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Theorem nmhmnghm 18815
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 18811 . . 3  |-  ( F  e.  ( S NMHom  T
)  <->  ( ( S  e. NrmMod  /\  T  e. NrmMod )  /\  ( F  e.  ( S LMHom  T )  /\  F  e.  ( S NGHom  T ) ) ) )
21simprbi 452 . 2  |-  ( F  e.  ( S NMHom  T
)  ->  ( F  e.  ( S LMHom  T )  /\  F  e.  ( S NGHom  T ) ) )
32simprd 451 1  |-  ( F  e.  ( S NMHom  T
)  ->  F  e.  ( S NGHom  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1727  (class class class)co 6110   LMHom clmhm 16126  NrmModcnlm 18659   NGHom cnghm 18771   NMHom cnmhm 18772
This theorem is referenced by:  nmhmghm  18816  nmhmcl  18818  nmhmco  18821  nmhmplusg  18822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-iota 5447  df-fun 5485  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-nmhm 18775
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