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Theorem islmhmd 16115
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 519 . 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 2798 . . 3  |-  ( ph  ->  A. x  e.  N  A. y  e.  X  ( F `  ( x 
.x.  y ) )  =  ( x  .X.  ( F `  y ) ) )
84, 5, 73jca 1134 . 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 16103 . 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 646 1  |-  ( ph  ->  F  e.  ( S LMHom 
T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   ` cfv 5454  (class class class)co 6081   Basecbs 13469  Scalarcsca 13532   .scvsca 13533    GrpHom cghm 15003   LModclmod 15950   LMHom clmhm 16095
This theorem is referenced by:  0lmhm  16116  idlmhm  16117  invlmhm  16118  lmhmco  16119  lmhmplusg  16120  lmhmvsca  16121  lmhmf1o  16122  reslmhm2  16129  reslmhm2b  16130  pwsdiaglmhm  16133  pwssplit3  27167  frlmup1  27227
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-lmhm 16098
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