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Theorem isnmhm 18785
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 18749 . . 3  |- NMHom  =  ( s  e. NrmMod ,  t  e. NrmMod  |->  ( ( s LMHom  t
)  i^i  ( s NGHom  t ) ) )
21elmpt2cl 6291 . 2  |-  ( F  e.  ( S NMHom  T
)  ->  ( S  e. NrmMod  /\  T  e. NrmMod )
)
3 oveq12 6093 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s LMHom  t )  =  ( S LMHom  T
) )
4 oveq12 6093 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s NGHom  t )  =  ( S NGHom  T
) )
53, 4ineq12d 3545 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( s LMHom  t
)  i^i  ( s NGHom  t ) )  =  ( ( S LMHom  T
)  i^i  ( S NGHom  T ) ) )
6 ovex 6109 . . . . . 6  |-  ( S LMHom 
T )  e.  _V
76inex1 4347 . . . . 5  |-  ( ( S LMHom  T )  i^i  ( S NGHom  T ) )  e.  _V
85, 1, 7ovmpt2a 6207 . . . 4  |-  ( ( S  e. NrmMod  /\  T  e. NrmMod
)  ->  ( S NMHom  T )  =  ( ( S LMHom  T )  i^i  ( S NGHom  T ) ) )
98eleq2d 2505 . . 3  |-  ( ( S  e. NrmMod  /\  T  e. NrmMod
)  ->  ( F  e.  ( S NMHom  T )  <-> 
F  e.  ( ( S LMHom  T )  i^i  ( S NGHom  T ) ) ) )
10 elin 3532 . . 3  |-  ( F  e.  ( ( S LMHom 
T )  i^i  ( S NGHom  T ) )  <->  ( F  e.  ( S LMHom  T )  /\  F  e.  ( S NGHom  T ) ) )
119, 10syl6bb 254 . 2  |-  ( ( S  e. NrmMod  /\  T  e. NrmMod
)  ->  ( F  e.  ( S NMHom  T )  <-> 
( F  e.  ( S LMHom  T )  /\  F  e.  ( S NGHom  T ) ) ) )
122, 11biadan2 625 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 178    /\ wa 360    = wceq 1653    e. wcel 1726    i^i cin 3321  (class class class)co 6084   LMHom clmhm 16100  NrmModcnlm 18633   NGHom cnghm 18745   NMHom cnmhm 18746
This theorem is referenced by:  nmhmrcl1  18786  nmhmrcl2  18787  nmhmlmhm  18788  nmhmnghm  18789  isnmhm2  18791  idnmhm  18793  0nmhm  18794  nmhmco  18795  nmhmplusg  18796  nmhmcn  19133
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 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-nmhm 18749
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