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Theorem isnmhm 18780
Description: A normed module homomorphism is a left module homomorphism which is also a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
Assertion
Ref Expression
isnmhm  |-  ( F  e.  ( S NMHom  T
)  <->  ( ( S  e. NrmMod  /\  T  e. NrmMod )  /\  ( F  e.  ( S LMHom  T )  /\  F  e.  ( S NGHom  T ) ) ) )

Proof of Theorem isnmhm
Dummy variables  t 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nmhm 18744 . . 3  |- NMHom  =  ( s  e. NrmMod ,  t  e. NrmMod  |->  ( ( s LMHom  t
)  i^i  ( s NGHom  t ) ) )
21elmpt2cl 6288 . 2  |-  ( F  e.  ( S NMHom  T
)  ->  ( S  e. NrmMod  /\  T  e. NrmMod )
)
3 oveq12 6090 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s LMHom  t )  =  ( S LMHom  T
) )
4 oveq12 6090 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s NGHom  t )  =  ( S NGHom  T
) )
53, 4ineq12d 3543 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( s LMHom  t
)  i^i  ( s NGHom  t ) )  =  ( ( S LMHom  T
)  i^i  ( S NGHom  T ) ) )
6 ovex 6106 . . . . . 6  |-  ( S LMHom 
T )  e.  _V
76inex1 4344 . . . . 5  |-  ( ( S LMHom  T )  i^i  ( S NGHom  T ) )  e.  _V
85, 1, 7ovmpt2a 6204 . . . 4  |-  ( ( S  e. NrmMod  /\  T  e. NrmMod
)  ->  ( S NMHom  T )  =  ( ( S LMHom  T )  i^i  ( S NGHom  T ) ) )
98eleq2d 2503 . . 3  |-  ( ( S  e. NrmMod  /\  T  e. NrmMod
)  ->  ( F  e.  ( S NMHom  T )  <-> 
F  e.  ( ( S LMHom  T )  i^i  ( S NGHom  T ) ) ) )
10 elin 3530 . . 3  |-  ( F  e.  ( ( S LMHom 
T )  i^i  ( S NGHom  T ) )  <->  ( F  e.  ( S LMHom  T )  /\  F  e.  ( S NGHom  T ) ) )
119, 10syl6bb 253 . 2  |-  ( ( S  e. NrmMod  /\  T  e. NrmMod
)  ->  ( F  e.  ( S NMHom  T )  <-> 
( F  e.  ( S LMHom  T )  /\  F  e.  ( S NGHom  T ) ) ) )
122, 11biadan2 624 1  |-  ( F  e.  ( S NMHom  T
)  <->  ( ( S  e. NrmMod  /\  T  e. NrmMod )  /\  ( F  e.  ( S LMHom  T )  /\  F  e.  ( S NGHom  T ) ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    i^i cin 3319  (class class class)co 6081   LMHom clmhm 16095  NrmModcnlm 18628   NGHom cnghm 18740   NMHom cnmhm 18741
This theorem is referenced by:  nmhmrcl1  18781  nmhmrcl2  18782  nmhmlmhm  18783  nmhmnghm  18784  isnmhm2  18786  idnmhm  18788  0nmhm  18789  nmhmco  18790  nmhmplusg  18791  nmhmcn  19128
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
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