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Theorem nmhmlmhm 18654
Description: A normed module homomorphism is a left module homomorphism (a linear operator). (Contributed by Mario Carneiro, 18-Oct-2015.)
Assertion
Ref Expression
nmhmlmhm  |-  ( F  e.  ( S NMHom  T
)  ->  F  e.  ( S LMHom  T ) )

Proof of Theorem nmhmlmhm
StepHypRef Expression
1 isnmhm 18651 . . 3  |-  ( F  e.  ( S NMHom  T
)  <->  ( ( S  e. NrmMod  /\  T  e. NrmMod )  /\  ( F  e.  ( S LMHom  T )  /\  F  e.  ( S NGHom  T ) ) ) )
21simprbi 451 . 2  |-  ( F  e.  ( S NMHom  T
)  ->  ( F  e.  ( S LMHom  T )  /\  F  e.  ( S NGHom  T ) ) )
32simpld 446 1  |-  ( F  e.  ( S NMHom  T
)  ->  F  e.  ( S LMHom  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717  (class class class)co 6020   LMHom clmhm 16022  NrmModcnlm 18499   NGHom cnghm 18611   NMHom cnmhm 18612
This theorem is referenced by:  nmhmco  18661  nmhmplusg  18662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-nmhm 18615
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