Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dia2dimlem2 Structured version   Unicode version

Theorem dia2dimlem2 35033
Description: Lemma for dia2dim 35045. Define a translation  G whose trace is atom  U. Part of proof of Lemma M in [Crawley] p. 121 line 4. (Contributed by NM, 8-Sep-2014.)
Hypotheses
Ref Expression
dia2dimlem2.l  |-  .<_  =  ( le `  K )
dia2dimlem2.j  |-  .\/  =  ( join `  K )
dia2dimlem2.m  |-  ./\  =  ( meet `  K )
dia2dimlem2.a  |-  A  =  ( Atoms `  K )
dia2dimlem2.h  |-  H  =  ( LHyp `  K
)
dia2dimlem2.t  |-  T  =  ( ( LTrn `  K
) `  W )
dia2dimlem2.r  |-  R  =  ( ( trL `  K
) `  W )
dia2dimlem2.q  |-  Q  =  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )
dia2dimlem2.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
dia2dimlem2.u  |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )
dia2dimlem2.v  |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )
dia2dimlem2.p  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
dia2dimlem2.f  |-  ( ph  ->  ( F  e.  T  /\  ( F `  P
)  =/=  P ) )
dia2dimlem2.rf  |-  ( ph  ->  ( R `  F
)  .<_  ( U  .\/  V ) )
dia2dimlem2.rv  |-  ( ph  ->  ( R `  F
)  =/=  V )
dia2dimlem2.g  |-  ( ph  ->  G  e.  T )
dia2dimlem2.gv  |-  ( ph  ->  ( G `  P
)  =  Q )
Assertion
Ref Expression
dia2dimlem2  |-  ( ph  ->  ( R `  G
)  =  U )

Proof of Theorem dia2dimlem2
StepHypRef Expression
1 dia2dimlem2.k . . . . . . . . 9  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
21simpld 459 . . . . . . . 8  |-  ( ph  ->  K  e.  HL )
3 hllat 33331 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 16 . . . . . . 7  |-  ( ph  ->  K  e.  Lat )
5 dia2dimlem2.p . . . . . . . . 9  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
65simpld 459 . . . . . . . 8  |-  ( ph  ->  P  e.  A )
7 eqid 2454 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
8 dia2dimlem2.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
97, 8atbase 33257 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
106, 9syl 16 . . . . . . 7  |-  ( ph  ->  P  e.  ( Base `  K ) )
11 dia2dimlem2.u . . . . . . . . 9  |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )
1211simpld 459 . . . . . . . 8  |-  ( ph  ->  U  e.  A )
137, 8atbase 33257 . . . . . . . 8  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
1412, 13syl 16 . . . . . . 7  |-  ( ph  ->  U  e.  ( Base `  K ) )
15 dia2dimlem2.l . . . . . . . 8  |-  .<_  =  ( le `  K )
16 dia2dimlem2.j . . . . . . . 8  |-  .\/  =  ( join `  K )
177, 15, 16latlej2 15349 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) )  ->  U  .<_  ( P  .\/  U
) )
184, 10, 14, 17syl3anc 1219 . . . . . 6  |-  ( ph  ->  U  .<_  ( P  .\/  U ) )
197, 16, 8hlatjcl 33334 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
202, 6, 12, 19syl3anc 1219 . . . . . . 7  |-  ( ph  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
21 dia2dimlem2.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
227, 15, 21latleeqm2 15368 . . . . . . 7  |-  ( ( K  e.  Lat  /\  U  e.  ( Base `  K )  /\  ( P  .\/  U )  e.  ( Base `  K
) )  ->  ( U  .<_  ( P  .\/  U )  <->  ( ( P 
.\/  U )  ./\  U )  =  U ) )
234, 14, 20, 22syl3anc 1219 . . . . . 6  |-  ( ph  ->  ( U  .<_  ( P 
.\/  U )  <->  ( ( P  .\/  U )  ./\  U )  =  U ) )
2418, 23mpbid 210 . . . . 5  |-  ( ph  ->  ( ( P  .\/  U )  ./\  U )  =  U )
25 dia2dimlem2.rf . . . . . . . 8  |-  ( ph  ->  ( R `  F
)  .<_  ( U  .\/  V ) )
26 dia2dimlem2.f . . . . . . . . . 10  |-  ( ph  ->  ( F  e.  T  /\  ( F `  P
