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Theorem dihmeetlem13N 36748
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 36708 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Rel  ( I `  Q ) )
433ad2ant1 1016 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  Rel  ( I `  Q
) )
5 relin1 5106 . . . 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 3669 . . . . . 6  |-  ( <.
f ,  s >.  e.  ( ( I `  Q )  i^i  (
I `  R )
)  <->  ( <. f ,  s >.  e.  ( I `  Q )  /\  <. f ,  s
>.  e.  ( I `  R ) ) )
8 simp1 995 . . . . . . . 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 1021 . . . . . . . 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 3096 . . . . . . . . 9  |-  f  e. 
_V
17 vex 3096 . . . . . . . . 9  |-  s  e. 
_V
1810, 11, 1, 12, 13, 14, 2, 15, 16, 17dihopelvalcqat 36675 . . . . . . . 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 661 . . . . . . 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 1022 . . . . . . . 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 36675 . . . . . . . 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 661 . . . . . . 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 710 . . . . . 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 257 . . . . 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 761 . . . . . . . 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 1000 . . . . . . . . . . 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 5851 . . . . . . . . . . . . 13  |-  ( F  =  G  ->  ( F `  P )  =  ( G `  P ) )
29 simpl1 998 . . . . . . . . . . . . . . 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 35445 . . . . . . . . . . . . . . . 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 1048 . . . . . . . . . . . . . . 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 36009 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( F `  P )  =  Q )
3429, 31, 32, 33syl3anc 1227 . . . . . . . . . . . . . 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 1049 . . . . . . . . . . . . . . 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 36009 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( G `  P )  =  R )
3729, 31, 35, 36syl3anc 1227 . . . . . . . . . . . . . 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 2463 . . . . . . . . . . . . 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 219 . . . . . . . . . . . 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 2665 . . . . . . . . . . 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 1062 . . . . . . . . . . . . . 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 1064 . . . . . . . . . . . . . 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 2484 . . . . . . . . . . . . 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 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 )  ->  ( K  e.  HL  /\  W  e.  H ) )
46 simp2rr 1065 . . . . . . . . . . . . . 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 997 . . . . . . . . . . . . . 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 1108 . . . . . . . . . . . . . . 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 36007 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  e.  T )
5145, 48, 49, 50syl3anc 1227 . . . . . . . . . . . . . 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 1109 . . . . . . . . . . . . . . 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 36007 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  G  e.  T )
5445, 48, 52, 53syl3anc 1227 . . . . . . . . . . . . . 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 36452 . . . . . . . . . . . . . 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 1236 . . . . . . . . . . . . 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 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 )
)  /\  s  =/=  O )  ->  F  =  G )
60593expia 1197 . . . . . . . . . . 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 2666 . . . . . . . . . 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 5854 . . . . . . . 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 1227 . . . . . . . . 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 36215 . . . . . . . . 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 2486 . . . . . . 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 532 . . . . . 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 434 . . . . 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 215 . . . 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 36540 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  =  <. (  _I  |`  B ) ,  O >. )
74733ad2ant1 1016 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  .0.  =  <. (  _I  |`  B ) ,  O >. )
7574sneqd 4022 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  {  .0.  }  =  { <. (  _I  |`  B ) ,  O >. } )
7675eleq2d 2511 . . . . 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 4697 . . . . . . 7  |-  <. f ,  s >.  e.  _V
7877elsnc 4034 . . . . . 6  |-  ( <.
f ,  s >.  e.  { <. (  _I  |`  B ) ,  O >. }  <->  <. f ,  s >.  =  <. (  _I  |`  B ) ,  O >. )
7916, 17opth 4707 . . . . . 6  |-  ( <.
f ,  s >.  =  <. (  _I  |`  B ) ,  O >.  <->  ( f  =  (  _I  |`  B )  /\  s  =  O ) )
8078, 79bitr2i 250 . . . . 5  |-  ( ( f  =  (  _I  |`  B )  /\  s  =  O )  <->  <. f ,  s >.  e.  { <. (  _I  |`  B ) ,  O >. } )
8176, 80syl6rbbr 264 . . . 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 214 . . 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 5081 . 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 36539 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  U  e.  LMod )
85 simp2ll 1062 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  Q  e.  A )
8655, 11atbase 34716 . . . . . 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 2441 . . . . . 6  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
8955, 1, 2, 71, 88dihlss 36679 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Q  e.  B
)  ->  ( I `  Q )  e.  (
LSubSp `  U ) )
908, 87, 89syl2anc 661 . . . 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 1064 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/= 
R )  ->  R  e.  A )
9255, 11atbase 34716 . . . . . 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 36679 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  B
)  ->  ( I `  R )  e.  (
LSubSp `  U ) )
958, 93, 94syl2anc 661 . . . 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 17479 . . . 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 1227 . . 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 17463 . . 3  |-  ( ( U  e.  LMod  /\  (
( I `  Q
)  i^i  ( I `  R ) )  e.  ( LSubSp `  U )
)  ->  {  .0.  } 
C_  ( ( I `
 Q )  i^i  ( I `  R
) ) )
9984, 97, 98syl2anc 661 . 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 3503 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 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802    =/= wne 2636    i^i cin 3457    C_ wss 3458   {csn 4010   <.cop 4016   class class class wbr 4433    |-> cmpt 4491    _I cid 4776    |` cres 4987   Rel wrel 4990   ` cfv 5574   iota_crio 6237   Basecbs 14504   lecple 14576   occoc 14577   0gc0g 14709   joincjn 15442   LModclmod 17380   LSubSpclss 17446   Atomscatm 34690   HLchlt 34777   LHypclh 35410   LTrncltrn 35527   TEndoctendo 36180   DVecHcdvh 36507   DIsoHcdih 36657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-riotaBAD 34386
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-iin 4314  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-tpos 6953  df-undef 7000  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-map 7420  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-n0 10797  df-z 10866  df-uz 11086  df-fz 11677  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-sca 14585  df-vsca 14586  df-0g 14711  df-preset 15426  df-poset 15444  df-plt 15457  df-lub 15473  df-glb 15474  df-join 15475  df-meet 15476  df-p0 15538  df-p1 15539  df-lat 15545  df-clat 15607  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-submnd 15836  df-grp 15926  df-minusg 15927  df-sbg 15928  df-subg 16067  df-cntz 16224  df-lsm 16525  df-cmn 16669  df-abl 16670  df-mgp 17010  df-ur 17022  df-ring 17068  df-oppr 17140  df-dvdsr 17158  df-unit 17159  df-invr 17189  df-dvr 17200  df-drng 17266  df-lmod 17382  df-lss 17447  df-lsp 17486  df-lvec 17617  df-oposet 34603  df-ol 34605  df-oml 34606  df-covers 34693  df-ats 34694  df-atl 34725  df-cvlat 34749  df-hlat 34778  df-llines 34924  df-lplanes 34925  df-lvols 34926  df-lines 34927  df-psubsp 34929  df-pmap 34930  df-padd 35222  df-lhyp 35414  df-laut 35415  df-ldil 35530  df-ltrn 35531  df-trl 35586  df-tendo 36183  df-edring 36185  df-disoa 36458  df-dvech 36508  df-dib 36568  df-dic 36602  df-dih 36658
This theorem is referenced by:  dihmeetlem15N  36750
  Copyright terms: Public domain W3C validator