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Theorem dihmeetlem13N 34805
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 34765 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Rel  ( I `  Q ) )
433ad2ant1 1026 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  Rel  ( I `  Q
) )
5 relin1 4966 . . . 4  |-  ( Rel  ( I `  Q
)  ->  Rel  ( ( I `  Q )  i^i  ( I `  R ) ) )
64, 5syl 17 . . 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 3649 . . . . . 6  |-  ( <.
f ,  s >.  e.  ( ( I `  Q )  i^i  (
I `  R )
)  <->  ( <. f ,  s >.  e.  ( I `  Q )  /\  <. f ,  s
>.  e.  ( I `  R ) ) )
8 simp1 1005 . . . . . . . 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 1031 . . . . . . . 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 3084 . . . . . . . . 9  |-  f  e. 
_V
17 vex 3084 . . . . . . . . 9  |-  s  e. 
_V
1810, 11, 1, 12, 13, 14, 2, 15, 16, 17dihopelvalcqat 34732 . . . . . . . 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 665 . . . . . . 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 1032 . . . . . . . 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 34732 . . . . . . . 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 665 . . . . . . 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 715 . . . . . 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 260 . . . . 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 770 . . . . . . . 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 1010 . . . . . . . . . . 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 5876 . . . . . . . . . . . . 13  |-  ( F  =  G  ->  ( F `  P )  =  ( G `  P ) )
29 simpl1 1008 . . . . . . . . . . . . . . 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 33502 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3129, 30syl 17 . . . . . . . . . . . . . . 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 1058 . . . . . . . . . . . . . . 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 34066 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( F `  P )  =  Q )
3429, 31, 32, 33syl3anc 1264 . . . . . . . . . . . . . 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 1059 . . . . . . . . . . . . . . 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 34066 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( G `  P )  =  R )
3729, 31, 35, 36syl3anc 1264 . . . . . . . . . . . . . 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 2444 . . . . . . . . . . . . 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 222 . . . . . . . . . . . 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 2648 . . . . . . . . . . 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 1072 . . . . . . . . . . . . . 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 1074 . . . . . . . . . . . . . 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 2465 . . . . . . . . . . . . 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 1035 . . . . . . . . . . . . . 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 1075 . . . . . . . . . . . . . 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 1007 . . . . . . . . . . . . . 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 17 . . . . . . . . . . . . . . 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 1118 . . . . . . . . . . . . . . 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 34064 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  e.  T )
5145, 48, 49, 50syl3anc 1264 . . . . . . . . . . . . . 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 1119 . . . . . . . . . . . . . . 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 34064 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  G  e.  T )
5445, 48, 52, 53syl3anc 1264 . . . . . . . . . . . . . 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 34509 . . . . . . . . . . . . . 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 1273 . . . . . . . . . . . . 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 213 . . . . . . . . . . . 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 1207 . . . . . . . . . . 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 2649 . . . . . . . . . 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 5879 . . . . . . . 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 1264 . . . . . . . . 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 34272 . . . . . . . . 9  |-  ( F  e.  T  ->  ( O `  F )  =  (  _I  |`  B ) )
6664, 65syl 17 . . . . . . . 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 2467 . . . . . . 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 534 . . . . . 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 435 . . . . 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 218 . . . 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 34597 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  =  <. (  _I  |`  B ) ,  O >. )
74733ad2ant1 1026 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  .0.  =  <. (  _I  |`  B ) ,  O >. )
7574sneqd 4008 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  {  .0.  }  =  { <. (  _I  |`  B ) ,  O >. } )
7675eleq2d 2492 . . . . 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 4681 . . . . . . 7  |-  <. f ,  s >.  e.  _V
7877elsnc 4020 . . . . . 6  |-  ( <.
f ,  s >.  e.  { <. (  _I  |`  B ) ,  O >. }  <->  <. f ,  s >.  =  <. (  _I  |`  B ) ,  O >. )
7916, 17opth 4691 . . . . . 6  |-  ( <.
