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Theorem dia2dimlem2 30523
Description: Lemma for dia2dim 30535. 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 28821 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 17 . . . . . . 7  |-  ( ph  ->  K  e.  Lat )
5 dia2dimlem2.p . . . . . . . . 9  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
65simpld 447 . . . . . . . 8  |-  ( ph  ->  P  e.  A )
7 eqid 2285 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
8 dia2dimlem2.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
97, 8atbase 28747 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
106, 9syl 17 . . . . . . 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 28747 . . . . . . . 8  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
1412, 13syl 17 . . . . . . 7  |-  ( ph  ->  U  e.  ( Base `  K ) )
15 dia2dimlem2.l . . . . . . . 8  |-  .<_  =  ( le `  K )
16 dia2dimlem2.j . . . . . . . 8  |-  .\/  =  ( join `  K )
177, 15, 16latlej2 14162 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) )  ->  U  .<_  ( P  .\/  U
) )
184, 10, 14, 17syl3anc 1184 . . . . . 6  |-  ( ph  ->  U  .<_  ( P  .\/  U ) )
197, 16, 8hlatjcl 28824 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
202, 6, 12, 19syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
21 dia2dimlem2.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
227, 15, 21latleeqm2 14181 . . . . . . 7  |-  ( ( K  e.  Lat  /\  U  e.  ( Base `  K )  /\  ( P  .\/  U )  e.  ( Base `  K
) )  ->  ( U  .<_  ( P  .\/  U )  <->  ( ( P 
.\/  U )  ./\  U )  =  U ) )
234, 14, 20, 22syl3anc 1184 . . . . . 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 29626 . . . . . . . . . 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 1184 . . . . . . . . 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 28853 . . . . . . . . 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 1197 . . . . . . . 8  |-  ( ph  ->  ( ( R `  F )  .<_  ( U 
.\/  V )  ->  U  .<_  ( ( R `
 F )  .\/  V ) ) )
3725, 36mpd 16 . . . . . . 7  |-  ( ph  ->  U  .<_  ( ( R `  F )  .\/  V ) )
3826simpld 447 . . . . . . . . . 10  |-  ( ph  ->  F  e.  T )
3915, 16, 21, 8, 27, 28, 29trlval2 29620 . . . . . . . . . 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 1184 . . . . . . . . 9  |-  ( ph  ->  ( R `  F
)  =  ( ( P  .\/  ( F `
 P ) ) 
./\  W ) )
4140oveq1d 5835 . . . . . . . 8  |-  ( ph  ->  ( ( R `  F )  .\/  V
)  =  ( ( ( P  .\/  ( F `  P )
)  ./\  W )  .\/  V ) )
4215, 8, 27, 28ltrnel 29596 . . . . . . . . . . . . 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 1184 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
4443simpld 447 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  P
)  e.  A )
457, 16, 8hlatjcl 28824 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  P )  e.  A )  -> 
( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
462, 6, 44, 45syl3anc 1184 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
471simprd 451 . . . . . . . . . . 11  |-  ( ph  ->  W  e.  H )
487, 27lhpbase 29455 . . . . . . . . . . 11  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
4947, 48syl 17 . . . . . . . . . 10  |-  ( ph  ->  W  e.  ( Base `  K ) )
5032simprd 451 . . . . . . . . . 10  |-  ( ph  ->  V  .<_  W )
517, 15, 16, 21, 8atmod4i1 29323 . . . . . . . . . 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 1197 . . . . . . . . 9  |-  ( ph  ->  ( ( ( P 
.\/  ( F `  P ) )  ./\  W )  .\/  V )  =  ( ( ( P  .\/  ( F `
 P ) ) 
.\/  V )  ./\  W ) )
