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Theorem islmhm 18193
Description: Property of being a homomorphism of left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof shortened by Mario Carneiro, 30-Apr-2015.)
Hypotheses
Ref Expression
islmhm.k  |-  K  =  (Scalar `  S )
islmhm.l  |-  L  =  (Scalar `  T )
islmhm.b  |-  B  =  ( Base `  K
)
islmhm.e  |-  E  =  ( Base `  S
)
islmhm.m  |-  .x.  =  ( .s `  S )
islmhm.n  |-  .X.  =  ( .s `  T )
Assertion
Ref Expression
islmhm  |-  ( F  e.  ( S LMHom  T
)  <->  ( ( S  e.  LMod  /\  T  e. 
LMod )  /\  ( F  e.  ( S  GrpHom  T )  /\  L  =  K  /\  A. x  e.  B  A. y  e.  E  ( F `  ( x  .x.  y
) )  =  ( x  .X.  ( F `  y ) ) ) ) )
Distinct variable groups:    x, B    y, E    x, y, S   
x, F, y    x, T, y
Allowed substitution hints:    B( y)    .x. ( x, y)   
.X. ( x, y)    E( x)    K( x, y)    L( x, y)

Proof of Theorem islmhm
Dummy variables  f 
s  t  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-lmhm 18188 . . 3  |- LMHom  =  ( s  e.  LMod ,  t  e.  LMod  |->  { f  e.  ( s  GrpHom  t )  |  [. (Scalar `  s )  /  w ]. ( (Scalar `  t
)  =  w  /\  A. x  e.  ( Base `  w ) A. y  e.  ( Base `  s
) ( f `  ( x ( .s
`  s ) y ) )  =  ( x ( .s `  t ) ( f `
 y ) ) ) } )
21elmpt2cl 6469 . 2  |-  ( F  e.  ( S LMHom  T
)  ->  ( S  e.  LMod  /\  T  e.  LMod ) )
3 oveq12 6258 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s  GrpHom  t )  =  ( S  GrpHom  T ) )
4 fvex 5835 . . . . . . . 8  |-  (Scalar `  s )  e.  _V
54a1i 11 . . . . . . 7  |-  ( ( s  =  S  /\  t  =  T )  ->  (Scalar `  s )  e.  _V )
6 simplr 760 . . . . . . . . . . 11  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  t  =  T )
76fveq2d 5829 . . . . . . . . . 10  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  (Scalar `  t
)  =  (Scalar `  T ) )
8 islmhm.l . . . . . . . . . 10  |-  L  =  (Scalar `  T )
97, 8syl6eqr 2480 . . . . . . . . 9  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  (Scalar `  t
)  =  L )
10 simpr 462 . . . . . . . . . . 11  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  w  =  (Scalar `  s ) )
11 simpll 758 . . . . . . . . . . . 12  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  s  =  S )
1211fveq2d 5829 . . . . . . . . . . 11  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  (Scalar `  s
)  =  (Scalar `  S ) )
1310, 12eqtrd 2462 . . . . . . . . . 10  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  w  =  (Scalar `  S ) )
14 islmhm.k . . . . . . . . . 10  |-  K  =  (Scalar `  S )
1513, 14syl6eqr 2480 . . . . . . . . 9  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  w  =  K )
169, 15eqeq12d 2443 . . . . . . . 8  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( (Scalar `  t )  =  w  <-> 
L  =  K ) )
1715fveq2d 5829 . . . . . . . . . 10  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( Base `  w )  =  (
Base `  K )
)
18 islmhm.b . . . . . . . . . 10  |-  B  =  ( Base `  K
)
1917, 18syl6eqr 2480 . . . . . . . . 9  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( Base `  w )  =  B )
2011fveq2d 5829 . . . . . . . . . . 11  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( Base `  s )  =  (
Base `  S )
)
21 islmhm.e . . . . . . . . . . 11  |-  E  =  ( Base `  S
)
