Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dihmeetlem4preN Structured version   Unicode version

Theorem dihmeetlem4preN 34582
Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihmeetlem4.b  |-  B  =  ( Base `  K
)
dihmeetlem4.l  |-  .<_  =  ( le `  K )
dihmeetlem4.m  |-  ./\  =  ( meet `  K )
dihmeetlem4.a  |-  A  =  ( Atoms `  K )
dihmeetlem4.h  |-  H  =  ( LHyp `  K
)
dihmeetlem4.i  |-  I  =  ( ( DIsoH `  K
) `  W )
dihmeetlem4.u  |-  U  =  ( ( DVecH `  K
) `  W )
dihmeetlem4.z  |-  .0.  =  ( 0g `  U )
dihmeetlem4.g  |-  G  =  ( iota_ g  e.  T  ( g `  P
)  =  Q )
dihmeetlem4.p  |-  P  =  ( ( oc `  K ) `  W
)
dihmeetlem4.t  |-  T  =  ( ( LTrn `  K
) `  W )
dihmeetlem4.r  |-  R  =  ( ( trL `  K
) `  W )
dihmeetlem4.e  |-  E  =  ( ( TEndo `  K
) `  W )
dihmeetlem4.o  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
Assertion
Ref Expression
dihmeetlem4preN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( (
I `  Q )  i^i  ( I `  ( X  ./\  W ) ) )  =  {  .0.  } )
Distinct variable groups:    .<_ , g    A, g    g, h, H    B, h    g, K, h    Q, g    T, g, h    g, W, h    P, g
Allowed substitution hints:    A( h)    B( g)    P( h)    Q( h)    R( g, h)    U( g, h)    E( g, h)    G( g, h)    I( g, h)    .<_ ( h)    ./\ ( g, h)    O( g, h)    X( g, h)    .0. ( g, h)

Proof of Theorem dihmeetlem4preN
Dummy variables  f 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dihmeetlem4.h . . . . 5  |-  H  =  ( LHyp `  K
)
2 dihmeetlem4.i . . . . 5  |-  I  =  ( ( DIsoH `  K
) `  W )
31, 2dihvalrel 34555 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Rel  ( I `  Q ) )
4 relin1 4971 . . . 4  |-  ( Rel  ( I `  Q
)  ->  Rel  ( ( I `  Q )  i^i  ( I `  ( X  ./\  W ) ) ) )
53, 4syl 17 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Rel  ( ( I `
 Q )  i^i  ( I `  ( X  ./\  W ) ) ) )
653ad2ant1 1026 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  Rel  ( ( I `  Q )  i^i  ( I `  ( X  ./\  W ) ) ) )
71, 2dihvalrel 34555 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Rel  ( I `  ( 0. `  K ) ) )
8 eqid 2429 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
9 dihmeetlem4.u . . . . . 6  |-  U  =  ( ( DVecH `  K
) `  W )
10 dihmeetlem4.z . . . . . 6  |-  .0.  =  ( 0g `  U )
118, 1, 2, 9, 10dih0 34556 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  ( 0. `  K ) )  =  {  .0.  }
)
1211releqd 4939 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Rel  ( I `
 ( 0. `  K ) )  <->  Rel  {  .0.  } ) )
137, 12mpbid 213 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Rel  {  .0.  }
)
14133ad2ant1 1026 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  Rel  {  .0.  } )
15 id 23 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
16 elin 3655 . . . 4  |-  ( <.
f ,  s >.  e.  ( ( I `  Q )  i^i  (
I `  ( X  ./\ 
W ) ) )  <-> 
( <. f ,  s
>.  e.  ( I `  Q )  /\  <. f ,  s >.  e.  ( I `  ( X 
./\  W ) ) ) )
17 dihmeetlem4.l . . . . . . . . . 10  |-  .<_  =  ( le `  K )
18 dihmeetlem4.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
19 dihmeetlem4.p . . . . . . . . . 10  |-  P  =  ( ( oc `  K ) `  W
)
20 dihmeetlem4.t . . . . . . . . . 10  |-  T  =  ( ( LTrn `  K
) `  W )
21 dihmeetlem4.e . . . . . . . . . 10  |-  E  =  ( ( TEndo `  K
) `  W )
22 dihmeetlem4.g . . . . . . . . . 10  |-  G  =  ( iota_ g  e.  T  ( g `  P
)  =  Q )
23 vex 3090 . . . . . . . . . 10  |-  f  e. 
_V
24 vex 3090 . . . . . . . . . 10  |-  s  e. 
