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

Theorem dia2dimlem2 29944
Description: Lemma for dia2dim 29956. 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 28242 . . . . . . . 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 2253 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
8 dia2dimlem2.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
97, 8atbase 28168 . . . . . . . 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 28168 . . . . . . . 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 14011 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) )  ->  U  .<_  ( P  .\/  U
) )
184, 10, 14, 17syl3anc 1187 . . . . . 6  |-  ( ph  ->  U  .<_  ( P  .\/  U ) )
197, 16, 8hlatjcl 28245 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
202, 6, 12, 19syl3anc 1187 . . . . . . 7  |-  ( ph  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
21 dia2dimlem2.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
227, 15, 21latleeqm2 14030 . . . . . . 7  |-  ( ( K  e.  Lat  /\  U  e.  ( Base `  K )  /\  ( P  .\/  U )  e.  ( Base `  K
) )  ->  ( U  .<_  ( P  .\/  U )  <->  ( ( P 
.\/  U )  ./\  U )  =  U ) )
234, 14, 20, 22syl3anc 1187 . . . . . 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 29047 . . . . . . . . . 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 1187 . . . . . . . . 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 28274 . . . . . . . . 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 1200 . . . . . . . 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 29041 . . . . . . . . . 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 1187 . . . . . . . . 9  |-  ( ph  ->  ( R `  F
)  =  ( ( P  .\/  ( F `
 P ) ) 
./\  W ) )
4140oveq1d 5725 . . . . . . . 8  |-  ( ph  ->  ( ( R `  F )  .\/  V
)  =  ( ( ( P  .\/  ( F `  P )
)  ./\  W )  .\/  V ) )
4215, 8, 27, 28ltrnel 29017 . . . . . . . . . . . . 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 1187 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
4443simpld 447 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  P
)  e.  A )
457, 16, 8hlatjcl 28245 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  P )  e.  A )  -> 
( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
462, 6, 44, 45syl3anc 1187 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  ( F `  P )
)  e.  ( Base `  K ) )
471simprd 451 . . . . . . . . . . 11  |-  ( ph  ->  W  e.  H )
487, 27lhpbase 28876 . . . . . . . . . . 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 28744 . . . . . . . . . 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 1200 . . . . . . . . 9  |-  ( ph  ->  ( ( ( P 
.\/  ( F `  P ) )  ./\  W )  .\/  V )  =  ( ( ( P  .\/  ( F `
 P ) ) 
.\/  V )  ./\  W ) )
5316, 8hlatjass 28248 . . . . . . . . . . 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 1189 . . . . . . . . . 10  |-  ( ph  ->  ( ( P  .\/  ( F `  P ) )  .\/  V )  =  ( P  .\/  ( ( F `  P )  .\/  V
) ) )
5554oveq1d 5725 . . . . . . . . 9  |-  ( ph  ->  ( ( ( P 
.\/  ( F `  P ) )  .\/  V )  ./\  W )  =  ( ( P 
.\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) )
5652, 55eqtrd 2285 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
.\/  ( F `  P ) )  ./\  W )  .\/  V )  =  ( ( P 
.\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) )
5741, 56eqtrd 2285 . . . . . . 7  |-  ( ph  ->  ( ( R `  F )  .\/  V
)  =  ( ( P  .\/  ( ( F `  P ) 
.\/  V ) ) 
./\  W ) )
5837, 57breqtrd 3944 . . . . . 6  |-  ( ph  ->  U  .<_  ( ( P  .\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) )
