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Theorem islmhmd 16117
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 520 . 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 2800 . . 3  |-  ( ph  ->  A. x  e.  N  A. y  e.  X  ( F `  ( x 
.x.  y ) )  =  ( x  .X.  ( F `  y ) ) )
84, 5, 73jca 1135 . 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 16105 . 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 647 1  |-  ( ph  ->  F  e.  ( S LMHom 
T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   ` cfv 5456  (class class class)co 6083   Basecbs 13471  Scalarcsca 13534   .scvsca 13535    GrpHom cghm 15005   LModclmod 15952   LMHom clmhm 16097
This theorem is referenced by:  0lmhm  16118  idlmhm  16119  invlmhm  16120  lmhmco  16121  lmhmplusg  16122  lmhmvsca  16123  lmhmf1o  16124  reslmhm2  16131  reslmhm2b  16132  pwsdiaglmhm  16135  pwssplit3  27169  frlmup1  27229
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-lmhm 16100
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