )
)  =  ( ( ( P  .\/  U
)  .\/  ( F `  P ) )  ./\  ( ( F `  P )  .\/  V
) ) )
6315, 7hlatj32 33335 . . . . . . . . . 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 1221 . . . . . . . . 9  |-  ( ph  ->  ( ( P  .\/  U )  .\/  ( F `
 P ) )  =  ( ( P 
.\/  ( F `  P ) )  .\/  U ) )
6564oveq1d 6210 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
.\/  U )  .\/  ( F `  P ) )  ./\  ( ( F `  P )  .\/  V ) )  =  ( ( ( P 
.\/  ( F `  P ) )  .\/  U )  ./\  ( ( F `  P )  .\/  V ) ) )
6655, 62, 653eqtrd 2497 . . . . . . 7  |-  ( ph  ->  ( Q  .\/  ( D `  Q )
)  =  ( ( ( P  .\/  ( F `  P )
)  .\/  U )  ./\  ( ( F `  P )  .\/  V
) ) )
6766oveq1d 6210 . . . . . 6  |-  ( ph  ->  ( ( Q  .\/  ( D `  Q ) )  ./\  W )  =  ( ( ( ( P  .\/  ( F `  P )
)  .\/  U )  ./\  ( ( F `  P )  .\/  V
) )  ./\  W
) )
68 hlol 33325 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OL )
692, 68syl 16 . . . . . . 7  |-  ( ph  ->  K  e.  OL )
7020, 15, 7hlatjcl 33330 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  P )  e.  A )  -> 
( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
712, 56, 12, 70syl3anc 1219 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
7220, 7atbase 33253 . . . . . . . . 9  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
7329, 72syl 16 . . . . . . . 8  |-  ( ph  ->  U  e.  ( Base `  K ) )
7420, 15latjcl 15335 . . . . . . . 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 1219 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  ( F `  P ) )  .\/  U )  e.  ( Base `  K
) )
761simprd 463 . . . . . . . 8  |-  ( ph  ->  W  e.  H )
7720, 8lhpbase 33961 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
7876, 77syl 16 . . . . . . 7  |-  ( ph  ->  W  e.  ( Base `  K ) )
7920, 32latm32 33195 . . . . . . 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 1221 . . . . . 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 34126 . . . . . . . . . 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 1219 . . . . . . . . 9  |-  ( ph  ->  ( R `  F
)  =  ( ( P  .\/  ( F `
 P ) ) 
./\  W ) )
8382oveq1d 6210 . . . . . . . 8  |-  ( ph  ->  ( ( R `  F )  .\/  U
)  =  ( ( ( P  .\/  ( F `  P )
)  ./\  W )  .\/  U ) )
8428simprd 463 . . . . . . . . 9  |-  ( ph  ->  U  .<_  W )
8520, 6, 15, 32, 7atmod4i1 33829 . . . . . . . . 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 1232 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
.\/  ( F `  P ) )  ./\  W )  .\/  U )  =  ( ( ( P  .\/  ( F `
 P ) ) 
.\/  U )  ./\  W ) )
8783, 86eqtr2d 2494 . . . . . . 7  |-  ( ph  ->  ( ( ( P 
.\/  ( F `  P ) )  .\/  U )  ./\  W )  =  ( ( R `
 F )  .\/  U ) )
8887oveq1d 6210 . . . . . 6  |-  ( ph  ->  ( ( ( ( P  .\/  ( F `
 P ) ) 
.\/  U )  ./\  W )  ./\  ( ( F `  P )  .\/  V ) )  =  ( ( ( R `
 F )  .\/  U )  ./\  ( ( F `  P )  .\/  V ) ) )
8967, 80, 883eqtrd 2497 . . . . 5  |-  ( ph  ->  ( ( Q  .\/  ( D `  Q ) )  ./\  W )  =  ( ( ( R `  F ) 
.\/  U )  ./\  ( ( F `  P )  .\/  V
) ) )
9052, 89eqtr2d 2494 . . . 4  |-  ( ph  ->  ( ( ( R `
 F )  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )  =  ( R `  D
) )
9135, 46, 903brtr3d 4424 . . 3  |-  ( ph  ->  V  .<_  ( R `  D ) )
92 hlatl 33324 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
932, 92syl 16 . . . 4  |-  ( ph  ->  K  e.  AtLat )
94 hlop 33326 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OP )
952, 94syl 16 . . . . . . . . 9  |-  ( ph  ->  K  e.  OP )
96 eqid 2452 . . . . . . . . . 10  |-  ( 0.
`  K )  =  ( 0. `  K
)
97 eqid 2452 . . . . . . . . . 10  |-  ( lt
`  K )  =  ( lt `  K
)
9896, 97, 70ltat 33255 . . . . . . . . 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 33329 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  Poset )
1012, 100syl 16 . . . . . . . . 9  |-  ( ph  ->  K  e.  Poset )
10220, 96op0cl 33148 . . . . . . . . . 10  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  ( Base `  K
) )
10395, 102syl 16 . . . . . . . . 9  |-  ( ph  ->  ( 0. `  K
)  e.  ( Base `  K ) )
10420, 8, 9, 25trlcl 34127 . . . . . . . . . 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 15255 . . . . . . . . 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 1221 . . . . . . . 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 33154 . . . . . . . 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 2652 . . . . 5  |-  ( ph  ->  -.  ( R `  D )  =  ( 0. `  K ) )
11396, 7, 8, 9, 25trlator0 34134 . . . . . . . 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 33275 . . . 4  |-  ( ( K  e.  AtLat  /\  V  e.  A  /\  ( R `  D )  e.  A )  ->  ( V  .<_  ( R `  D )  <->  V  =  ( R `  D ) ) )
11993, 14, 117, 118syl3anc 1219 . . 3  |-  ( ph  ->  ( V  .<_  ( R `
 D )  <->  V  =  ( R `  D ) ) )
12091, 119mpbid 210 . 2  |-  ( ph  ->  V  =  ( R `
 D ) )
121120eqcomd 2460 1  |-  ( ph  ->  ( R `  D
)  =  V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2645   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   Basecbs 14287   lecple 14359   Posetcpo 15224   ltcplt 15225   joincjn 15228   meetcmee 15229   0.cp0 15321   Latclat 15329   OPcops 33136   OLcol 33138   Atomscatm 33227   AtLatcal 33228   HLchlt 33314   LHypclh 33947   LTrncltrn 34064   trLctrl 34121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-1st 6682  df-2nd 6683  df-map 7321  df-poset 15230  df-plt 15242  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-p0 15323  df-p1 15324  df-lat 15330  df-clat 15392  df-oposet 33140  df-ol 33142  df-oml 33143  df-covers 33230  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315  df-llines 33461  df-psubsp 33466  df-pmap 33467  df-padd 33759  df-lhyp 33951  df-laut 33952  df-ldil 34067  df-ltrn 34068  df-trl 34122
This theorem is referenced by:  dia2dimlem5  35032
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