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Theorem islmhmd 15812
Description: Deduction for a module homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.)
Hypotheses
Ref Expression
islmhmd.x  |-  X  =  ( Base `  S
)
islmhmd.a  |-  .x.  =  ( .s `  S )
islmhmd.b  |-  .X.  =  ( .s `  T )
islmhmd.k  |-  K  =  (Scalar `  S )
islmhmd.j  |-  J  =  (Scalar `  T )
islmhmd.n  |-  N  =  ( Base `  K
)
islmhmd.s  |-  ( ph  ->  S  e.  LMod )
islmhmd.t  |-  ( ph  ->  T  e.  LMod )
islmhmd.c  |-  ( ph  ->  J  =  K )
islmhmd.f  |-  ( ph  ->  F  e.  ( S 
GrpHom  T ) )
islmhmd.l  |-  ( (
ph  /\  ( x  e.  N  /\  y  e.  X ) )  -> 
( F `  (
x  .x.  y )
)  =  ( x 
.X.  ( F `  y ) ) )
Assertion
Ref Expression
islmhmd  |-  ( ph  ->  F  e.  ( S LMHom 
T ) )
Distinct variable groups:    ph, x, y   
x, F, y    x, S, y    x, T, y   
x, X, y    x, J, y    x, N, y   
x, K, y
Allowed substitution hints:    .x. ( x, y)    .X. ( x, y)

Proof of Theorem islmhmd
StepHypRef Expression
1 islmhmd.s . . 3  |-  ( ph  ->  S  e.  LMod )
2 islmhmd.t . . 3  |-  ( ph  ->  T  e.  LMod )
31, 2jca 518 . 2  |-  ( ph  ->  ( S  e.  LMod  /\  T  e.  LMod )
)
4 islmhmd.f . . 3  |-  ( ph  ->  F  e.  ( S 
GrpHom  T ) )
5 islmhmd.c . . 3  |-  ( ph  ->  J  =  K )
6 islmhmd.l . . . 4  |-  ( (
ph  /\  ( x  e.  N  /\  y  e.  X ) )  -> 
( F `  (
x  .x.  y )
)  =  ( x 
.X.  ( F `  y ) ) )
76ralrimivva 2648 . . 3  |-  ( ph  ->  A. x  e.  N  A. y  e.  X  ( F `  ( x 
.x.  y ) )  =  ( x  .X.  ( F `  y ) ) )
84, 5, 73jca 1132 . 2  |-  ( ph  ->  ( F  e.  ( S  GrpHom  T )  /\  J  =  K  /\  A. x  e.  N  A. y  e.  X  ( F `  ( x  .x.  y ) )  =  ( x  .X.  ( F `  y )
) ) )
9 islmhmd.k . . 3  |-  K  =  (Scalar `  S )
10 islmhmd.j . . 3  |-  J  =  (Scalar `  T )
11 islmhmd.n . . 3  |-  N  =  ( Base `  K
)
12 islmhmd.x . . 3  |-  X  =  ( Base `  S
)
13 islmhmd.a . . 3  |-  .x.  =  ( .s `  S )
14 islmhmd.b . . 3  |-  .X.  =  ( .s `  T )
159, 10, 11, 12, 13, 14islmhm 15800 . 2  |-  ( F  e.  ( S LMHom  T
)  <->  ( ( S  e.  LMod  /\  T  e. 
LMod )  /\  ( F  e.  ( S  GrpHom  T )  /\  J  =  K  /\  A. x  e.  N  A. y  e.  X  ( F `  ( x  .x.  y
) )  =  ( x  .X.  ( F `  y ) ) ) ) )
163, 8, 15sylanbrc 645 1  |-  ( ph  ->  F  e.  ( S LMHom 
T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ 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:  0lmhm  15813  idlmhm  15814  invlmhm  15815  lmhmco  15816  lmhmplusg  15817  lmhmvsca  15818  lmhmf1o  15819  reslmhm2  15826  reslmhm2b  15827  pwsdiaglmhm  15830  pwssplit3  27293  frlmup1  27353
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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