)  =/=  P ) )
27 dia2dimlem2.h . . . . . . . . . . 11  |-  H  =  ( LHyp `  K
)
28 dia2dimlem2.t . . . . . . . . . . 11  |-  T  =  ( ( LTrn `  K
) `  W )
29 dia2dimlem2.r . . . . . . . . . . 11  |-  R  =  ( ( trL `  K
) `  W )
3015, 8, 27, 28, 29trlat 34136 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( R `  F )  e.  A
)
311, 5, 26, 30syl3anc 1219 . . . . . . . . 9  |-  ( ph  ->  ( R `  F
)  e.  A )
32 dia2dimlem2.v . . . . . . . . . 10  |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )
3332simpld 459 . . . . . . . . 9  |-  ( ph  ->  V  e.  A )
34 dia2dimlem2.rv . . . . . . . . 9  |-  ( ph  ->  ( R `  F
)  =/=  V )
3515, 16, 8hlatexch2 33363 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( ( R `  F )  e.  A  /\  U  e.  A  /\  V  e.  A
)  /\  ( R `  F )  =/=  V
)  ->  ( ( R `  F )  .<_  ( U  .\/  V
)  ->  U  .<_  ( ( R `  F
)  .\/  V )
) )
362, 31, 12, 33, 34, 35syl131anc 1232 . . . . . . . 8  |-  ( ph  ->  ( ( R `  F )  .<_  ( U 
.\/  V )  ->  U  .<_  ( ( R `
 F )  .\/  V ) ) )
3725, 36mpd 15 . . . . . . 7  |-  ( ph  ->  U  .<_  ( ( R `  F )  .\/  V ) )
3826simpld 459 . . . . . . . . . 10  |-  ( ph  ->  F  e.  T )
3915, 16, 21, 8, 27, 28, 29trlval2 34130 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  F )  =  ( ( P  .\/  ( F `  P )
)  ./\  W )
)
401, 38, 5, 39syl3anc 1219 . . . . . . . . 9  |-  ( ph  ->  ( R `  F
)  =  ( ( P  .\/  ( F `
 P ) ) 
./\  W ) )
4140oveq1d 6214 . . . . . . . 8  |-  ( ph  ->  ( ( R `  F )  .\/  V
)  =  ( ( ( P  .\/  ( F `  P )
)  ./\  W )  .\/  V ) )
4215, 8, 27, 28ltrnel 34106 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
431, 38, 5, 42syl3anc 1219 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
4443simpld 459 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  P
)  e.  A )
457, 16, 8hlatjcl 33334 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  P )  e.  A )  -> 
( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
462, 6, 44, 45syl3anc 1219 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
471simprd 463 . . . . . . . . . . 11  |-  ( ph  ->  W  e.  H )
487, 27lhpbase 33965 . . . . . . . . . . 11  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
4947, 48syl 16 . . . . . . . . . 10  |-  ( ph  ->  W  e.  ( Base `  K ) )
5032simprd 463 . . . . . . . . . 10  |-  ( ph  ->  V  .<_  W )
517, 15, 16, 21, 8atmod4i1 33833 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( V  e.  A  /\  ( P  .\/  ( F `  P )
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  V  .<_  W )  ->  (
( ( P  .\/  ( F `  P ) )  ./\  W )  .\/  V )  =  ( ( ( P  .\/  ( F `  P ) )  .\/  V ) 
./\  W ) )
522, 33, 46, 49, 50, 51syl131anc 1232 . . . . . . . . 9  |-  ( ph  ->  ( ( ( P 
.\/  ( F `  P ) )  ./\  W )  .\/  V )  =  ( ( ( P  .\/  ( F `
 P ) ) 
.\/  V )  ./\  W ) )
5316, 8hlatjass 33337 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  ( F `  P
)  e.  A  /\  V  e.  A )
)  ->  ( ( P  .\/  ( F `  P ) )  .\/  V )  =  ( P 
.\/  ( ( F `
 P )  .\/  V ) ) )
542, 6, 44, 33, 53syl13anc 1221 . . . . . . . . . 10  |-  ( ph  ->  ( ( P  .\/  ( F `  P ) )  .\/  V )  =  ( P  .\/  ( ( F `  P )  .\/  V
) ) )
5554oveq1d 6214 . . . . . . . . 9  |-  ( ph  ->  ( ( ( P 
.\/  ( F `  P ) )  .\/  V )  ./\  W )  =  ( ( P 
.\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) )
5652, 55eqtrd 2495 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
.\/  ( F `  P ) )  ./\  W )  .\/  V )  =  ( ( P 