5316, 8hlatjass 28827 . . . . . . . . . . 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 1186 . . . . . . . . . 10  |-  ( ph  ->  ( ( P  .\/  ( F `  P ) )  .\/  V )  =  ( P  .\/  ( ( F `  P )  .\/  V
) ) )
5554oveq1d 5835 . . . . . . . . 9  |-  ( ph  ->  ( ( ( P 
.\/  ( F `  P ) )  .\/  V )  ./\  W )  =  ( ( P 
.\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) )
5652, 55eqtrd 2317 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
.\/  ( F `  P ) )  ./\  W )  .\/  V )  =  ( ( P 
.\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) )
5741, 56eqtrd 2317 . . . . . . 7  |-  ( ph  ->  ( ( R `  F )  .\/  V
)  =  ( ( P  .\/  ( ( F `  P ) 
.\/  V ) ) 
./\  W ) )
5837, 57breqtrd 4049 . . . . . 6  |-  ( ph  ->  U  .<_  ( ( P  .\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) )
597, 16, 8hlatjcl 28824 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  V  e.  A )  ->  (
( F `  P
)  .\/  V )  e.  ( Base `  K
) )
602, 44, 33, 59syl3anc 1184 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  P )  .\/  V
)  e.  ( Base `  K ) )
617, 16latjcl 14151 . . . . . . . . 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 1184 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  (
( F `  P
)  .\/  V )
)  e.  ( Base `  K ) )
637, 21latmcl 14152 . . . . . . . 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 1184 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  ( ( F `  P )  .\/  V
) )  ./\  W
)  e.  ( Base `  K ) )
657, 15, 21latmlem2 14183 . . . . . . 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 1186 . . . . . 6  |-  ( ph  ->  ( U  .<_  ( ( P  .\/  ( ( F `  P ) 
.\/  V ) ) 
./\  W )  -> 
( ( P  .\/  U )  ./\  U )  .<_  ( ( P  .\/  U )  ./\  ( ( P  .\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) ) ) )
6758, 66mpd 16 . . . . 5  |-  ( ph  ->  ( ( P  .\/  U )  ./\  U )  .<_  ( ( P  .\/  U )  ./\  ( ( P  .\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) ) )
6824, 67eqbrtrrd 4047 . . . 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 29620 . . . . . . 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 1184 . . . . . 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 2333 . . . . . . . . 9  |-  ( ph  ->  ( G `  P
)  =  ( ( P  .\/  U ) 
./\  ( ( F `
 P )  .\/  V ) ) )
7574oveq2d 5836 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  ( G `  P )
)  =  ( P 
.\/  ( ( P 
.\/  U )  ./\  ( ( F `  P )  .\/  V
) ) ) )
7675oveq1d 5835 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  ( G `  P ) )  ./\  W )  =  ( ( P 
.\/  ( ( P 
.\/  U )  ./\  ( ( F `  P )  .\/  V
) ) )  ./\  W ) )
7715, 16, 8hlatlej1 28832 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  ->  P  .<_  ( P  .\/  U ) )
782, 6, 12, 77syl3anc 1184 . . . . . . . . . 10  |-  ( ph  ->  P  .<_  ( P  .\/  U ) )
797, 15, 16, 21, 8atmod3i1 29321 . . . . . . . . . 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 1197 . . . . . . . . 9  |-  ( ph  ->  ( P  .\/  (
( P  .\/  U
)  ./\  ( ( F `  P )  .\/  V ) ) )  =  ( ( P 
.\/  U )  ./\  ( P  .\/  ( ( F `  P ) 
.\/  V ) ) ) )
8180oveq1d 5835 . . . . . . . 8  |-  ( ph  ->  ( ( P  .\/  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) ) ) 
./\  W )  =  ( ( ( P 
.\/  U )  ./\  ( P  .\/  ( ( F `  P ) 
.\/  V ) ) )  ./\  W )
)
82 hlol 28819 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OL )
832, 82syl 17 . . . . . . . . 9  |-  ( ph  ->  K  e.  OL )
847, 21latmassOLD 28687 . . . . . . . . 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 1186 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
