Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dihmeetlem13N Structured version   Unicode version

Theorem dihmeetlem13N 32018
Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihmeetlem13.b  |-  B  =  ( Base `  K
)
dihmeetlem13.l  |-  .<_  =  ( le `  K )
dihmeetlem13.j  |-  .\/  =  ( join `  K )
dihmeetlem13.a  |-  A  =  ( Atoms `  K )
dihmeetlem13.h  |-  H  =  ( LHyp `  K
)
dihmeetlem13.p  |-  P  =  ( ( oc `  K ) `  W
)
dihmeetlem13.t  |-  T  =  ( ( LTrn `  K
) `  W )
dihmeetlem13.e  |-  E  =  ( ( TEndo `  K
) `  W )
dihmeetlem13.o  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
dihmeetlem13.i  |-  I  =  ( ( DIsoH `  K
) `  W )
dihmeetlem13.u  |-  U  =  ( ( DVecH `  K
) `  W )
dihmeetlem13.z  |-  .0.  =  ( 0g `  U )
dihmeetlem13.f  |-  F  =  ( iota_ h  e.  T
( h `  P
)  =  Q )
dihmeetlem13.g  |-  G  =  ( iota_ h  e.  T
( h `  P
)  =  R )
Assertion
Ref Expression
dihmeetlem13N  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  (
( I `  Q
)  i^i  ( I `  R ) )  =  {  .0.  } )
Distinct variable groups:    .<_ , h    A, h    B, h    h, H   
h, K    P, h    Q, h    R, h    T, h   
h, W
Allowed substitution hints:    U( h)    E( h)    F( h)    G( h)    I( h)    .\/ ( h)    O( h)    .0. (
h)

Proof of Theorem dihmeetlem13N
Dummy variables  f 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dihmeetlem13.h . . . . . 6  |-  H  =  ( LHyp `  K
)
2 dihmeetlem13.i . . . . . 6  |-  I  =  ( ( DIsoH `  K
) `  W )
31, 2dihvalrel 31978 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Rel  ( I `  Q ) )
433ad2ant1 978 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  Rel  ( I `  Q
) )
5 relin1 4984 . . . 4  |-  ( Rel  ( I `  Q
)  ->  Rel  ( ( I `  Q )  i^i  ( I `  R ) ) )
64, 5syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  Rel  ( ( I `  Q )  i^i  (
I `  R )
) )
7 elin 3522 . . . . . 6  |-  ( <.
f ,  s >.  e.  ( ( I `  Q )  i^i  (
I `  R )
)  <->  ( <. f ,  s >.  e.  ( I `  Q )  /\  <. f ,  s
>.  e.  ( I `  R ) ) )
8 simp1 957 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  ( K  e.  HL  /\  W  e.  H ) )
9 simp2l 983 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
10 dihmeetlem13.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
11 dihmeetlem13.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
12 dihmeetlem13.p . . . . . . . . 9  |-  P  =  ( ( oc `  K ) `  W
)
13 dihmeetlem13.t . . . . . . . . 9  |-  T  =  ( ( LTrn `  K
) `  W )
14 dihmeetlem13.e . . . . . . . . 9  |-  E  =  ( ( TEndo `  K
) `  W )
15 dihmeetlem13.f . . . . . . . . 9  |-  F  =  ( iota_ h  e.  T
( h `  P
)  =  Q )
16 vex 2951 . . . . . . . . 9  |-  f  e. 
_V
17 vex 2951 . . . . . . . . 9  |-  s  e. 