.\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) )
5741, 56eqtrd 2495 . . . . . . 7  |-  ( ph  ->  ( ( R `  F )  .\/  V
)  =  ( ( P  .\/  ( ( F `  P ) 
.\/  V ) ) 
./\  W ) )
5837, 57breqtrd 4423 . . . . . 6  |-  ( ph  ->  U  .<_  ( ( P  .\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) )
597, 16, 8hlatjcl 33334 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  V  e.  A )  ->  (
( F `  P
)  .\/  V )  e.  ( Base `  K
) )
602, 44, 33, 59syl3anc 1219 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  P )  .\/  V
)  e.  ( Base `  K ) )
617, 16latjcl 15339 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  (
( F `  P
)  .\/  V )  e.  ( Base `  K
) )  ->  ( P  .\/  ( ( F `
 P )  .\/  V ) )  e.  (
Base `  K )
)
624, 10, 60, 61syl3anc 1219 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  (
( F `  P
)  .\/  V )
)  e.  ( Base `  K ) )
637, 21latmcl 15340 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  ( ( F `  P ) 
.\/  V ) )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  ( ( F `
 P )  .\/  V ) )  ./\  W
)  e.  ( Base `  K ) )
644, 62, 49, 63syl3anc 1219 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  ( ( F `  P )  .\/  V
) )  ./\  W
)  e.  ( Base `  K ) )
657, 15, 21latmlem2 15370 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( U  e.  ( Base `  K )  /\  ( ( P  .\/  ( ( F `  P )  .\/  V
) )  ./\  W
)  e.  ( Base `  K )  /\  ( P  .\/  U )  e.  ( Base `  K
) ) )  -> 
( U  .<_  ( ( P  .\/  ( ( F `  P ) 
.\/  V ) ) 
./\  W )  -> 
( ( P  .\/  U )  ./\  U )  .<_  ( ( P  .\/  U )  ./\  ( ( P  .\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) ) ) )
664, 14, 64, 20, 65syl13anc 1221 . . . . . 6  |-  ( ph  ->  ( U  .<_  ( ( P  .\/  ( ( F `  P ) 
.\/  V ) ) 
./\  W )  -> 
( ( P  .\/  U )  ./\  U )  .<_  ( ( P  .\/  U )  ./\  ( ( P  .\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) ) ) )
6758, 66mpd 15 . . . . 5  |-  ( ph  ->  ( ( P  .\/  U )  ./\  U )  .<_  ( ( P  .\/  U )  ./\  ( ( P  .\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) ) )
6824, 67eqbrtrrd 4421 . . . 4  |-  ( ph  ->  U  .<_  ( ( P  .\/  U )  ./\  ( ( P  .\/  ( ( F `  P )  .\/  V
) )  ./\  W
) ) )
69 dia2dimlem2.g . . . . . . 7  |-  ( ph  ->  G  e.  T )
7015, 16, 21, 8, 27, 28, 29trlval2 34130 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  G )  =  ( ( P  .\/  ( G `  P )
)  ./\  W )
)
711, 69, 5, 70syl3anc 1219 . . . . . 6  |-  ( ph  ->  ( R `  G
)  =  ( ( P  .\/  ( G `
 P ) ) 
./\  W ) )
72 dia2dimlem2.gv . . . . . . . . . 10  |-  ( ph  ->  ( G `  P
)  =  Q )
73 dia2dimlem2.q . . . . . . . . . 10  |-  Q  =  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )
7472, 73syl6eq 2511 . . . . . . . . 9  |-  ( ph  ->  ( G `  P
)  =  ( ( P  .\/  U ) 
./\  ( ( F `
 P )  .\/  V ) ) )
7574oveq2d 6215 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  ( G `  P )
)  =  ( P 
.\/  ( ( P 
.\/  U )  ./\  ( ( F `  P )  .\/  V
) ) ) )
7675oveq1d 6214 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  ( G `  P ) )  ./\  W )  =  ( ( P 
.\/  ( ( P 
.\/  U )  ./\  ( ( F `  P )  .\/  V
) ) )  ./\  W ) )
7715, 16, 8hlatlej1 33342 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  ->  P  .<_  ( P  .\/  U ) )
782, 6, 12, 77syl3anc 1219 . . . . . . . . . 10  |-  ( ph  ->  P  .<_  ( P  .\/  U ) )
797, 15, 16, 21, 8atmod3i1 33831 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  ( P  .\/  U
)  e.  ( Base `  K )  /\  (
( F `  P
)  .\/  V )  e.  ( Base `  K
) )  /\  P  .<_  ( P  .\/  U
