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Theorem lmhmsca 17244
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 17243 . 2  |-  ( F  e.  ( S LMHom  T
)  ->  ( ( S  e.  LMod  /\  T  e.  LMod )  /\  ( F  e.  ( S  GrpHom  T )  /\  L  =  K ) ) )
4 simprr 756 . 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 1370    e. wcel 1758   ` cfv 5529  (class class class)co 6203  Scalarcsca 14364    GrpHom cghm 15867   LModclmod 17081   LMHom clmhm 17233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-lmhm 17236
This theorem is referenced by:  islmhm2  17252  lmhmco  17257  lmhmplusg  17258  lmhmvsca  17259  lmhmf1o  17260  lmhmima  17261  lmhmpreima  17262  reslmhm  17266  reslmhm2  17267  reslmhm2b  17268  lindfmm  18391  lmhmclm  20800  nmoleub2lem3  20812  nmoleub3  20816
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