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Theorem dia2dimlem2 31864
Description: Lemma for dia2dim 31876. 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 447 . . . . . . . 8  |-  ( ph  ->  K  e.  HL )
3 hllat 30162 . . . . . . . 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 447 . . . . . . . 8  |-  ( ph  ->  P  e.  A )
7 eqid 2437 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
8 dia2dimlem2.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
97, 8atbase 30088 . . . . . . . 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 447 . . . . . . . 8  |-  ( ph  ->  U  e.  A )
137, 8atbase 30088 . . . . . . . 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 14491 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) )  ->  U  .<_  ( P  .\/  U
) )
184, 10, 14, 17syl3anc 1185 . . . . . 6  |-  ( ph  ->  U  .<_  ( P  .\/  U ) )
197, 16, 8hlatjcl 30165 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
202, 6, 12, 19syl3anc 1185 . . . . . . 7  |-  ( ph  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
21 dia2dimlem2.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
227, 15, 21latleeqm2 14510 . . . . . . 7  |-  ( ( K  e.  Lat  /\  U  e.  ( Base `  K )  /\  ( P  .\/  U )  e.  ( Base `  K
) )  ->  ( U  .<_  ( P  .\/  U )  <->  ( ( P 
.\/  U )  ./\  U )  =  U ) )
234, 14, 20, 22syl3anc 1185 . . . . . 6  |-  ( ph  ->  ( U  .<_  ( P 
.\/  U )  <->  ( ( P  .\/  U )  ./\  U )  =  U ) )
2418, 23mpbid 203 . . . . 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 30967 . . . . . . . . . 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 1185 . . . . . . . . 9  |-  ( ph  ->  ( R `  F
)  e.  A )
32 dia2dimlem2.v . . . . . . . . . 10  |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )
3332simpld 447 . . . . . . . . 9  |-  ( ph  ->  V  e.  A )
34 dia2dimlem2.rv . . . . . . . . 9  |-  ( ph  ->  ( R `  F
)  =/=  V )
3515, 16, 8hlatexch2 30194 . . . . . . . . 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 1198 . . . . . . . 8  |-  ( ph  ->  ( ( R `  F )  .<_  ( U 
.\/  V )  ->  U  .<_  ( ( R `
 F )  .\/  V ) ) )
3725, 36mpd 15 . . . . . . 7  |-  ( ph  ->  U  .<_  ( ( R `  F )  .\/  V ) )
3826simpld 447 . . . . . . . . . 10  |-  ( ph  ->  F  e.  T )
3915, 16, 21, 8, 27, 28, 29trlval2 30961 . . . . . . . . . 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 1185 . . . . . . . . 9  |-  ( ph  ->  ( R `  F
)  =  ( ( P  .\/  ( F `
 P ) ) 
./\  W ) )
4140oveq1d 6097 . . . . . . . 8  |-  ( ph  ->  ( ( R `  F )  .\/  V
)  =  ( ( ( P  .\/  ( F `  P )
)  ./\  W )  .\/  V ) )
4215, 8, 27, 28ltrnel 30937 . . . . . . . . . . . . 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 1185 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
4443simpld 447 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  P
)  e.  A )
457, 16, 8hlatjcl 30165 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  P )  e.  A )  -> 
( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
462, 6, 44, 45syl3anc 1185 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
471simprd 451 . . . . . . . . . . 11  |-  ( ph  ->  W  e.  H )
487, 27lhpbase 30796 . . . . . . . . . . 11  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
4947, 48syl 16 . . . . . . . . . 10  |-  ( ph  ->  W  e.  ( Base `  K ) )
5032simprd 451 . . . . . . . . . 10  |-  ( ph  ->  V  .<_  W )
517, 15, 16, 21, 8atmod4i1 30664 . . . . . . . . . 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 1198 . . . . . . . . 9  |-  ( ph  ->  ( ( ( P 
.\/  ( F `  P ) )  ./\  W )  .\/  V )  =  ( ( ( P  .\/  ( F `
 P ) ) 
.\/  V )  ./\  W ) )
5316, 8hlatjass 30168 . . . . . . . . . . 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 1187 . . . . . . . . . 10  |-  ( ph  ->  ( ( P  .\/  ( F `  P ) )  .\/  V )  =  ( P  .\/  ( ( F `  P )  .\/  V
) ) )
5554oveq1d 6097 . . . . . . . . 9  |-  ( ph  ->  ( ( ( P 
.\/  ( F `  P ) )  .\/  V )  ./\  W )  =  ( ( P 
.\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) )