) )  ->  ( P  .\/  ( ( P 
.\/  U )  ./\  ( ( F `  P )  .\/  V
) ) )  =  ( ( P  .\/  U )  ./\  ( P  .\/  ( ( F `  P )  .\/  V
) ) ) )
802, 6, 20, 60, 78, 79syl131anc 1232 . . . . . . . . 9  |-  ( ph  ->  ( P  .\/  (
( P  .\/  U
)  ./\  ( ( F `  P )  .\/  V ) ) )  =  ( ( P 
.\/  U )  ./\  ( P  .\/  ( ( F `  P ) 
.\/  V ) ) ) )
8180oveq1d 6214 . . . . . . . 8  |-  ( ph  ->  ( ( P  .\/  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) ) ) 
./\  W )  =  ( ( ( P 
.\/  U )  ./\  ( P  .\/  ( ( F `  P ) 
.\/  V ) ) )  ./\  W )
)
82 hlol 33329 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OL )
832, 82syl 16 . . . . . . . . 9  |-  ( ph  ->  K  e.  OL )
847, 21latmassOLD 33197 . . . . . . . . 9  |-  ( ( K  e.  OL  /\  ( ( P  .\/  U )  e.  ( Base `  K )  /\  ( P  .\/  ( ( F `
 P )  .\/  V ) )  e.  (
Base `  K )  /\  W  e.  ( Base `  K ) ) )  ->  ( (
( P  .\/  U
)  ./\  ( P  .\/  ( ( F `  P )  .\/  V
) ) )  ./\  W )  =  ( ( P  .\/  U ) 
./\  ( ( P 
.\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) ) )
8583, 20, 62, 49, 84syl13anc 1221 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
.\/  U )  ./\  ( P  .\/  ( ( F `  P ) 
.\/  V ) ) )  ./\  W )  =  ( ( P 
.\/  U )  ./\  ( ( P  .\/  ( ( F `  P )  .\/  V
) )  ./\  W
) ) )
8681, 85eqtrd 2495 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) ) ) 
./\  W )  =  ( ( P  .\/  U )  ./\  ( ( P  .\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) ) )
8776, 86eqtrd 2495 . . . . . 6  |-  ( ph  ->  ( ( P  .\/  ( G `  P ) )  ./\  W )  =  ( ( P 
.\/  U )  ./\  ( ( P  .\/  ( ( F `  P )  .\/  V
) )  ./\  W
) ) )
8871, 87eqtrd 2495 . . . . 5  |-  ( ph  ->  ( R `  G
)  =  ( ( P  .\/  U ) 
./\  ( ( P 
.\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) ) )
8988eqcomd 2462 . . . 4  |-  ( ph  ->  ( ( P  .\/  U )  ./\  ( ( P  .\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) )  =  ( R `  G ) )
9068, 89breqtrd 4423 . . 3  |-  ( ph  ->  U  .<_  ( R `  G ) )
91 hlatl 33328 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
922, 91syl 16 . . . 4  |-  ( ph  ->  K  e.  AtLat )
93 hlop 33330 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OP )
942, 93syl 16 . . . . . . . . 9  |-  ( ph  ->  K  e.  OP )
95 eqid 2454 . . . . . . . . . 10  |-  ( 0.
`  K )  =  ( 0. `  K
)
96 eqid 2454 . . . . . . . . . 10  |-  ( lt
`  K )  =  ( lt `  K
)
9795, 96, 80ltat 33259 . . . . . . . . 9  |-  ( ( K  e.  OP  /\  U  e.  A )  ->  ( 0. `  K
) ( lt `  K ) U )
9894, 12, 97syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( 0. `  K
) ( lt `  K ) U )
99 hlpos 33333 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  Poset )
1002, 99syl 16 . . . . . . . . 9  |-  ( ph  ->  K  e.  Poset )
1017, 95op0cl 33152 . . . . . . . . . 10  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  ( Base `  K
) )
10294, 101syl 16 . . . . . . . . 9  |-  ( ph  ->  ( 0. `  K
)  e.  ( Base `  K ) )
1037, 27, 28, 29trlcl 34131 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( R `  G )  e.  (
Base `  K )
)
1041, 69, 103syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( R `  G
)  e.  ( Base `  K ) )
1057, 15, 96pltletr 15259 . . . . . . . . 9  |-  ( ( K  e.  Poset  /\  (
( 0. `  K
)  e.  ( Base `  K )  /\  U  e.  ( Base `  K
)  /\  ( R `  G )  e.  (
Base `  K )
) )  ->  (
( ( 0. `  K ) ( lt
`  K ) U  /\  U  .<_  ( R `
 G ) )  ->  ( 0. `  K ) ( lt
`  K ) ( R `  G ) ) )
106100, 102, 14, 104, 105syl13anc 1221 . . . . . . . 8  |-  ( ph  ->  ( ( ( 0.
