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Theorem islmhmd 17880
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 530 . 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 2875 . . 3  |-  ( ph  ->  A. x  e.  N  A. y  e.  X  ( F `  ( x 
.x.  y ) )  =  ( x  .X.  ( F `  y ) ) )
84, 5, 73jca 1174 . 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 17868 . 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 662 1  |-  ( ph  ->  F  e.  ( S LMHom 
T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   ` cfv 5570  (class class class)co 6270   Basecbs 14716  Scalarcsca 14787   .scvsca 14788    GrpHom cghm 16463   LModclmod 17707   LMHom clmhm 17860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-lmhm 17863
This theorem is referenced by:  0lmhm  17881  idlmhm  17882  invlmhm  17883  lmhmco  17884  lmhmplusg  17885  lmhmvsca  17886  lmhmf1o  17887  reslmhm2  17894  reslmhm2b  17895  pwsdiaglmhm  17898  pwssplit3  17902  frlmup1  19000
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