_V
1810, 11, 1, 12, 13, 14, 2, 15, 16, 17dihopelvalcqat 31945 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( <. f ,  s
>.  e.  ( I `  Q )  <->  ( f  =  ( s `  F )  /\  s  e.  E ) ) )
198, 9, 18syl2anc 643 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  ( <. f ,  s >.  e.  ( I `  Q
)  <->  ( f  =  ( s `  F
)  /\  s  e.  E ) ) )
20 simp2r 984 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
21 dihmeetlem13.g . . . . . . . . 9  |-  G  =  ( iota_ h  e.  T
( h `  P
)  =  R )
2210, 11, 1, 12, 13, 14, 2, 21, 16, 17dihopelvalcqat 31945 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  -> 
( <. f ,  s
>.  e.  ( I `  R )  <->  ( f  =  ( s `  G )  /\  s  e.  E ) ) )
238, 20, 22syl2anc 643 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  ( <. f ,  s >.  e.  ( I `  R
)  <->  ( f  =  ( s `  G
)  /\  s  e.  E ) ) )
2419, 23anbi12d 692 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  (
( <. f ,  s
>.  e.  ( I `  Q )  /\  <. f ,  s >.  e.  ( I `  R ) )  <->  ( ( f  =  ( s `  F )  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
) ) )
257, 24syl5bb 249 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  ( <. f ,  s >.  e.  ( ( I `  Q )  i^i  (
I `  R )
)  <->  ( ( f  =  ( s `  F )  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
) ) )
26 simprll 739 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
) )  ->  f  =  ( s `  F ) )
27 simpl3 962 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
) )  ->  Q  =/=  R )
28 fveq1 5719 . . . . . . . . . . . . 13  |-  ( F  =  G  ->  ( F `  P )  =  ( G `  P ) )
29 simpl1 960 . . . . . . . . . . . . . . 15  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3010, 11, 1, 12lhpocnel2 30717 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3129, 30syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
32 simpl2l 1010 . . . . . . . . . . . . . . 15  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
3310, 11, 1, 13, 15ltrniotaval 31279 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( F `  P )  =  Q )
3429, 31, 32, 33syl3anc 1184 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
) )  ->  ( F `  P )  =  Q )
35 simpl2r 1011 . . . . . . . . . . . . . . 15  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
) )  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
3610, 11, 1, 13, 21ltrniotaval 31279 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( G `  P )  =  R )
3729, 31, 35, 36syl3anc 1184 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
) )  ->  ( G `  P )  =  R )
3834, 37eqeq12d 2449 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
) )  ->  (
( F `  P
)  =  ( G `
 P )  <->  Q  =  R ) )
3928, 38syl5ib 211 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
) )  ->  ( F  =  G  ->  Q  =  R ) )
4039necon3d 2636 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
) )  ->  ( Q  =/=  R  ->  F  =/=  G ) )
4127, 40mpd 15 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
) )  ->  F  =/=  G )
42 simp2ll 1024 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
)  /\  s  =/=  O )  ->  f  =  ( s `  F
) )
43 simp2rl 1026 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
)  /\  s  =/=  O )  ->  f  =  ( s `  G
) )
4442, 43eqtr3d 2469 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
)  /\  s  =/=  O )  ->  ( s `  F )  =  ( s `  G ) )
45 simp11 987 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
)  /\  s  =/=  O )  ->  ( K  e.  HL  /\  W  e.  H ) )
46 simp2rr 1027 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
)  /\  s  =/=  O )  ->  s  e.  E )
47 simp3 959 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
)  /\  s  =/=  O )  ->  s  =/=  O )
4845, 30syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