`  K ) ( lt `  K ) U  /\  U  .<_  ( R `  G ) )  ->  ( 0. `  K ) ( lt
`  K ) ( R `  G ) ) )
10798, 90, 106mp2and 679 . . . . . . 7  |-  ( ph  ->  ( 0. `  K
) ( lt `  K ) ( R `
 G ) )
1087, 96, 95opltn0 33158 . . . . . . . 8  |-  ( ( K  e.  OP  /\  ( R `  G )  e.  ( Base `  K
) )  ->  (
( 0. `  K
) ( lt `  K ) ( R `
 G )  <->  ( R `  G )  =/=  ( 0. `  K ) ) )
10994, 104, 108syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( ( 0. `  K ) ( lt
`  K ) ( R `  G )  <-> 
( R `  G
)  =/=  ( 0.
`  K ) ) )
110107, 109mpbid 210 . . . . . 6  |-  ( ph  ->  ( R `  G
)  =/=  ( 0.
`  K ) )
111110neneqd 2654 . . . . 5  |-  ( ph  ->  -.  ( R `  G )  =  ( 0. `  K ) )
11295, 8, 27, 28, 29trlator0 34138 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( ( R `  G )  e.  A  \/  ( R `  G )  =  ( 0. `  K ) ) )
1131, 69, 112syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( ( R `  G )  e.  A  \/  ( R `  G
)  =  ( 0.
`  K ) ) )
114113orcomd 388 . . . . . 6  |-  ( ph  ->  ( ( R `  G )  =  ( 0. `  K )  \/  ( R `  G )  e.  A
) )
115114ord 377 . . . . 5  |-  ( ph  ->  ( -.  ( R `
 G )  =  ( 0. `  K
)  ->  ( R `  G )  e.  A
) )
116111, 115mpd 15 . . . 4  |-  ( ph  ->  ( R `  G
)  e.  A )
11715, 8atcmp 33279 . . . 4  |-  ( ( K  e.  AtLat  /\  U  e.  A  /\  ( R `  G )  e.  A )  ->  ( U  .<_  ( R `  G )  <->  U  =  ( R `  G ) ) )
11892, 12, 116, 117syl3anc 1219 . . 3  |-  ( ph  ->  ( U  .<_  ( R `
 G )  <->  U  =  ( R `  G ) ) )
11990, 118mpbid 210 . 2  |-  ( ph  ->  U  =  ( R `
 G ) )
120119eqcomd 2462 1  |-  ( ph  ->  ( R `  G
)  =  U )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2647   class class class wbr 4399   ` cfv 5525  (class class class)co 6199   Basecbs 14291   lecple 14363   Posetcpo 15228   ltcplt 15229   joincjn 15232   meetcmee 15233   0.cp0 15325   Latclat 15333   OPcops 33140   OLcol 33142   Atomscatm 33231   AtLatcal 33232   HLchlt 33318   LHypclh 33951   LTrncltrn 34068   trLctrl 34125
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 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-1st 6686  df-2nd 6687  df-map 7325  df-poset 15234  df-plt 15246  df-lub 15262  df-glb 15263  df-join 15264  df-meet 15265  df-p0 15327  df-p1 15328  df-lat 15334  df-clat 15396  df-oposet 33144  df-ol 33146  df-oml 33147  df-covers 33234  df-ats 33235  df-atl 33266  df-cvlat 33290  df-hlat 33319  df-psubsp 33470  df-pmap 33471  df-padd 33763  df-lhyp 33955  df-laut 33956  df-ldil 34071  df-ltrn 34072  df-trl 34126
This theorem is referenced by:  dia2dimlem5  35036
  Copyright terms: Public domain W3C validator