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Theorem isnmhm 21121
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 21085 . . 3  |- NMHom  =  ( s  e. NrmMod ,  t  e. NrmMod  |->  ( ( s LMHom  t
)  i^i  ( s NGHom  t ) ) )
21elmpt2cl 6512 . 2  |-  ( F  e.  ( S NMHom  T
)  ->  ( S  e. NrmMod  /\  T  e. NrmMod )
)
3 oveq12 6304 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s LMHom  t )  =  ( S LMHom  T
) )
4 oveq12 6304 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s NGHom  t )  =  ( S NGHom  T
) )
53, 4ineq12d 3706 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( s LMHom  t
)  i^i  ( s NGHom  t ) )  =  ( ( S LMHom  T
)  i^i  ( S NGHom  T ) ) )
6 ovex 6320 . . . . . 6  |-  ( S LMHom 
T )  e.  _V
76inex1 4594 . . . . 5  |-  ( ( S LMHom  T )  i^i  ( S NGHom  T ) )  e.  _V
85, 1, 7ovmpt2a 6428 . . . 4  |-  ( ( S  e. NrmMod  /\  T  e. NrmMod
)  ->  ( S NMHom  T )  =  ( ( S LMHom  T )  i^i  ( S NGHom  T ) ) )
98eleq2d 2537 . . 3  |-  ( ( S  e. NrmMod  /\  T  e. NrmMod
)  ->  ( F  e.  ( S NMHom  T )  <-> 
F  e.  ( ( S LMHom  T )  i^i  ( S NGHom  T ) ) ) )
10 elin 3692 . . 3  |-  ( F  e.  ( ( S LMHom 
T )  i^i  ( S NGHom  T ) )  <->  ( F  e.  ( S LMHom  T )  /\  F  e.  ( S NGHom  T ) ) )
119, 10syl6bb 261 . 2  |-  ( ( S  e. NrmMod  /\  T  e. NrmMod
)  ->  ( F  e.  ( S NMHom  T )  <-> 
( F  e.  ( S LMHom  T )  /\  F  e.  ( S NGHom  T ) ) ) )
122, 11biadan2 642 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 setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    i^i cin 3480  (class class class)co 6295   LMHom clmhm 17536  NrmModcnlm 20969   NGHom cnghm 21081   NMHom cnmhm 21082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-nmhm 21085
This theorem is referenced by:  nmhmrcl1  21122  nmhmrcl2  21123  nmhmlmhm  21124  nmhmnghm  21125  isnmhm2  21127  idnmhm  21129  0nmhm  21130  nmhmco  21131  nmhmplusg  21132  nmhmcn  21471
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