597, 16, 8hlatjcl 28245 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( F `  P )  e.  A  /\  V  e.  A )  ->  (
( F `  P
)  .\/  V )  e.  ( Base `  K
) )
602, 44, 33, 59syl3anc 1187 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  P )  .\/  V
)  e.  ( Base `  K ) )
617, 16latjcl 14000 . . . . . . . . 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 1187 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  (
( F `  P
)  .\/  V )
)  e.  ( Base `  K ) )
637, 21latmcl 14001 . . . . . . . 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 1187 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  ( ( F `  P )  .\/  V
) )  ./\  W
)  e.  ( Base `  K ) )
657, 15, 21latmlem2 14032 . . . . . . 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 1189 . . . . . 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 3942 . . . 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 29041 . . . . . . 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 1187 . . . . . 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 2301 . . . . . . . . 9  |-  ( ph  ->  ( G `  P
)  =  ( ( P  .\/  U ) 
./\  ( ( F `
 P )  .\/  V ) ) )
7574oveq2d 5726 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  ( G `  P )
)  =  ( P 
.\/  ( ( P 
.\/  U )  ./\  ( ( F `  P )  .\/  V
) ) ) )
7675oveq1d 5725 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  ( G `  P ) )  ./\  W )  =  ( ( P 
.\/  ( ( P 
.\/  U )  ./\  ( ( F `  P )  .\/  V
) ) )  ./\  W ) )
7715, 16, 8hlatlej1 28253 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  ->  P  .<_  ( P  .\/  U ) )
782, 6, 12, 77syl3anc 1187 . . . . . . . . . 10  |-  ( ph  ->  P  .<_  ( P  .\/  U ) )
797, 15, 16, 21, 8atmod3i1 28742 . . . . . . . . . 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 1200 . . . . . . . . 9  |-  ( ph  ->  ( P  .\/  (
( P  .\/  U
)  ./\  ( ( F `  P )  .\/  V ) ) )  =  ( ( P 
.\/  U )  ./\  ( P  .\/  ( ( F `  P ) 
.\/  V ) ) ) )
8180oveq1d 5725 . . . . . . . 8  |-  ( ph  ->  ( ( P  .\/  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) ) ) 
./\  W )  =  ( ( ( P 
.\/  U )  ./\  ( P  .\/  ( ( F `  P ) 
.\/  V ) ) )  ./\  W )
)
82 hlol 28240 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OL )
832, 82syl 17 . . . . . . . . 9  |-  ( ph  ->  K  e.  OL )
847, 21latmassOLD 28108 . . . . . . . . 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 1189 . . . . . . . 8  |-  ( ph  ->  ( ( ( P 
.\/  U )  ./\  ( P  .\/  ( ( F `  P ) 
.\/  V ) ) )  ./\  W )  =  ( ( P 
.\/  U )  ./\  ( ( P  .\/  ( ( F `  P )  .\/  V
) )  ./\  W
) ) )
8681, 85eqtrd 2285 . . . . . . 7  |-  ( ph  ->  ( ( P  .\/  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) ) ) 
./\  W )  =  ( ( P  .\/  U )  ./\  ( ( P  .\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) ) )
8776, 86eqtrd 2285 . . . . . 6  |-  ( ph  ->  ( ( P  .\/  ( G `  P ) )  ./\  W )  =  ( ( P 
.\/  U )  ./\  ( ( P  .\/  ( ( F `  P )  .\/  V
) )  ./\  W
) ) )
8871, 87eqtrd 2285 . . . . 5  |-  ( ph  ->  ( R `  G
)  =  ( ( P  .\/  U ) 
./\  ( ( P 
.\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) ) )
8988eqcomd 2258 . . . 4  |-  ( ph  ->  ( ( P  .\/  U )  ./\  ( ( P  .\/  ( ( F `
 P )  .\/  V ) )  ./\  W
) )  =  ( R `  G ) )
9068, 89breqtrd 3944 . . 3  |-  ( ph  ->  U  .<_  ( R `  G ) )
91 hlatl 28239 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
922, 91syl 17 . . . 4  |-  ( ph  ->  K  e.  AtLat )
93 hlop 28241 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OP )
942, 93syl 17 . . . . . . . . 9  |-  ( ph  ->  K  e.  OP )
95 eqid 2253 . . . . . . . . . 10  |-  ( 0.
