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Theorem isnmhm 18733
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 18697 . . 3  |- NMHom  =  ( s  e. NrmMod ,  t  e. NrmMod  |->  ( ( s LMHom  t
)  i^i  ( s NGHom  t ) ) )
21elmpt2cl 6247 . 2  |-  ( F  e.  ( S NMHom  T
)  ->  ( S  e. NrmMod  /\  T  e. NrmMod )
)
3 oveq12 6049 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s LMHom  t )  =  ( S LMHom  T
) )
4 oveq12 6049 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s NGHom  t )  =  ( S NGHom  T
) )
53, 4ineq12d 3503 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( s LMHom  t
)  i^i  ( s NGHom  t ) )  =  ( ( S LMHom  T
)  i^i  ( S NGHom  T ) ) )
6 ovex 6065 . . . . . 6  |-  ( S LMHom 
T )  e.  _V
76inex1 4304 . . . . 5  |-  ( ( S LMHom  T )  i^i  ( S NGHom  T ) )  e.  _V
85, 1, 7ovmpt2a 6163 . . . 4  |-  ( ( S  e. NrmMod  /\  T  e. NrmMod
)  ->  ( S NMHom  T )  =  ( ( S LMHom  T )  i^i  ( S NGHom  T ) ) )
98eleq2d 2471 . . 3  |-  ( ( S  e. NrmMod  /\  T  e. NrmMod
)  ->  ( F  e.  ( S NMHom  T )  <-> 
F  e.  ( ( S LMHom  T )  i^i  ( S NGHom  T ) ) ) )
10 elin 3490 . . 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 1649    e. wcel 1721    i^i cin 3279  (class class class)co 6040   LMHom clmhm 16050  NrmModcnlm 18581   NGHom cnghm 18693   NMHom cnmhm 18694
This theorem is referenced by:  nmhmrcl1  18734  nmhmrcl2  18735  nmhmlmhm  18736  nmhmnghm  18737  isnmhm2  18739  idnmhm  18741  0nmhm  18742  nmhmco  18743  nmhmplusg  18744  nmhmcn  19081
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 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-nmhm 18697
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