5652, 55eqtrd 2469 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
.\/  ( F `  P ) )  ./\  W )  .\/  V )  =  ( ( P 
.\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) )
5741, 56eqtrd 2469 . . . . . . 7  |-  ( ph  ->  ( ( R `  F )  .\/  V
)  =  ( ( P  .\/  ( ( F `  P ) 
.\/  V ) ) 
./\  W ) )
5837, 57breqtrd 4237 . . . . . 6  |-  ( ph  ->  U  .<_  ( ( P  .\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) )
597, 16, 8hlatjcl 30165 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  V  e.  A )  ->  (
( F `  P
)  .\/  V )  e.  ( Base `  K
) )
602, 44, 33, 59syl3anc 1185 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  P )  .\/  V
)  e.  ( Base `  K ) )
617, 16latjcl 14480 . . . . . . . . 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 1185 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  (
( F `  P
)  .\/  V )
)  e.  ( Base `  K ) )
637, 21latmcl 14481 . . . . . . . 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 1185 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  ( ( F `  P )  .\/  V
) )  ./\  W
)  e.  ( Base `  K ) )
657, 15, 21latmlem2 14512 . . . . . . 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 1187 . . . . . 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 4235 . . . 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 30961 . . . . . . 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 1185 . . . . . 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 2485 . . . . . . . . 9  |-  ( ph  ->  ( G `  P
)  =  ( ( P  .\/  U ) 
./\  ( ( F `
 P )  .\/  V ) ) )
7574oveq2d 6098 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  ( G `  P )
)  =  ( P 
.\/  ( ( P 
.\/  U )  ./\  ( ( F `  P )  .\/  V
) ) ) )
7675oveq1d 6097 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  ( G `  P ) )  ./\  W )  =  ( ( P 
.\/  ( ( P 
.\/  U )  ./\  ( ( F `  P )  .\/  V
) ) )  ./\  W ) )
7715, 16, 8hlatlej1 30173 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  ->  P  .<_  ( P  .\/  U ) )
782, 6, 12, 77syl3anc 1185 . . . . . . . . . 10  |-  ( ph  ->  P  .<_  ( P  .\/  U ) )
797, 15, 16, 21, 8atmod3i1 30662 . . . . . . . . . 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 1198 . . . . . . . . 9  |-  ( ph  ->  ( P  .\/  (
( P  .\/  U
)  ./\  ( ( F `  P )  .\/  V ) ) )  =  ( ( P 
.\/  U )  ./\  ( P  .\/  ( ( F `  P ) 
.\/  V ) ) ) )
8180oveq1d 6097 . . . . . . . 8  |-  ( ph  ->  ( ( P  .\/  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) ) ) 
./\  W )  =  ( ( ( P 
.\/  U )  ./\  ( P  .\/  ( ( F `  P ) 
.\/  V ) ) )  ./\  W )
)
82 hlol 30160 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OL )
832, 82syl 16 . . . . . . . . 9  |-  ( ph  ->  K  e.  OL )
847, 21latmassOLD 30028 . . . . . . . . 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 1187 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
.\/  U )  ./\  ( P  .\/  ( ( F `  P ) 
.\/  V ) ) )  ./\  W )  =  ( ( P 
.\/  U )  ./\  ( ( P  .\/  ( ( F `  P )  .\/  V
) )  ./\  W
) ) )
8681, 85eqtrd 2469 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) ) ) 
./\  W )  =  ( ( P  .\/  U )  ./\  ( ( P  .\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) ) )
8776, 86eqtrd 2469 . . . . . 6  |-  ( ph  ->  ( ( P  .\/  ( G `  P ) )  ./\  W )  =  ( ( P 
.\/  U )  ./\  ( ( P  .\/  ( ( F `  P )  .\/  V
) )  ./\  W
) ) )
8871, 87eqtrd 2469 . . . . 5  |-  ( ph  ->  ( R `  G
)  =  ( ( P  .\/  U ) 
./\  ( ( P 
.\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) ) )
8988eqcomd 2442 . . . 4  |-  ( ph  ->  ( ( P  .\/  U )  ./\  ( ( P  .\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) )  =  ( R `  G ) )
9068, 89breqtrd 4237 . . 3  |-  ( ph  ->  U  .<_  ( R `  G ) )
91 hlatl 30159 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
922, 91syl 16 . . . 4  |-  ( ph  ->  K  e.  AtLat )
93 hlop 30161 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OP )
942, 93syl 16 . . . . . . . . 9  |-  ( ph  ->  K  e.  OP )
95 eqid 2437 . . . . . . . . . 10  |-  ( 0.
