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Theorem islmhm 17800
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 17795 . . 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 6516 . 2  |-  ( F  e.  ( S LMHom  T
)  ->  ( S  e.  LMod  /\  T  e.  LMod ) )
3 oveq12 6305 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s  GrpHom  t )  =  ( S  GrpHom  T ) )
4 fvex 5882 . . . . . . . 8  |-  (Scalar `  s )  e.  _V
54a1i 11 . . . . . . 7  |-  ( ( s  =  S  /\  t  =  T )  ->  (Scalar `  s )  e.  _V )
6 simplr 755 . . . . . . . . . . 11  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  t  =  T )
76fveq2d 5876 . . . . . . . . . 10  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  (Scalar `  t
)  =  (Scalar `  T ) )
8 islmhm.l . . . . . . . . . 10  |-  L  =  (Scalar `  T )
97, 8syl6eqr 2516 . . . . . . . . 9  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  (Scalar `  t
)  =  L )
10 simpr 461 . . . . . . . . . . 11  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  w  =  (Scalar `  s ) )
11 simpll 753 . . . . . . . . . . . 12  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  s  =  S )
1211fveq2d 5876 . . . . . . . . . . 11  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  (Scalar `  s
)  =  (Scalar `  S ) )
1310, 12eqtrd 2498 . . . . . . . . . 10  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  w  =  (Scalar `  S ) )
14 islmhm.k . . . . . . . . . 10  |-  K  =  (Scalar `  S )
1513, 14syl6eqr 2516 . . . . . . . . 9  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  w  =  K )
169, 15eqeq12d 2479 . . . . . . . 8  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( (Scalar `  t )  =  w  <-> 
L  =  K ) )
1715fveq2d 5876 . . . . . . . . . 10  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( Base `  w )  =  (
Base `  K )
)
18 islmhm.b . . . . . . . . . 10  |-  B  =  ( Base `  K
)
1917, 18syl6eqr 2516 . . . . . . . . 9  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( Base `  w )  =  B )
2011fveq2d 5876 . . . . . . . . . . 11  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( Base `  s )  =  (
Base `  S )
)
21 islmhm.e . . . . . . . . . . 11  |-  E  =  ( Base `  S
)
2220, 21syl6eqr 2516 . . . . . . . . . 10  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( Base `  s )  =  E )
2311fveq2d 5876 . . . . . . . . . . . . . 14  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( .s `  s )  =  ( .s `  S ) )
24 islmhm.m . . . . . . . . . . . . . 14  |-  .x.  =  ( .s `  S )
2523, 24syl6eqr 2516 . . . . . . . . . . . . 13  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( .s `  s )  =  .x.  )
2625oveqd 6313 . . . . . . . . . . . 12  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( x
( .s `  s
) y )  =  ( x  .x.  y
) )
2726fveq2d 5876 . . . . . . . . . . 11  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( f `  ( x ( .s
`  s ) y ) )  =  ( f `  ( x 
.x.  y ) ) )
286fveq2d 5876 . . . . . . . . . . . . 13  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( .s `  t )  =  ( .s `  T ) )
29 islmhm.n . . . . . . . . . . . . 13  |-  .X.  =  ( .s `  T )
3028, 29syl6eqr 2516 . . . . . . . . . . . 12  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( .s `  t )  =  .X.  )
3130oveqd 6313 . . . . . . . . . . 11  |-  ( ( ( s  =  S  /\  t  =  T )  /\  w  =  (Scalar `  s )
)  ->  ( x
( .s `  t
) ( f `  y ) )  =  ( x  .X.  (
f `  y )
) )
3227, 31eqeq12d 2479 . . . . . . . . . 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 3068 . . . . . . . . 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 3068 . . . . . . . 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 710 . . . . . . 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 3364 . . . . . 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 3104 . . . . 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 6324 . . . . . 6  |-  ( S 
GrpHom  T )  e.  _V
3938rabex 4607 . . . . 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 6432 . . . 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 2527 . . 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 5871 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  ( x  .x.  y ) )  =  ( F `  (
x  .x.  y )
) )
43 fveq1 5871 . . . . . . . . 9  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
4443oveq2d 6312 . . . . . . . 8  |-  ( f  =  F  ->  (
x  .X.  ( f `  y ) )  =  ( x  .X.  ( F `  y )
) )
4542, 44eqeq12d 2479 . . . . . . 7  |-  ( f  =  F  ->  (
( f `  (
x  .x.  y )
)  =  ( x 
.X.  ( f `  y ) )  <->  ( F `  ( x  .x.  y
) )  =  ( x  .X.  ( F `  y ) ) ) )
46452ralbidv 2901 . . . . . 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 703 . . . . 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 3257 . . . 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 977 . . . 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 252 . . 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 261 . 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 642 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 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   {crab 2811   _Vcvv 3109   [.wsbc 3327   ` cfv 5594  (class class class)co 6296   Basecbs 14644  Scalarcsca 14715   .scvsca 14716    GrpHom cghm 16391   LModclmod 17639   LMHom clmhm 17792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-lmhm 17795
This theorem is referenced by:  islmhm3  17801  lmhmlem  17802  lmhmlin  17808  islmhmd  17812  reslmhm  17825  lmhmpropd  17846
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