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Theorem dia2dimlem3 34281
Description: Lemma for dia2dim 34292. 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 456 . . . . . 6  |-  ( ph  ->  K  e.  HL )
3 dia2dimlem3.f . . . . . . . . 9  |-  ( ph  ->  ( F  e.  T  /\  ( F `  P
)  =/=  P ) )
43simpld 456 . . . . . . . 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 33353 . . . . . . . 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 1211 . . . . . . 7  |-  ( ph  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
1211simpld 456 . . . . . 6  |-  ( ph  ->  ( F `  P
)  e.  A )
13 dia2dimlem3.v . . . . . . 7  |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )
1413simpld 456 . . . . . 6  |-  ( ph  ->  V  e.  A )
15 dia2dimlem3.j . . . . . . 7  |-  .\/  =  ( join `  K )
166, 15, 7hlatlej2 32590 . . . . . 6  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  V  e.  A )  ->  V  .<_  ( ( F `  P )  .\/  V
) )
172, 12, 14, 16syl3anc 1211 . . . . 5  |-  ( ph  ->  V  .<_  ( ( F `  P )  .\/  V ) )
18 hllat 32578 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
192, 18syl 16 . . . . . 6  |-  ( ph  ->  K  e.  Lat )
20 eqid 2433 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2120, 7atbase 32504 . . . . . . 7  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
2214, 21syl 16 . . . . . 6  |-  ( ph  ->  V  e.  ( Base `  K ) )
2320, 15, 7hlatjcl 32581 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  V  e.  A )  ->  (
( F `  P
)  .\/  V )  e.  ( Base `  K
) )
242, 12, 14, 23syl3anc 1211 . . . . . 6  |-  ( ph  ->  ( ( F `  P )  .\/  V
)  e.  ( Base `  K ) )
25 dia2dimlem3.r . . . . . . . . 9  |-  R  =  ( ( trL `  K
) `  W )
266, 7, 8, 9, 25trlat 33383 . . . . . . . 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 1211 . . . . . . 7  |-  ( ph  ->  ( R `  F
)  e.  A )
28 dia2dimlem3.u . . . . . . . 8  |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )
2928simpld 456 . . . . . . 7  |-  ( ph  ->  U  e.  A )
3020, 15, 7hlatjcl 32581 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( R `  F )  e.  A  /\  U  e.  A )  ->  (
( R `  F
)  .\/  U )  e.  ( Base `  K
) )
312, 27, 29, 30syl3anc 1211 . . . . . 6  |-  ( ph  ->  ( ( R `  F )  .\/  U
)  e.  ( Base `  K ) )
32 dia2dimlem3.m . . . . . . 7  |-  ./\  =  ( meet `  K )
3320, 6, 32latmlem2 15234 . . . . . 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 1213 . . . . 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 32582 . . . . . . . 8  |-  ( ( K  e.  HL  /\  U  e.  A  /\  V  e.  A )  ->  ( U  .\/  V
)  =  ( V 
.\/  U ) )
382, 29, 14, 37syl3anc 1211 . . . . . . 7  |-  ( ph  ->  ( U  .\/  V
)  =  ( V 
.\/  U ) )
3936, 38breqtrd 4304 . . . . . 6  |-  ( ph  ->  ( R `  F
)  .<_  ( V  .\/  U ) )
40 dia2dimlem3.ru . . . . . . 7  |-  ( ph  ->  ( R `  F
)  =/=  U )
416, 15, 7hlatexch2 32610 . . . . . . 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 1224 . . . . . 6  |-  ( ph  ->  ( ( R `  F )  .<_  ( V 
.\/  U )  ->  V  .<_  ( ( R `
 F )  .\/  U ) ) )
4339, 42mpd 15 . . . . 5  |-  ( ph  ->  V  .<_  ( ( R `  F )  .\/  U ) )
4420, 6, 32latleeqm2 15232 . . . . . 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 1211 . . . . 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 34279 . . . . . 6  |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
516, 15, 32, 7, 8, 9, 25trlval2 33377 . . . . . 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 1211 . . . . 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 6098 . . . . . . . 8  |-  ( ph  ->  ( Q  .\/  ( D `  Q )
)  =  ( ( ( P  .\/  U
)  ./\  ( ( F `  P )  .\/  V ) )  .\/  ( F `  P ) ) )
565simpld 456 . . . . . . . . . 10  |-  ( ph  ->  P  e.  A )
5720, 15, 7hlatjcl 32581 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
582, 56, 29, 57syl3anc 1211 . . . . . . . . 9  |-  ( ph  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