.\/  U )  ./\  ( P  .\/  ( ( F `  P ) 
.\/  V ) ) )  ./\  W )  =  ( ( P 
.\/  U )  ./\  ( ( P  .\/  ( ( F `  P )  .\/  V
) )  ./\  W
) ) )
8681, 85eqtrd 2317 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) ) ) 
./\  W )  =  ( ( P  .\/  U )  ./\  ( ( P  .\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) ) )
8776, 86eqtrd 2317 . . . . . 6  |-  ( ph  ->  ( ( P  .\/  ( G `  P ) )  ./\  W )  =  ( ( P 
.\/  U )  ./\  ( ( P  .\/  ( ( F `  P )  .\/  V
) )  ./\  W
) ) )
8871, 87eqtrd 2317 . . . . 5  |-  ( ph  ->  ( R `  G
)  =  ( ( P  .\/  U ) 
./\  ( ( P 
.\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) ) )
8988eqcomd 2290 . . . 4  |-  ( ph  ->  ( ( P  .\/  U )  ./\  ( ( P  .\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) )  =  ( R `  G ) )
9068, 89breqtrd 4049 . . 3  |-  ( ph  ->  U  .<_  ( R `  G ) )
91 hlatl 28818 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
922, 91syl 17 . . . 4  |-  ( ph  ->  K  e.  AtLat )
93 hlop 28820 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OP )
942, 93syl 17 . . . . . . . . 9  |-  ( ph  ->  K  e.  OP )
95 eqid 2285 . . . . . . . . . 10  |-  ( 0.
`  K )  =  ( 0. `  K
)
96 eqid 2285 . . . . . . . . . 10  |-  ( lt
`  K )  =  ( lt `  K
)
9795, 96, 80ltat 28749 . . . . . . . . 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 28823 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  Poset )
1002, 99syl 17 . . . . . . . . 9  |-  ( ph  ->  K  e.  Poset )
1017, 95op0cl 28642 . . . . . . . . . 10  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  ( Base `  K
) )
10294, 101syl 17 . . . . . . . . 9  |-  ( ph  ->  ( 0. `  K
)  e.  ( Base `  K ) )
1037, 27, 28, 29trlcl 29621 . . . . . . . . . 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 14100 . . . . . . . . 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 1186 . . . . . . . 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 28648 . . . . . . . 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 2464 . . . . 5  |-  ( ph  ->  -.  ( R `  G )  =  ( 0. `  K ) )
11295, 8, 27, 28, 29trlator0 29628 . . . . . . . 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 16 . . . 4  |-  ( ph  ->  ( R `  G
)  e.  A )
11715, 8atcmp 28769 . . . 4  |-  ( ( K  e.  AtLat  /\  U  e.  A  /\  ( R `  G )  e.  A )  ->  ( U  .<_  ( R `  G )  <->  U  =  ( R `  G ) ) )
11892, 12, 116, 117syl3anc 1184 . . 3  |-  ( ph  ->  ( U  .<_  ( R `
 G )  <->  U  =  ( R `  G ) ) )
11990, 118mpbid 203 . 2  |-  ( ph  ->  U  =  ( R `
 G ) )
120119eqcomd 2290 1  |-  ( ph  ->  ( R `  G
)  =  U )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1624    e. wcel 1685    =/= wne 2448   class class class wbr 4025   ` cfv 5222  (class class class)co 5820   Basecbs 13143   lecple 13210   Posetcpo 14069   ltcplt 14070   joincjn 14073   meetcmee 14074   0.cp0 14138   Latclat 14146   OPcops 28630   OLcol 28632   Atomscatm 28721   AtLatcal 28722   HLchlt 28808   LHypclh 29441   LTrncltrn 29558   trLctrl 29615
This theorem is referenced by:  dia2dimlem5  30526
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-map 6770  df-poset 14075  df-plt 14087  df-lub 14103  df-glb 14104  df-join 14105  df-meet 14106  df-p0 14140  df-p1 14141  df-lat 14147  df-clat 14209  df-oposet 28634  df-ol 28636  df-oml 28637  df-covers 28724  df-ats 28725  df-atl 28756  df-cvlat 28780  df-hlat 28809  df-psubsp 28960  df-pmap 28961  df-padd 29253  df-lhyp 29445  df-laut 29446  df-ldil 29561  df-ltrn 29562  df-trl 29616
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