`  K )  =  ( 0. `  K
)
96 eqid 2437 . . . . . . . . . 10  |-  ( lt
`  K )  =  ( lt `  K
)
9795, 96, 80ltat 30090 . . . . . . . . 9  |-  ( ( K  e.  OP  /\  U  e.  A )  ->  ( 0. `  K
) ( lt `  K ) U )
9894, 12, 97syl2anc 644 . . . . . . . 8  |-  ( ph  ->  ( 0. `  K
) ( lt `  K ) U )
99 hlpos 30164 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  Poset )
1002, 99syl 16 . . . . . . . . 9  |-  ( ph  ->  K  e.  Poset )
1017, 95op0cl 29983 . . . . . . . . . 10  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  ( Base `  K
) )
10294, 101syl 16 . . . . . . . . 9  |-  ( ph  ->  ( 0. `  K
)  e.  ( Base `  K ) )
1037, 27, 28, 29trlcl 30962 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( R `  G )  e.  (
Base `  K )
)
1041, 69, 103syl2anc 644 . . . . . . . . 9  |-  ( ph  ->  ( R `  G
)  e.  ( Base `  K ) )
1057, 15, 96pltletr 14429 . . . . . . . . 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 1187 . . . . . . . 8  |-  ( ph  ->  ( ( ( 0.
`  K ) ( lt `  K ) U  /\  U  .<_  ( R `  G ) )  ->  ( 0. `  K ) ( lt
`  K ) ( R `  G ) ) )
10798, 90, 106mp2and 662 . . . . . . 7  |-  ( ph  ->  ( 0. `  K
) ( lt `  K ) ( R `
 G ) )
1087, 96, 95opltn0 29989 . . . . . . . 8  |-  ( ( K  e.  OP  /\  ( R `  G )  e.  ( Base `  K
) )  ->  (
( 0. `  K
) ( lt `  K ) ( R `
 G )  <->  ( R `  G )  =/=  ( 0. `  K ) ) )
10994, 104, 108syl2anc 644 . . . . . . 7  |-  ( ph  ->  ( ( 0. `  K ) ( lt
`  K ) ( R `  G )  <-> 
( R `  G
)  =/=  ( 0.
`  K ) ) )
110107, 109mpbid 203 . . . . . 6  |-  ( ph  ->  ( R `  G
)  =/=  ( 0.
`  K ) )
111110neneqd 2618 . . . . 5  |-  ( ph  ->  -.  ( R `  G )  =  ( 0. `  K ) )
11295, 8, 27, 28, 29trlator0 30969 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( ( R `  G )  e.  A  \/  ( R `  G )  =  ( 0. `  K ) ) )
1131, 69, 112syl2anc 644 . . . . . . 7  |-  ( ph  ->  ( ( R `  G )  e.  A  \/  ( R `  G
)  =  ( 0.
`  K ) ) )
114113orcomd 379 . . . . . 6  |-  ( ph  ->  ( ( R `  G )  =  ( 0. `  K )  \/  ( R `  G )  e.  A
) )
115114ord 368 . . . . 5  |-  ( ph  ->  ( -.  ( R `
 G )  =  ( 0. `  K
)  ->  ( R `  G )  e.  A
) )
116111, 115mpd 15 . . . 4  |-  ( ph  ->  ( R `  G
)  e.  A )
11715, 8atcmp 30110 . . . 4  |-  ( ( K  e.  AtLat  /\  U  e.  A  /\  ( R `  G )  e.  A )  ->  ( U  .<_  ( R `  G )  <->  U  =  ( R `  G ) ) )
11892, 12, 116, 117syl3anc 1185 . . 3  |-  ( ph  ->  ( U  .<_  ( R `
 G )  <->  U  =  ( R `  G ) ) )
11990, 118mpbid 203 . 2  |-  ( ph  ->  U  =  ( R `
 G ) )
120119eqcomd 2442 1  |-  ( ph  ->  ( R `  G
)  =  U )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2600   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   Basecbs 13470   lecple 13537   Posetcpo 14398   ltcplt 14399   joincjn 14402   meetcmee 14403   0.cp0 14467   Latclat 14475   OPcops 29971   OLcol 29973   Atomscatm 30062   AtLatcal 30063   HLchlt 30149   LHypclh 30782   LTrncltrn 30899   trLctrl 30956
This theorem is referenced by:  dia2dimlem5  31867
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-iin 4097  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-undef 6544  df-riota 6550  df-map 7021  df-poset 14404  df-plt 14416  df-lub 14432  df-glb 14433  df-join 14434  df-meet 14435  df-p0 14469  df-p1 14470  df-lat 14476  df-clat 14538  df-oposet 29975  df-ol 29977  df-oml 29978  df-covers 30065  df-ats 30066  df-atl 30097  df-cvlat 30121  df-hlat 30150  df-psubsp 30301  df-pmap 30302  df-padd 30594  df-lhyp 30786  df-laut 30787  df-ldil 30902  df-ltrn 30903  df-trl 30957
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