596, 15, 7hlatlej1 32589 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  V  e.  A )  ->  ( F `  P )  .<_  ( ( F `  P )  .\/  V
) )
602, 12, 14, 59syl3anc 1211 . . . . . . . . 9  |-  ( ph  ->  ( F `  P
)  .<_  ( ( F `
 P )  .\/  V ) )
6120, 6, 15, 32, 7atmod4i1 33080 . . . . . . . . 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 1224 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
.\/  U )  ./\  ( ( F `  P )  .\/  V
) )  .\/  ( F `  P )
)  =  ( ( ( P  .\/  U
)  .\/  ( F `  P ) )  ./\  ( ( F `  P )  .\/  V
) ) )
6315, 7hlatj32 32586 . . . . . . . . . 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 1213 . . . . . . . . 9  |-  ( ph  ->  ( ( P  .\/  U )  .\/  ( F `
 P ) )  =  ( ( P 
.\/  ( F `  P ) )  .\/  U ) )
6564oveq1d 6095 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
.\/  U )  .\/  ( F `  P ) )  ./\  ( ( F `  P )  .\/  V ) )  =  ( ( ( P 
.\/  ( F `  P ) )  .\/  U )  ./\  ( ( F `  P )  .\/  V ) ) )
6655, 62, 653eqtrd 2469 . . . . . . 7  |-  ( ph  ->  ( Q  .\/  ( D `  Q )
)  =  ( ( ( P  .\/  ( F `  P )
)  .\/  U )  ./\  ( ( F `  P )  .\/  V
) ) )
6766oveq1d 6095 . . . . . 6  |-  ( ph  ->  ( ( Q  .\/  ( D `  Q ) )  ./\  W )  =  ( ( ( ( P  .\/  ( F `  P )
)  .\/  U )  ./\  ( ( F `  P )  .\/  V
) )  ./\  W
) )
68 hlol 32576 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OL )
692, 68syl 16 . . . . . . 7  |-  ( ph  ->  K  e.  OL )
7020, 15, 7hlatjcl 32581 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  P )  e.  A )  -> 
( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
712, 56, 12, 70syl3anc 1211 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
7220, 7atbase 32504 . . . . . . . . 9  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
7329, 72syl 16 . . . . . . . 8  |-  ( ph  ->  U  e.  ( Base `  K ) )
7420, 15latjcl 15203 . . . . . . . 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 1211 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  ( F `  P ) )  .\/  U )  e.  ( Base `  K
) )
761simprd 460 . . . . . . . 8  |-  ( ph  ->  W  e.  H )
7720, 8lhpbase 33212 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
7876, 77syl 16 . . . . . . 7  |-  ( ph  ->  W  e.  ( Base `  K ) )
7920, 32latm32 32446 . . . . . . 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 1213 . . . . . 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 33377 . . . . . . . . . 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 1211 . . . . . . . . 9  |-  ( ph  ->  ( R `  F
)  =  ( ( P  .\/  ( F `
 P ) ) 
./\  W ) )
8382oveq1d 6095 . . . . . . . 8  |-  ( ph  ->  ( ( R `  F )  .\/  U
)  =  ( ( ( P  .\/  ( F `  P )
)  ./\  W )  .\/  U ) )
8428simprd 460 . . . . . . . . 9  |-  ( ph  ->  U  .<_  W )
8520, 6, 15, 32, 7atmod4i1 33080 . . . . . . . . 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 1224 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
.\/  ( F `  P ) )  ./\  W )  .\/  U )  =  ( ( ( P  .\/  ( F `
 P ) ) 
.\/  U )  ./\  W ) )
8783, 86eqtr2d 2466 . . . . . . 7  |-  ( ph  ->  ( ( ( P 
.\/  ( F `  P ) )  .\/  U )  ./\  W )  =  ( ( R `
 F )  .\/  U ) )
8887oveq1d 6095 . . . . . 6  |-  ( ph  ->  ( ( ( ( P  .\/  ( F `
 P ) ) 
.\/  U )  ./\  W )  ./\  ( ( F `  P )  .\/  V ) )  =  ( ( ( R `
 F )  .\/  U )  ./\  ( ( F `  P )  .\/  V ) ) )
8967, 80, 883eqtrd 2469 . . . . 5  |-  ( ph  ->  ( ( Q  .\/  ( D `  Q ) )  ./\  W )  =  ( ( ( R `  F ) 
.\/  U )  ./\  ( ( F `  P )  .\/  V
) ) )
9052, 89eqtr2d 2466 . . . 4  |-  ( ph  ->  ( ( ( R `
 F )  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )  =  ( R `  D
) )
9135, 46, 903brtr3d 4309 . . 3  |-  ( ph  ->  V  .<_  ( R `  D ) )
92 hlatl 32575 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
932, 92syl 16 . . . 4  |-  ( ph  ->  K  e.  AtLat )
94 hlop 32577 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OP )
952, 94syl 16 . . . . . . . . 9  |-  ( ph  ->  K  e.  OP )
96 eqid 2433 . . . . . . . . . 10  |-  ( 0.