`  K )  =  ( 0. `  K
)
96 eqid 2253 . . . . . . . . . 10  |-  ( lt
`  K )  =  ( lt `  K
)
9795, 96, 80ltat 28170 . . . . . . . . 9  |-  ( ( K  e.  OP  /\  U  e.  A )  ->  ( 0. `  K
) ( lt `  K ) U )
9894, 12, 97syl2anc 645 . . . . . . . 8  |-  ( ph  ->  ( 0. `  K
) ( lt `  K ) U )
99 hlpos 28244 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  Poset )
1002, 99syl 17 . . . . . . . . 9  |-  ( ph  ->  K  e.  Poset )
1017, 95op0cl 28063 . . . . . . . . . 10  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  ( Base `  K
) )
10294, 101syl 17 . . . . . . . . 9  |-  ( ph  ->  ( 0. `  K
)  e.  ( Base `  K ) )
1037, 27, 28, 29trlcl 29042 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( R `  G )  e.  (
Base `  K )
)
1041, 69, 103syl2anc 645 . . . . . . . . 9  |-  ( ph  ->  ( R `  G
)  e.  ( Base `  K ) )
1057, 15, 96pltletr 13949 . . . . . . . . 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 1189 . . . . . . . 8  |-  ( ph  ->  ( ( ( 0.
`  K ) ( lt `  K ) U  /\  U  .<_  ( R `  G ) )  ->  ( 0. `  K ) ( lt
`  K ) ( R `  G ) ) )
10798, 90, 106mp2and 663 . . . . . . 7  |-  ( ph  ->  ( 0. `  K
) ( lt `  K ) ( R `
 G ) )
1087, 96, 95opltn0 28069 . . . . . . . 8  |-  ( ( K  e.  OP  /\  ( R `  G )  e.  ( Base `  K
) )  ->  (
( 0. `  K
) ( lt `  K ) ( R `
 G )  <->  ( R `  G )  =/=  ( 0. `  K ) ) )
10994, 104, 108syl2anc 645 . . . . . . 7  |-  ( ph  ->  ( ( 0. `  K ) ( lt
`  K ) ( R `  G )  <-> 
( R `  G
)  =/=  ( 0.
`  K ) ) )
110107, 109mpbid 203 . . . . . 6  |-  ( ph  ->  ( R `  G
)  =/=  ( 0.
`  K ) )
111110neneqd 2428 . . . . 5  |-  ( ph  ->  -.  ( R `  G )  =  ( 0. `  K ) )
11295, 8, 27, 28, 29trlator0 29049 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( ( R `  G )  e.  A  \/  ( R `  G )  =  ( 0. `  K ) ) )
1131, 69, 112syl2anc 645 . . . . . . 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 28190 . . . 4  |-  ( ( K  e.  AtLat  /\  U  e.  A  /\  ( R `  G )  e.  A )  ->  ( U  .<_  ( R `  G )  <->  U  =  ( R `  G ) ) )
11892, 12, 116, 117syl3anc 1187 . . 3  |-  ( ph  ->  ( U  .<_  ( R `
 G )  <->  U  =  ( R `  G ) ) )
11990, 118mpbid 203 . 2  |-  ( ph  ->  U  =  ( R `
 G ) )
120119eqcomd 2258 1  |-  ( ph  ->  ( R `  G
)  =  U )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2412   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   Basecbs 13022   lecple 13089   Posetcpo 13918   ltcplt 13919   joincjn 13922   meetcmee 13923   0.cp0 13987   Latclat 13995   OPcops 28051   OLcol 28053   Atomscatm 28142   AtLatcal 28143   HLchlt 28229   LHypclh 28862   LTrncltrn 28979   trLctrl 29036
This theorem is referenced by:  dia2dimlem5  29947
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-map 6660  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-psubsp 28381  df-pmap 28382  df-padd 28674  df-lhyp 28866  df-laut 28867  df-ldil 28982  df-ltrn 28983  df-trl 29037
  Copyright terms: Public domain W3C validator