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Theorem isnmhm 21765
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 21713 . . 3  |- NMHom  =  ( s  e. NrmMod ,  t  e. NrmMod  |->  ( ( s LMHom  t
)  i^i  ( s NGHom  t ) ) )
21elmpt2cl 6525 . 2  |-  ( F  e.  ( S NMHom  T
)  ->  ( S  e. NrmMod  /\  T  e. NrmMod )
)
3 oveq12 6314 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s LMHom  t )  =  ( S LMHom  T
) )
4 oveq12 6314 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s NGHom  t )  =  ( S NGHom  T
) )
53, 4ineq12d 3665 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( s LMHom  t
)  i^i  ( s NGHom  t ) )  =  ( ( S LMHom  T
)  i^i  ( S NGHom  T ) ) )
6 ovex 6333 . . . . . 6  |-  ( S LMHom 
T )  e.  _V
76inex1 4565 . . . . 5  |-  ( ( S LMHom  T )  i^i  ( S NGHom  T ) )  e.  _V
85, 1, 7ovmpt2a 6441 . . . 4  |-  ( ( S  e. NrmMod  /\  T  e. NrmMod
)  ->  ( S NMHom  T )  =  ( ( S LMHom  T )  i^i  ( S NGHom  T ) ) )
98eleq2d 2492 . . 3  |-  ( ( S  e. NrmMod  /\  T  e. NrmMod
)  ->  ( F  e.  ( S NMHom  T )  <-> 
F  e.  ( ( S LMHom  T )  i^i  ( S NGHom  T ) ) ) )
10 elin 3649 . . 3  |-  ( F  e.  ( ( S LMHom 
T )  i^i  ( S NGHom  T ) )  <->  ( F  e.  ( S LMHom  T )  /\  F  e.  ( S NGHom  T ) ) )
119, 10syl6bb 264 . 2  |-  ( ( S  e. NrmMod  /\  T  e. NrmMod
)  ->  ( F  e.  ( S NMHom  T )  <-> 
( F  e.  ( S LMHom  T )  /\  F  e.  ( S NGHom  T ) ) ) )
122, 11biadan2 646 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 187    /\ wa 370    = wceq 1437    e. wcel 1872    i^i cin 3435  (class class class)co 6305   LMHom clmhm 18241  NrmModcnlm 21593   NGHom cnghm 21706   NMHom cnmhm 21708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-nmhm 21713
This theorem is referenced by:  nmhmrcl1  21766  nmhmrcl2  21767  nmhmlmhm  21768  nmhmnghm  21769  isnmhm2  21771  idnmhm  21773  0nmhm  21774  nmhmco  21775  nmhmplusg  21776  nmhmcn  22132
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