`  K )  =  ( 0. `  K
)
97 eqid 2433 . . . . . . . . . 10  |-  ( lt
`  K )  =  ( lt `  K
)
9896, 97, 70ltat 32506 . . . . . . . . 9  |-  ( ( K  e.  OP  /\  V  e.  A )  ->  ( 0. `  K
) ( lt `  K ) V )
9995, 14, 98syl2anc 654 . . . . . . . 8  |-  ( ph  ->  ( 0. `  K
) ( lt `  K ) V )
100 hlpos 32580 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  Poset )
1012, 100syl 16 . . . . . . . . 9  |-  ( ph  ->  K  e.  Poset )
10220, 96op0cl 32399 . . . . . . . . . 10  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  ( Base `  K
) )
10395, 102syl 16 . . . . . . . . 9  |-  ( ph  ->  ( 0. `  K
)  e.  ( Base `  K ) )
10420, 8, 9, 25trlcl 33378 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  D  e.  T
)  ->  ( R `  D )  e.  (
Base `  K )
)
1051, 47, 104syl2anc 654 . . . . . . . . 9  |-  ( ph  ->  ( R `  D
)  e.  ( Base `  K ) )
10620, 6, 97pltletr 15123 . . . . . . . . 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 1213 . . . . . . . 8  |-  ( ph  ->  ( ( ( 0.
`  K ) ( lt `  K ) V  /\  V  .<_  ( R `  D ) )  ->  ( 0. `  K ) ( lt
`  K ) ( R `  D ) ) )
10899, 91, 107mp2and 672 . . . . . . 7  |-  ( ph  ->  ( 0. `  K
) ( lt `  K ) ( R `
 D ) )
10920, 97, 96opltn0 32405 . . . . . . . 8  |-  ( ( K  e.  OP  /\  ( R `  D )  e.  ( Base `  K
) )  ->  (
( 0. `  K
) ( lt `  K ) ( R `
 D )  <->  ( R `  D )  =/=  ( 0. `  K ) ) )
11095, 105, 109syl2anc 654 . . . . . . 7  |-  ( ph  ->  ( ( 0. `  K ) ( lt
`  K ) ( R `  D )  <-> 
( R `  D
)  =/=  ( 0.
`  K ) ) )
111108, 110mpbid 210 . . . . . 6  |-  ( ph  ->  ( R `  D
)  =/=  ( 0.
`  K ) )
112111neneqd 2614 . . . . 5  |-  ( ph  ->  -.  ( R `  D )  =  ( 0. `  K ) )
11396, 7, 8, 9, 25trlator0 33385 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  D  e.  T
)  ->  ( ( R `  D )  e.  A  \/  ( R `  D )  =  ( 0. `  K ) ) )
1141, 47, 113syl2anc 654 . . . . . . 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 32526 . . . 4  |-  ( ( K  e.  AtLat  /\  V  e.  A  /\  ( R `  D )  e.  A )  ->  ( V  .<_  ( R `  D )  <->  V  =  ( R `  D ) ) )
11993, 14, 117, 118syl3anc 1211 . . 3  |-  ( ph  ->  ( V  .<_  ( R `
 D )  <->  V  =  ( R `  D ) ) )
12091, 119mpbid 210 . 2  |-  ( ph  ->  V  =  ( R `
 D ) )
121120eqcomd 2438 1  |-  ( ph  ->  ( R `  D
)  =  V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1362    e. wcel 1755    =/= wne 2596   class class class wbr 4280   ` cfv 5406  (class class class)co 6080   Basecbs 14156   lecple 14227   Posetcpo 15092   ltcplt 15093   joincjn 15096   meetcmee 15097   0.cp0 15189   Latclat 15197   OPcops 32387   OLcol 32389   Atomscatm 32478   AtLatcal 32479   HLchlt 32565   LHypclh 33198   LTrncltrn 33315   trLctrl 33372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-1st 6566  df-2nd 6567  df-map 7204  df-poset 15098  df-plt 15110  df-lub 15126  df-glb 15127  df-join 15128  df-meet 15129  df-p0 15191  df-p1 15192  df-lat 15198  df-clat 15260  df-oposet 32391  df-ol 32393  df-oml 32394  df-covers 32481  df-ats 32482  df-atl 32513  df-cvlat 32537  df-hlat 32566  df-llines 32712  df-psubsp 32717  df-pmap 32718  df-padd 33010  df-lhyp 33202  df-laut 33203  df-ldil 33318  df-ltrn 33319  df-trl 33373
This theorem is referenced by:  dia2dimlem5  34283
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