MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nmhmrcl1 Structured version   Unicode version

Theorem nmhmrcl1 18781
Description: Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
Assertion
Ref Expression
nmhmrcl1  |-  ( F  e.  ( S NMHom  T
)  ->  S  e. NrmMod )

Proof of Theorem nmhmrcl1
StepHypRef Expression
1 isnmhm 18780 . . 3  |-  ( F  e.  ( S NMHom  T
)  <->  ( ( S  e. NrmMod  /\  T  e. NrmMod )  /\  ( F  e.  ( S LMHom  T )  /\  F  e.  ( S NGHom  T ) ) ) )
21simplbi 447 . 2  |-  ( F  e.  ( S NMHom  T
)  ->  ( S  e. NrmMod  /\  T  e. NrmMod )
)
32simpld 446 1  |-  ( F  e.  ( S NMHom  T
)  ->  S  e. NrmMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725  (class class class)co 6081   LMHom clmhm 16095  NrmModcnlm 18628   NGHom cnghm 18740   NMHom cnmhm 18741
This theorem is referenced by:  nmhmco  18790  nmhmplusg  18791
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-nmhm 18744
  Copyright terms: Public domain W3C validator