)  /\  s  =/=  O )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
49 simp12l 1070 . . . . . . . . . . . . . . 15  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
)  /\  s  =/=  O )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
5010, 11, 1, 13, 15ltrniotacl 31277 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  e.  T )
5145, 48, 49, 50syl3anc 1184 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
)  /\  s  =/=  O )  ->  F  e.  T )
52 simp12r 1071 . . . . . . . . . . . . . . 15  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
)  /\  s  =/=  O )  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
5310, 11, 1, 13, 21ltrniotacl 31277 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  G  e.  T )
5445, 48, 52, 53syl3anc 1184 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
)  /\  s  =/=  O )  ->  G  e.  T )
55 dihmeetlem13.b . . . . . . . . . . . . . . 15  |-  B  =  ( Base `  K
)
56 dihmeetlem13.o . . . . . . . . . . . . . . 15  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
5755, 1, 13, 14, 56tendospcanN 31722 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  s  =/= 
O )  /\  ( F  e.  T  /\  G  e.  T )
)  ->  ( (
s `  F )  =  ( s `  G )  <->  F  =  G ) )
5845, 46, 47, 51, 54, 57syl122anc 1193 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
)  /\  s  =/=  O )  ->  ( (
s `  F )  =  ( s `  G )  <->  F  =  G ) )
5944, 58mpbid 202 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
)  /\  s  =/=  O )  ->  F  =  G )
60593expia 1155 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
) )  ->  (
s  =/=  O  ->  F  =  G )
)
6160necon1d 2667 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
) )  ->  ( F  =/=  G  ->  s  =  O ) )
6241, 61mpd 15 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
) )  ->  s  =  O )
6362fveq1d 5722 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
) )  ->  (
s `  F )  =  ( O `  F ) )
6429, 31, 32, 50syl3anc 1184 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
) )  ->  F  e.  T )
6556, 55tendo02 31485 . . . . . . . . 9  |-  ( F  e.  T  ->  ( O `  F )  =  (  _I  |`  B ) )
6664, 65syl 16 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
) )  ->  ( O `  F )  =  (  _I  |`  B ) )
6726, 63, 663eqtrd 2471 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
) )  ->  f  =  (  _I  |`  B ) )
6867, 62jca 519 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  /\  ( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
) )  ->  (
f  =  (  _I  |`  B )  /\  s  =  O ) )
6968ex 424 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  (
( ( f  =  ( s `  F
)  /\  s  e.  E )  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
)  ->  ( f  =  (  _I  |`  B )  /\  s  =  O ) ) )
7025, 69sylbid 207 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  ( <. f ,  s >.  e.  ( ( I `  Q )  i^i  (
I `  R )
)  ->  ( f  =  (  _I  |`  B )  /\  s  =  O ) ) )
71 dihmeetlem13.u . . . . . . . . 9  |-  U  =  ( ( DVecH `  K
) `  W )
72 dihmeetlem13.z . . . . . . . . 9  |-  .0.  =  ( 0g `  U )
7355, 1, 13, 71, 72, 56dvh0g 31810 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  =  <. (  _I  |`  B ) ,  O >. )
74733ad2ant1 978 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  .0.  =  <. (  _I  |`  B ) ,  O >. )
7574sneqd 3819 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  {  .0.  }  =  { <. (  _I  |`  B ) ,  O >. } )
7675eleq2d 2502 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  ( <. f ,  s >.  e.  {  .0.  }  <->  <. f ,  s >.  e.  { <. (  _I  |`  B ) ,  O >. } ) )
77 opex 4419 . . . . . . 7  |-  <. f ,  s >.  e.  _V
7877elsnc 3829 . . . . . 6  |-  ( <.
f ,  s >.  e.  { <. (  _I  |`  B ) ,  O >. }  <->  <. f ,  s >.  =  <. (  _I  |`  B ) ,  O >. )
7916, 17opth 4427 . . . . . 6  |-  ( <.