2220, 21syl6eqr 2480 . . . . . . . . . 10  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( Base `  s )  =  E )
2311fveq2d 5829 . . . . . . . . . . . . . 14  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( .s `  s )  =  ( .s `  S ) )
24 islmhm.m . . . . . . . . . . . . . 14  |-  .x.  =  ( .s `  S )
2523, 24syl6eqr 2480 . . . . . . . . . . . . 13  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( .s `  s )  =  .x.  )
2625oveqd 6266 . . . . . . . . . . . 12  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( x
( .s `  s
) y )  =  ( x  .x.  y
) )
2726fveq2d 5829 . . . . . . . . . . 11  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( f `  ( x ( .s
`  s ) y ) )  =  ( f `  ( x 
.x.  y ) ) )
286fveq2d 5829 . . . . . . . . . . . . 13  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( .s `  t )  =  ( .s `  T ) )
29 islmhm.n . . . . . . . . . . . . 13  |-  .X.  =  ( .s `  T )
3028, 29syl6eqr 2480 . . . . . . . . . . . 12  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( .s `  t )  =  .X.  )
3130oveqd 6266 . . . . . . . . . . 11  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( x
( .s `  t
) ( f `  y ) )  =  ( x  .X.  (
f `  y )
) )
3227, 31eqeq12d 2443 . . . . . . . . . 10  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( (
f `  ( x
( .s `  s
) y ) )  =  ( x ( .s `  t ) ( f `  y
) )  <->  ( f `  ( x  .x.  y
) )  =  ( x  .X.  ( f `  y ) ) ) )
3322, 32raleqbidv 2978 . . . . . . . . 9  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( A. y  e.  ( Base `  s ) ( f `
 ( x ( .s `  s ) y ) )  =  ( x ( .s
`  t ) ( f `  y ) )  <->  A. y  e.  E  ( f `  (
x  .x.  y )
)  =  ( x 
.X.  ( f `  y ) ) ) )
3419, 33raleqbidv 2978 . . . . . . . 8  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( A. x  e.  ( Base `  w ) A. y  e.  ( Base `  s
) ( f `  ( x ( .s
`  s ) y ) )  =  ( x ( .s `  t ) ( f `
 y ) )  <->  A. x  e.  B  A. y  e.  E  ( f `  (
x  .x.  y )
)  =  ( x 
.X.  ( f `  y ) ) ) )
3516, 34anbi12d 715 . . . . . . 7  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( (
(Scalar `  t )  =  w  /\  A. x  e.  ( Base `  w
) A. y  e.  ( Base `  s
) ( f `  ( x ( .s
`  s ) y ) )  =  ( x ( .s `  t ) ( f `
 y ) ) )  <->  ( L  =  K  /\  A. x  e.  B  A. y  e.  E  ( f `  ( x  .x.  y
) )  =  ( x  .X.  ( f `  y ) ) ) ) )
365, 35sbcied 3279 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  ( [. (Scalar `  s )  /  w ]. ( (Scalar `  t
)  =  w  /\  A. x  e.  ( Base `  w ) A. y  e.  ( Base `  s
) ( f `  ( x ( .s
`  s ) y ) )  =  ( x ( .s `  t ) ( f `
 y ) ) )  <->  ( L  =  K  /\  A. x  e.  B  A. y  e.  E  ( f `  ( x  .x.  y
) )  =  ( x  .X.  ( f `  y ) ) ) ) )
373, 36rabeqbidv 3017 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  { f  e.  ( s  GrpHom  t )  | 
[. (Scalar `  s )  /  w ]. ( (Scalar `  t )  =  w  /\  A. x  e.  ( Base `  w
) A. y  e.  ( Base `  s
) ( f `  ( x ( .s
`  s ) y ) )  =  ( x ( .s `  t ) ( f `
 y ) ) ) }  =  {
f  e.  ( S 
GrpHom  T )  |  ( L  =  K  /\  A. x  e.  B  A. y  e.  E  (
f `  ( x  .x.  y ) )  =  ( x  .X.  (
f `  y )
) ) } )
38 ovex 6277 . . . . . 6  |-  ( S 
GrpHom  T )  e.  _V
3938rabex 4518 . . . . 5  |-  { f  e.  ( S  GrpHom  T )  |  ( L  =  K  /\  A. x  e.  B  A. y  e.  E  (
f `  ( x  .x.  y ) )  =  ( x  .X.  (
f `  y )
) ) }  e.  _V
