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Theorem lmhmsca 17994
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 17993 . 2  |-  ( F  e.  ( S LMHom  T
)  ->  ( ( S  e.  LMod  /\  T  e.  LMod )  /\  ( F  e.  ( S  GrpHom  T )  /\  L  =  K ) ) )
43simprrd 759 1  |-  ( F  e.  ( S LMHom  T
)  ->  L  =  K )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   ` cfv 5568  (class class class)co 6277  Scalarcsca 14910    GrpHom cghm 16586   LModclmod 17830   LMHom clmhm 17983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-iota 5532  df-fun 5570  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-lmhm 17986
This theorem is referenced by:  islmhm2  18002  lmhmco  18007  lmhmplusg  18008  lmhmvsca  18009  lmhmf1o  18010  lmhmima  18011  lmhmpreima  18012  reslmhm  18016  reslmhm2  18017  reslmhm2b  18018  lindfmm  19152  lmhmclm  21876  nmoleub2lem3  21888  nmoleub3  21892
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