f ,  s >.  =  <. (  _I  |`  B ) ,  O >.  <->  ( f  =  (  _I  |`  B )  /\  s  =  O ) )
8078, 79bitr2i 253 . . . . 5  |-  ( ( f  =  (  _I  |`  B )  /\  s  =  O )  <->  <. f ,  s >.  e.  { <. (  _I  |`  B ) ,  O >. } )
8176, 80syl6rbbr 267 . . . 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 217 . . 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 4942 . 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 34596 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  U  e.  LMod )
85 simp2ll 1072 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  Q  e.  A )
8655, 11atbase 32773 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  B )
8785, 86syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  Q  e.  B )
88 eqid 2422 . . . . . 6  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
8955, 1, 2, 71, 88dihlss 34736 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Q  e.  B
)  ->  ( I `  Q )  e.  (
LSubSp `  U ) )
908, 87, 89syl2anc 665 . . . 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 1074 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  R  e.  A )
9255, 11atbase 32773 . . . . . 6  |-  ( R  e.  A  ->  R  e.  B )
9391, 92syl 17 . . . . 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 34736 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  B
)  ->  ( I `  R )  e.  (
LSubSp `  U ) )
958, 93, 94syl2anc 665 . . . 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 18175 . . . 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 1264 . . 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 18159 . . 3  |-  ( ( U  e.  LMod  /\  (
( I `  Q
)  i^i  ( I `  R ) )  e.  ( LSubSp `  U )
)  ->  {  .0.  } 
C_  ( ( I `
 Q )  i^i  ( I `  R
) ) )
9984, 97, 98syl2anc 665 . 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 3481 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618    i^i cin 3435    C_ wss 3436   {csn 3996   <.cop 4002   class class class wbr 4420    |-> cmpt 4479    _I cid 4759    |` cres 4851   Rel wrel 4854   ` cfv 5597   iota_crio 6262   Basecbs 15108   lecple 15184   occoc 15185   0gc0g 15325   joincjn 16176   LModclmod 18078   LSubSpclss 18142   Atomscatm 32747   HLchlt 32834   LHypclh 33467   LTrncltrn 33584   TEndoctendo 34237   DVecHcdvh 34564   DIsoHcdih 34714
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 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-riotaBAD 32443
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 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-1st 6803  df-2nd 6804  df-tpos 6977  df-undef 7024  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-oadd 7190  df-er 7367  df-map 7478  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  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 11785  df-struct 15110  df-ndx 15111  df-slot 15112  df-base 15113  df-sets 15114  df-ress 15115  df-plusg 15190  df-mulr 15191  df-sca 15193  df-vsca 15194  df-0g 15327  df-preset 16160  df-poset 16178  df-plt 16191  df-lub 16207  df-glb 16208  df-join 16209  df-meet 16210  df-p0 16272  df-p1 16273  df-lat 16279  df-clat 16341  df-mgm 16475  df-sgrp 16514  df-mnd 16524  df-submnd 16570  df-grp 16660  df-minusg 16661  df-sbg 16662  df-subg 16801  df-cntz 16958  df-lsm 17275  df-cmn 17419  df-abl 17420  df-mgp 17711  df-ur 17723  df-ring 17769  df-oppr 17838  df-dvdsr 17856  df-unit 17857  df-invr 17887  df-dvr 17898  df-drng 17964  df-lmod 18080  df-lss 18143  df-lsp 18182  df-lvec 18313  df-oposet 32660  df-ol 32662  df-oml 32663  df-covers 32750  df-ats 32751  df-atl 32782  df-cvlat 32806  df-hlat 32835  df-llines 32981  df-lplanes 32982  df-lvols 32983  df-lines 32984  df-psubsp 32986  df-pmap 32987  df-padd 33279  df-lhyp 33471  df-laut 33472  df-ldil 33587  df-ltrn 33588  df-trl 33643  df-tendo 34240  df-edring 34242  df-disoa 34515  df-dvech 34565  df-dib 34625  df-dic 34659  df-dih 34715
This theorem is referenced by:  dihmeetlem15N  34807
  Copyright terms: Public domain W3C validator