4037, 1, 39ovmpt2a 6385 . . . 4  |-  ( ( S  e.  LMod  /\  T  e.  LMod )  ->  ( S LMHom  T )  =  {
f  e.  ( S 
GrpHom  T )  |  ( L  =  K  /\  A. x  e.  B  A. y  e.  E  (
f `  ( x  .x.  y ) )  =  ( x  .X.  (
f `  y )
) ) } )
4140eleq2d 2491 . . 3  |-  ( ( S  e.  LMod  /\  T  e.  LMod )  ->  ( F  e.  ( S LMHom  T )  <->  F  e.  { f  e.  ( S  GrpHom  T )  |  ( L  =  K  /\  A. x  e.  B  A. y  e.  E  (
f `  ( x  .x.  y ) )  =  ( x  .X.  (
f `  y )
) ) } ) )
42 fveq1 5824 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  ( x  .x.  y ) )  =  ( F `  (
x  .x.  y )
) )
43 fveq1 5824 . . . . . . . . 9  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
4443oveq2d 6265 . . . . . . . 8  |-  ( f  =  F  ->  (
x  .X.  ( f `  y ) )  =  ( x  .X.  ( F `  y )
) )
4542, 44eqeq12d 2443 . . . . . . 7  |-  ( f  =  F  ->  (
( f `  (
x  .x.  y )
)  =  ( x 
.X.  ( f `  y ) )  <->  ( F `  ( x  .x.  y
) )  =  ( x  .X.  ( F `  y ) ) ) )
46452ralbidv 2809 . . . . . 6  |-  ( f  =  F  ->  ( A. x  e.  B  A. y  e.  E  ( f `  (
x  .x.  y )
)  =  ( x 
.X.  ( f `  y ) )  <->  A. x  e.  B  A. y  e.  E  ( F `  ( x  .x.  y
) )  =  ( x  .X.  ( F `  y ) ) ) )
4746anbi2d 708 . . . . 5  |-  ( f  =  F  ->  (
( L  =  K  /\  A. x  e.  B  A. y  e.  E  ( f `  ( x  .x.  y ) )  =  ( x 
.X.  ( f `  y ) ) )  <-> 
( L  =  K  /\  A. x  e.  B  A. y  e.  E  ( F `  ( x  .x.  y ) )  =  ( x 
.X.  ( F `  y ) ) ) ) )
4847elrab 3171 . . . 4  |-  ( F  e.  { f  e.  ( S  GrpHom  T )  |  ( L  =  K  /\  A. x  e.  B  A. y  e.  E  ( f `  ( x  .x.  y
) )  =  ( x  .X.  ( f `  y ) ) ) }  <->  ( F  e.  ( S  GrpHom  T )  /\  ( L  =  K  /\  A. x  e.  B  A. y  e.  E  ( F `  ( x  .x.  y
) )  =  ( x  .X.  ( F `  y ) ) ) ) )
49 3anass 986 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  L  =  K  /\  A. x  e.  B  A. y  e.  E  ( F `  ( x  .x.  y
) )  =  ( x  .X.  ( F `  y ) ) )  <-> 
( F  e.  ( S  GrpHom  T )  /\  ( L  =  K  /\  A. x  e.  B  A. y  e.  E  ( F `  ( x 
.x.  y ) )  =  ( x  .X.  ( F `  y ) ) ) ) )
5048, 49bitr4i 255 . . 3  |-  ( F  e.  { f  e.  ( S  GrpHom  T )  |  ( L  =  K  /\  A. x  e.  B  A. y  e.  E  ( f `  ( x  .x.  y
) )  =  ( x  .X.  ( f `  y ) ) ) }  <->  ( F  e.  ( S  GrpHom  T )  /\  L  =  K  /\  A. x  e.  B  A. y  e.  E  ( F `  ( x  .x.  y ) )  =  ( x 
.X.  ( F `  y ) ) ) )
5141, 50syl6bb 264 . 2  |-  ( ( S  e.  LMod  /\  T  e.  LMod )  ->  ( F  e.  ( S LMHom  T )  <->  ( F  e.  ( S  GrpHom  T )  /\  L  =  K  /\  A. x  e.  B  A. y  e.  E  ( F `  ( x  .x.  y ) )  =  ( x 
.X.  ( F `  y ) ) ) ) )
522, 51biadan2 646 1  |-  ( F  e.  ( S LMHom  T
)  <->  ( ( S  e.  LMod  /\  T  e. 
LMod )  /\  ( F  e.  ( S  GrpHom  T )  /\  L  =  K  /\  A. x  e.  B  A. y  e.  E  ( F `  ( x  .x.  y
) )  =  ( x  .X.  ( F `  y ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   A.wral 2714   {crab 2718   _Vcvv 3022   [.wsbc 3242   ` cfv 5544  (class class class)co 6249   Basecbs 15064  Scalarcsca 15136   .scvsca 15137    GrpHom cghm 16823   LModclmod 18034   LMHom clmhm 18185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-iota 5508  df-fun 5546  df-fv 5552  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-lmhm 18188
This theorem is referenced by:  islmhm3  18194  lmhmlem  18195  lmhmlin  18201  islmhmd  18205  reslmhm  18218  lmhmpropd  18239
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