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Theorem lmhmsca 17033
Description: A homomorphism of left modules constrains both modules to the same ring of scalars. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Hypotheses
Ref Expression
lmhmlem.k  |-  K  =  (Scalar `  S )
lmhmlem.l  |-  L  =  (Scalar `  T )
Assertion
Ref Expression
lmhmsca  |-  ( F  e.  ( S LMHom  T
)  ->  L  =  K )

Proof of Theorem lmhmsca
StepHypRef Expression
1 lmhmlem.k . . 3  |-  K  =  (Scalar `  S )
2 lmhmlem.l . . 3  |-  L  =  (Scalar `  T )
31, 2lmhmlem 17032 . 2  |-  ( F  e.  ( S LMHom  T
)  ->  ( ( S  e.  LMod  /\  T  e.  LMod )  /\  ( F  e.  ( S  GrpHom  T )  /\  L  =  K ) ) )
4 simprr 749 . 2  |-  ( ( ( S  e.  LMod  /\  T  e.  LMod )  /\  ( F  e.  ( S  GrpHom  T )  /\  L  =  K )
)  ->  L  =  K )
53, 4syl 16 1  |-  ( F  e.  ( S LMHom  T
)  ->  L  =  K )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755   ` cfv 5406  (class class class)co 6080  Scalarcsca 14224    GrpHom cghm 15724   LModclmod 16872   LMHom clmhm 17022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-iota 5369  df-fun 5408  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-lmhm 17025
This theorem is referenced by:  islmhm2  17041  lmhmco  17046  lmhmplusg  17047  lmhmvsca  17048  lmhmf1o  17049  lmhmima  17050  lmhmpreima  17051  reslmhm  17055  reslmhm2  17056  reslmhm2b  17057  lindfmm  18098  lmhmclm  20500  nmoleub2lem3  20512  nmoleub3  20516
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