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Theorem nmhmnghm 18745
Description: A normed module homomorphism is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
Assertion
Ref Expression
nmhmnghm  |-  ( F  e.  ( S NMHom  T
)  ->  F  e.  ( S NGHom  T ) )

Proof of Theorem nmhmnghm
StepHypRef Expression
1 isnmhm 18741 . . 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 ) ) )
32simprd 450 1  |-  ( F  e.  ( S NMHom  T
)  ->  F  e.  ( S NGHom  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1721  (class class class)co 6048   LMHom clmhm 16058  NrmModcnlm 18589   NGHom cnghm 18701   NMHom cnmhm 18702
This theorem is referenced by:  nmhmghm  18746  nmhmcl  18748  nmhmco  18751  nmhmplusg  18752
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-nmhm 18705
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