f ,  s >.  =  <. (  _I  |`  B ) ,  O >.  <->  ( f  =  (  _I  |`  B )  /\  s  =  O ) )
8078, 79bitr2i 242 . . . . 5  |-  ( ( f  =  (  _I  |`  B )  /\  s  =  O )  <->  <. f ,  s >.  e.  { <. (  _I  |`  B ) ,  O >. } )
8176, 80syl6rbbr 256 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  (
( f  =  (  _I  |`  B )  /\  s  =  O
)  <->  <. f ,  s
>.  e.  {  .0.  }
) )
8270, 81sylibd 206 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  ( <. f ,  s >.  e.  ( ( I `  Q )  i^i  (
I `  R )
)  ->  <. f ,  s >.  e.  {  .0.  } ) )
836, 82relssdv 4960 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  (
( I `  Q
)  i^i  ( I `  R ) )  C_  {  .0.  } )
841, 71, 8dvhlmod 31809 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  U  e.  LMod )
85 simp2ll 1024 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  Q  e.  A )
8655, 11atbase 29988 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  B )
8785, 86syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  Q  e.  B )
88 eqid 2435 . . . . . 6  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
8955, 1, 2, 71, 88dihlss 31949 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Q  e.  B
)  ->  ( I `  Q )  e.  (
LSubSp `  U ) )
908, 87, 89syl2anc 643 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  (
I `  Q )  e.  ( LSubSp `  U )
)
91 simp2rl 1026 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  R  e.  A )
9255, 11atbase 29988 . . . . . 6  |-  ( R  e.  A  ->  R  e.  B )
9391, 92syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  R  e.  B )
9455, 1, 2, 71, 88dihlss 31949 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  B
)  ->  ( I `  R )  e.  (
LSubSp `  U ) )
958, 93, 94syl2anc 643 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  (
I `  R )  e.  ( LSubSp `  U )
)
9688lssincl 16031 . . . 4  |-  ( ( U  e.  LMod  /\  (
I `  Q )  e.  ( LSubSp `  U )  /\  ( I `  R
)  e.  ( LSubSp `  U ) )  -> 
( ( I `  Q )  i^i  (
I `  R )
)  e.  ( LSubSp `  U ) )
9784, 90, 95, 96syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  (
( I `  Q
)  i^i  ( I `  R ) )  e.  ( LSubSp `  U )
)
9872, 88lss0ss 16015 . . 3  |-  ( ( U  e.  LMod  /\  (
( I `  Q
)  i^i  ( I `  R ) )  e.  ( LSubSp `  U )
)  ->  {  .0.  } 
C_  ( ( I `
 Q )  i^i  ( I `  R
) ) )
9984, 97, 98syl2anc 643 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  {  .0.  } 
C_  ( ( I `
 Q )  i^i  ( I `  R
) ) )
10083, 99eqssd 3357 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  (
( I `  Q
)  i^i  ( I `  R ) )  =  {  .0.  } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598    i^i cin 3311    C_ wss 3312   {csn 3806   <.cop 3809   class class class wbr 4204    e. cmpt 4258    _I cid 4485    |` cres 4872   Rel wrel 4875   ` cfv 5446   iota_crio 6534   Basecbs 13459   lecple 13526   occoc 13527   0gc0g 13713   joincjn 14391   LModclmod 15940   LSubSpclss 15998   Atomscatm 29962   HLchlt 30049   LHypclh 30682   LTrncltrn 30799   TEndoctendo 31450   DVecHcdvh 31777   DIsoHcdih 31927
This theorem is referenced by:  dihmeetlem15N  32020
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-undef 6535  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-n0 10212  df-z 10273  df-uz 10479  df-fz 11034  df-struct 13461  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-ress 13466  df-plusg 13532  df-mulr 13533  df-sca 13535  df-vsca 13536  df-0g 13717  df-poset 14393  df-plt 14405  df-lub 14421  df-glb 14422  df-join 14423  df-meet 14424  df-p0 14458  df-p1 14459  df-lat 14465  df-clat 14527  df-mnd 14680  df-submnd 14729  df-grp 14802  df-minusg 14803  df-sbg 14804  df-subg 14931  df-cntz 15106  df-lsm 15260  df-cmn 15404  df-abl 15405  df-mgp 15639  df-rng 15653  df-ur 15655  df-oppr 15718  df-dvdsr 15736  df-unit 15737  df-invr 15767  df-dvr 15778  df-drng 15827  df-lmod 15942  df-lss 15999  df-lsp 16038  df-lvec 16165  df-oposet 29875  df-ol 29877  df-oml 29878  df-covers 29965  df-ats 29966  df-atl 29997  df-cvlat 30021  df-hlat 30050  df-llines 30196  df-lplanes 30197  df-lvols 30198  df-lines 30199  df-psubsp 30201  df-pmap 30202  df-padd 30494  df-lhyp 30686  df-laut 30687  df-ldil 30802  df-ltrn 30803  df-trl 30857  df-tendo 31453  df-edring 31455  df-disoa 31728  df-dvech 31778  df-dib 31838  df-dic 31872  df-dih 31928
  Copyright terms: Public domain W3C validator