_V
2517, 18, 1, 19, 20, 21, 2, 22, 23, 24dihopelvalcqat 34522 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( <. f ,  s
>.  e.  ( I `  Q )  <->  ( f  =  ( s `  G )  /\  s  e.  E ) ) )
26253adant2 1024 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( <. f ,  s >.  e.  ( I `  Q )  <-> 
( f  =  ( s `  G )  /\  s  e.  E
) ) )
27 simp1 1005 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
28 simp1l 1029 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  K  e.  HL )
29 hllat 32637 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Lat )
3028, 29syl 17 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  K  e.  Lat )
31 simp2l 1031 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  X  e.  B )
32 simp1r 1030 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  W  e.  H )
33 dihmeetlem4.b . . . . . . . . . . . 12  |-  B  =  ( Base `  K
)
3433, 1lhpbase 33271 . . . . . . . . . . 11  |-  ( W  e.  H  ->  W  e.  B )
3532, 34syl 17 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  W  e.  B )
36 dihmeetlem4.m . . . . . . . . . . 11  |-  ./\  =  ( meet `  K )
3733, 36latmcl 16249 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
3830, 31, 35, 37syl3anc 1264 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( X  ./\ 
W )  e.  B
)
3933, 17, 36latmle2 16274 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  .<_  W )
4030, 31, 35, 39syl3anc 1264 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( X  ./\ 
W )  .<_  W )
41 dihmeetlem4.r . . . . . . . . . 10  |-  R  =  ( ( trL `  K
) `  W )
42 dihmeetlem4.o . . . . . . . . . 10  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
4333, 17, 1, 20, 41, 42, 2dihopelvalbN 34514 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( X 
./\  W )  e.  B  /\  ( X 
./\  W )  .<_  W ) )  -> 
( <. f ,  s
>.  e.  ( I `  ( X  ./\  W ) )  <->  ( ( f  e.  T  /\  ( R `  f )  .<_  ( X  ./\  W
) )  /\  s  =  O ) ) )
4427, 38, 40, 43syl12anc 1262 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( <. f ,  s >.  e.  ( I `  ( X 
./\  W ) )  <-> 
( ( f  e.  T  /\  ( R `
 f )  .<_  ( X  ./\  W ) )  /\  s  =  O ) ) )
4526, 44anbi12d 715 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( ( <. f ,  s >.  e.  ( I `  Q
)  /\  <. f ,  s >.  e.  (
I `  ( X  ./\ 
W ) ) )  <-> 
( ( f  =  ( s `  G
)  /\  s  e.  E )  /\  (
( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) )  /\  s  =  O ) ) ) )
46 simprll 770 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( f  =  ( s `  G
)  /\  s  e.  E )  /\  (
( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) )  /\  s  =  O ) ) )  ->  f  =  ( s `  G ) )
47 simprrr 773 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( f  =  ( s `  G
)  /\  s  e.  E )  /\  (
( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) )  /\  s  =  O ) ) )  ->  s  =  O )
4847fveq1d 5883 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( f  =  ( s `  G
)  /\  s  e.  E )  /\  (
( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) )  /\  s  =  O ) ) )  ->  ( s `  G )  =  ( O `  G ) )
49 simpl1 1008 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( f  =  ( s `  G
)  /\  s  e.  E )  /\  (
( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) )  /\  s  =  O ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
5017, 18, 1, 19lhpocnel2 33292 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
5149, 50syl 17 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( f  =  ( s `  G
)  /\  s  e.  E )  /\  (
( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) )  /\  s  =  O ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
52 simpl3 1010 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( f  =  ( s `  G
)  /\  s  e.  E )  /\  (
( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) )  /\  s  =  O ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
5317, 18, 1, 20, 22ltrniotacl 33854 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  G  e.  T )
5449, 51, 52, 53syl3anc 1264 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( f  =  ( s `  G
)  /\  s  e.  E )  /\  (
( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) )  /\  s  =  O ) ) )  ->  G  e.  T
)
5542, 33tendo02 34062 . . . . . . . . . . 11  |-  ( G  e.  T  ->  ( O `  G )  =  (  _I  |`  B ) )
5654, 55syl 17 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( f  =  ( s `  G
)  /\  s  e.  E )  /\  (
( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) )  /\  s  =  O ) ) )  ->  ( O `  G )  =  (  _I  |`  B )
)
5746, 48, 563eqtrd 2474 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( f  =  ( s `  G
)  /\  s  e.  E )  /\  (
( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) )  /\  s  =  O ) ) )  ->  f  =  (  _I  |`  B )
)
5857, 47jca 534 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( f  =  ( s `  G
)  /\  s  e.  E )  /\  (
( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) )  /\  s  =  O ) ) )  ->  ( f  =  (  _I  |`  B )  /\  s  =  O ) )
59 simpl1 1008 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  (  _I  |`  B )  /\  s  =  O
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
6059, 50syl 17 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  (  _I  |`  B )  /\  s  =  O
) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
61 simpl3 1010 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  (  _I  |`  B )  /\  s  =  O
) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
6259, 60, 61, 53syl3anc 1264 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  (  _I  |`  B )  /\  s  =  O
) )  ->  G  e.  T )
6362, 55syl 17 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  (  _I  |`  B )  /\  s  =  O
) )  ->  ( O `  G )  =  (  _I  |`  B ) )
64 simprr 764 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  (  _I  |`  B )  /\  s  =  O
) )  ->  s  =  O )
6564fveq1d 5883 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  (  _I  |`  B )  /\  s  =  O
) )  ->  (
s `  G )  =  ( O `  G ) )
66 simprl 762 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  (  _I  |`  B )  /\  s  =  O
) )  ->  f  =  (  _I  |`  B ) )
6763, 65, 663eqtr4rd 2481 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  (  _I  |`  B )  /\  s  =  O
) )  ->  f  =  ( s `  G ) )
6833, 1, 20, 21, 42tendo0cl 34065 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  E )
6959, 68syl 17 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  (  _I  |`  B )  /\  s  =  O
) )  ->  O  e.  E )
7064, 69eqeltrd 2517 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  (  _I  |`  B )  /\  s  =  O
) )  ->  s  e.  E )
7133, 1, 20idltrn 33423 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
7259, 71syl 17 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  (  _I  |`  B )  /\  s  =  O
) )  ->  (  _I  |`  B )  e.  T )
7366, 72eqeltrd 2517 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  (  _I  |`  B )  /\  s  =  O
) )  ->  f  e.  T )
7466fveq2d 5885 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  (  _I  |`  B )  /\  s  =  O
) )  ->  ( R `  f )  =  ( R `  (  _I  |`  B ) ) )
7533, 8, 1, 41trlid0 33450 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( R `  (  _I  |`  B ) )  =  ( 0. `  K ) )
7659, 75syl 17 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  (  _I  |`  B )  /\  s  =  O
) )  ->  ( R `  (  _I  |`  B ) )  =  ( 0. `  K
) )
7774, 76eqtrd 2470 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  (  _I  |`  B )  /\  s  =  O
) )  ->  ( R `  f )  =  ( 0. `  K ) )
78 simpl1l 1056 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  (  _I  |`  B )  /\  s  =  O
) )  ->  K  e.  HL )
79 hlatl 32634 . . . . . . . . . . . . 13  |-  ( K  e.  HL  ->  K  e.  AtLat )
8078, 79syl 17 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  (  _I  |`  B )  /\  s  =  O
) )  ->  K  e.  AtLat )
8138adantr 466 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  (  _I  |`  B )  /\  s  =  O
) )  ->  ( X  ./\  W )  e.  B )
8233, 17, 8atl0le 32578 . . . . . . . . . . . 12  |-  ( ( K  e.  AtLat  /\  ( X  ./\  W )  e.  B )  ->  ( 0. `  K )  .<_  ( X  ./\  W ) )
8380, 81, 82syl2anc 665 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  (  _I  |`  B )  /\  s  =  O
) )  ->  ( 0. `  K )  .<_  ( X  ./\  W ) )
8477, 83eqbrtrd 4446 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  (  _I  |`  B )  /\  s  =  O
) )  ->  ( R `  f )  .<_  ( X  ./\  W
) )
8573, 84, 64jca31 536 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  (  _I  |`  B )  /\  s  =  O
) )  ->  (
( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) )  /\  s  =  O ) )
8667, 70, 85jca31 536 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( f  =  (  _I  |`  B )  /\  s  =  O
) )  ->  (
( f  =  ( s `  G )  /\  s  e.  E
)  /\  ( (
f  e.  T  /\  ( R `  f ) 
.<_  ( X  ./\  W
) )  /\  s  =  O ) ) )
8758, 86impbida 840 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( (
( f  =  ( s `  G )  /\  s  e.  E
)  /\  ( (
f  e.  T  /\  ( R `  f ) 
.<_  ( X  ./\  W
) )  /\  s  =  O ) )  <->  ( f  =  (  _I  |`  B )  /\  s  =  O ) ) )
8845, 87bitrd 256 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( ( <. f ,  s >.  e.  ( I `  Q
)  /\  <. f ,  s >.  e.  (
I `  ( X  ./\ 
W ) ) )  <-> 
( f  =  (  _I  |`  B )  /\  s  =  O
) ) )
89 opex 4686 . . . . . . . 8  |-  <. f ,  s >.  e.  _V
9089elsnc 4026 . . . . . . 7  |-  ( <.
