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Theorem dihmeetlem13N 31509
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 31469 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Rel  ( I `  Q ) )
433ad2ant1 976 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  Rel  ( I `  Q
) )
5 relin1 4803 . . . 4  |-  ( Rel  ( I `  Q
)  ->  Rel  ( ( I `  Q )  i^i  ( I `  R ) ) )
64, 5syl 15 . . 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 3358 . . . . . 6  |-  ( <.
f ,  s >.  e.  ( ( I `  Q )  i^i  (
I `  R )
)  <->  ( <. f ,  s >.  e.  ( I `  Q )  /\  <. f ,  s
>.  e.  ( I `  R ) ) )
8 simp1 955 . . . . . . . 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 981 . . . . . . . 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 2791 . . . . . . . . 9  |-  f  e. 
_V
17 vex 2791 . . . . . . . . 9  |-  s  e. 
_V
1810, 11, 1, 12, 13, 14, 2, 15, 16, 17dihopelvalcqat 31436 . . . . . . . 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 642 . . . . . . 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 982 . . . . . . . 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 31436 . . . . . . . 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 642 . . . . . . 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 691 . . . . . 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 248 . . . . 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 738 . . . . . . . 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 960 . . . . . . . . . . 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 5524 . . . . . . . . . . . . 13  |-  ( F  =  G  ->  ( F `  P )  =  ( G `  P ) )
29 simpl1 958 . . . . . . . . . . . . . . 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 30208 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3129, 30syl 15 . . . . . . . . . . . . . . 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 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 )
) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
3310, 11, 1, 13, 15ltrniotaval 30770 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( F `  P )  =  Q )
3429, 31, 32, 33syl3anc 1182 . . . . . . . . . . . . . 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 1009 . . . . . . . . . . . . . . 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 30770 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( G `  P )  =  R )
3729, 31, 35, 36syl3anc 1182 . . . . . . . . . . . . . 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 2297 . . . . . . . . . . . . 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 210 . . . . . . . . . . . 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 2484 . . . . . . . . . . 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 14 . . . . . . . . . 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 1022 . . . . . . . . . . . . . 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 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 `  G
) )
4442, 43eqtr3d 2317 . . . . . . . . . . . . 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 985 . . . . . . . . . . . . . 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 1025 . . . . . . . . . . . . . 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 957 . . . . . . . . . . . . . 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 15 . . . . . . . . . . . . . . 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 1068 . . . . . . . . . . . . . . 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 30768 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  e.  T )
5145, 48, 49, 50syl3anc 1182 . . . . . . . . . . . . . 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 1069 . . . . . . . . . . . . . . 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 30768 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  G  e.  T )
5445, 48, 52, 53syl3anc 1182 . . . . . . . . . . . . . 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 31213 . . . . . . . . . . . . . 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 1191 . . . . . . . . . . . . 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 201 . . . . . . . . . . . 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 1153 . . . . . . . . . . 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 2515 . . . . . . . . . 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 14 . . . . . . . . 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 5527 . . . . . . . 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 1182 . . . . . . . . 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 30976 . . . . . . . . 9  |-  ( F  e.  T  ->  ( O `  F )  =  (  _I  |`  B ) )
6664, 65syl 15 . . . . . . . 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 2319 . . . . . . 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 518 . . . . . 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 423 . . . . 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 206 . . . 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 31301 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  =  <. (  _I  |`  B ) ,  O >. )
74733ad2ant1 976 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  .0.  =  <. (  _I  |`  B ) ,  O >. )
7574sneqd 3653 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  {  .0.  }  =  { <. (  _I  |`  B ) ,  O >. } )
7675eleq2d 2350 . . . . 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 4237 . . . . . . 7  |-  <. f ,  s >.  e.  _V
7877elsnc 3663 . . . . . 6  |-  ( <.
f ,  s >.  e.  { <. (  _I  |`  B ) ,  O >. }  <->  <. f ,  s >.  =  <. (  _I  |`  B ) ,  O >. )
7916, 17opth 4245 . . . . . 6  |-  ( <.
f ,  s >.  =  <. (  _I  |`  B ) ,  O >.  <->  ( f  =  (  _I  |`  B )  /\  s  =  O ) )
8078, 79bitr2i 241 . . . . 5  |-  ( ( f  =  (  _I  |`  B )  /\  s  =  O )  <->  <. f ,  s >.  e.  { <. (  _I  |`  B ) ,  O >. } )
8176, 80syl6rbbr 255 . . . 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 205 . . 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 4779 . 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 31300 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  U  e.  LMod )
85 simp2ll 1022 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  Q  e.  A )
8655, 11atbase 29479 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  B )
8785, 86syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  Q  e.  B )
88 eqid 2283 . . . . . 6  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
8955, 1, 2, 71, 88dihlss 31440 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Q  e.  B
)  ->  ( I `  Q )  e.  (
LSubSp `  U ) )
908, 87, 89syl2anc 642 . . . 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 1024 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  R  e.  A )
9255, 11atbase 29479 . . . . . 6  |-  ( R  e.  A  ->  R  e.  B )
9391, 92syl 15 . . . . 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 31440 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  B
)  ->  ( I `  R )  e.  (
LSubSp `  U ) )
958, 93, 94syl2anc 642 . . . 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 15722 . . . 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 1182 . . 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 15706 . . 3  |-  ( ( U  e.  LMod  /\  (
( I `  Q
)  i^i  ( I `  R ) )  e.  ( LSubSp `  U )
)  ->  {  .0.  } 
C_  ( ( I `
 Q )  i^i  ( I `  R
) ) )
9984, 97, 98syl2anc 642 . 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 3196 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    i^i cin 3151    C_ wss 3152   {csn 3640   <.cop 3643   class class class wbr 4023    e. cmpt 4077    _I cid 4304    |` cres 4691   Rel wrel 4694   ` cfv 5255   iota_crio 6297   Basecbs 13148   lecple 13215   occoc 13216   0gc0g 13400   joincjn 14078   LModclmod 15627   LSubSpclss 15689   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   TEndoctendo 30941   DVecHcdvh 31268   DIsoHcdih 31418
This theorem is referenced by:  dihmeetlem15N  31511
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-undef 6298  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tendo 30944  df-edring 30946  df-disoa 31219  df-dvech 31269  df-dib 31329  df-dic 31363  df-dih 31419
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