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Theorem islmhm3 15801
Description: Property of a module homomorphism, similar to ismhm 14433. (Contributed by Stefan O'Rear, 7-Mar-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
islmhm3  |-  ( ( 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 ) ) ) ) )
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 islmhm3
StepHypRef Expression
1 islmhm.k . . 3  |-  K  =  (Scalar `  S )
2 islmhm.l . . 3  |-  L  =  (Scalar `  T )
3 islmhm.b . . 3  |-  B  =  ( Base `  K
)
4 islmhm.e . . 3  |-  E  =  ( Base `  S
)
5 islmhm.m . . 3  |-  .x.  =  ( .s `  S )
6 islmhm.n . . 3  |-  .X.  =  ( .s `  T )
71, 2, 3, 4, 5, 6islmhm 15800 . 2  |-  ( 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 ) ) ) ) )
87baib 871 1  |-  ( ( 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 ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   ` cfv 5271  (class class class)co 5874   Basecbs 13164  Scalarcsca 13227   .scvsca 13228    GrpHom cghm 14696   LModclmod 15643   LMHom clmhm 15792
This theorem is referenced by:  islmhm2  15811  pj1lmhm  15869
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-lmhm 15795
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