f ,  s >.  e.  { <. (  _I  |`  B ) ,  O >. }  <->  <. f ,  s >.  =  <. (  _I  |`  B ) ,  O >. )
9123, 24opth 4696 . . . . . . 7  |-  ( <.
f ,  s >.  =  <. (  _I  |`  B ) ,  O >.  <->  ( f  =  (  _I  |`  B )  /\  s  =  O ) )
9290, 91bitr2i 253 . . . . . 6  |-  ( ( f  =  (  _I  |`  B )  /\  s  =  O )  <->  <. f ,  s >.  e.  { <. (  _I  |`  B ) ,  O >. } )
9388, 92syl6bb 264 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( ( <. f ,  s >.  e.  ( I `  Q
)  /\  <. f ,  s >.  e.  (
I `  ( X  ./\ 
W ) ) )  <->  <. f ,  s >.  e.  { <. (  _I  |`  B ) ,  O >. } ) )
9433, 1, 20, 9, 10, 42dvh0g 34387 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  =  <. (  _I  |`  B ) ,  O >. )
95943ad2ant1 1026 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  .0.  =  <. (  _I  |`  B ) ,  O >. )
9695sneqd 4014 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  {  .0.  }  =  { <. (  _I  |`  B ) ,  O >. } )
9796eleq2d 2499 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( <. f ,  s >.  e.  {  .0.  }  <->  <. f ,  s
>.  e.  { <. (  _I  |`  B ) ,  O >. } ) )
9893, 97bitr4d 259 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( ( <. f ,  s >.  e.  ( I `  Q
)  /\  <. f ,  s >.  e.  (
I `  ( X  ./\ 
W ) ) )  <->  <. f ,  s >.  e.  {  .0.  } ) )
9916, 98syl5bb 260 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( <. f ,  s >.  e.  ( ( I `  Q
)  i^i  ( I `  ( X  ./\  W
) ) )  <->  <. f ,  s >.  e.  {  .0.  } ) )
10099eqrelrdv2 4954 . 2  |-  ( ( ( Rel  ( ( I `  Q )  i^i  ( I `  ( X  ./\  W ) ) )  /\  Rel  {  .0.  } )  /\  ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( ( I `
 Q )  i^i  ( I `  ( X  ./\  W ) ) )  =  {  .0.  } )
1016, 14, 15, 100syl21anc 1263 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( (
I `  Q )  i^i  ( I `  ( X  ./\  W ) ) )  =  {  .0.  } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    i^i cin 3441   {csn 4002   <.cop 4008   class class class wbr 4426    |-> cmpt 4484    _I cid 4764    |` cres 4856   Rel wrel 4859   ` cfv 5601   iota_crio 6266  (class class class)co 6305   Basecbs 15084   lecple 15159   occoc 15160   0gc0g 15297   meetcmee 16141   0.cp0 16234   Latclat 16242   Atomscatm 32537   AtLatcal 32538   HLchlt 32624   LHypclh 33257   LTrncltrn 33374   trLctrl 33432   TEndoctendo 34027   DVecHcdvh 34354   DIsoHcdih 34504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-riotaBAD 32233
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-tpos 6981  df-undef 7028  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-sca 15168  df-vsca 15169  df-0g 15299  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-p1 16237  df-lat 16243  df-clat 16305  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-grp 16624  df-minusg 16625  df-sbg 16626  df-subg 16765  df-cntz 16922  df-lsm 17223  df-cmn 17367  df-abl 17368  df-mgp 17659  df-ur 17671  df-ring 17717  df-oppr 17786  df-dvdsr 17804  df-unit 17805  df-invr 17835  df-dvr 17846  df-drng 17912  df-lmod 18028  df-lss 18091  df-lsp 18130  df-lvec 18261  df-oposet 32450  df-ol 32452  df-oml 32453  df-covers 32540  df-ats 32541  df-atl 32572  df-cvlat 32596  df-hlat 32625  df-llines 32771  df-lplanes 32772  df-lvols 32773  df-lines 32774  df-psubsp 32776  df-pmap 32777  df-padd 33069  df-lhyp 33261  df-laut 33262  df-ldil 33377  df-ltrn 33378  df-trl 33433  df-tendo 34030  df-edring 34032  df-disoa 34305  df-dvech 34355  df-dib 34415  df-dic 34449  df-dih 34505
This theorem is referenced by:  dihmeetlem4N  34583
  Copyright terms: Public domain W3C validator