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Theorem isnghm 21338
Description: A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
Hypothesis
Ref Expression
nmofval.1  |-  N  =  ( S normOp T )
Assertion
Ref Expression
isnghm  |-  ( F  e.  ( S NGHom  T
)  <->  ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( F  e.  ( S  GrpHom  T )  /\  ( N `  F )  e.  RR ) ) )

Proof of Theorem isnghm
StepHypRef Expression
1 nmofval.1 . . . 4  |-  N  =  ( S normOp T )
21nghmfval 21337 . . 3  |-  ( S NGHom 
T )  =  ( `' N " RR )
32eleq2i 2474 . 2  |-  ( F  e.  ( S NGHom  T
)  <->  F  e.  ( `' N " RR ) )
4 n0i 3733 . . . 4  |-  ( F  e.  ( `' N " RR )  ->  -.  ( `' N " RR )  =  (/) )
5 nmoffn 21326 . . . . . . . . . . 11  |-  normOp  Fn  (NrmGrp  X. NrmGrp
)
6 fndm 5605 . . . . . . . . . . 11  |-  ( normOp  Fn  (NrmGrp  X. NrmGrp )  ->  dom  normOp  =  (NrmGrp  X. NrmGrp )
)
75, 6ax-mp 5 . . . . . . . . . 10  |-  dom  normOp  =  (NrmGrp  X. NrmGrp )
87ndmov 6380 . . . . . . . . 9  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( S normOp T )  =  (/) )
91, 8syl5eq 2449 . . . . . . . 8  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  N  =  (/) )
109cnveqd 5108 . . . . . . 7  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  `' N  =  `' (/) )
11 cnv0 5336 . . . . . . 7  |-  `' (/)  =  (/)
1210, 11syl6eq 2453 . . . . . 6  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  `' N  =  (/) )
1312imaeq1d 5265 . . . . 5  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( `' N " RR )  =  ( (/) " RR ) )
14 0ima 5282 . . . . 5  |-  ( (/) " RR )  =  (/)
1513, 14syl6eq 2453 . . . 4  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( `' N " RR )  =  (/) )
164, 15nsyl2 127 . . 3  |-  ( F  e.  ( `' N " RR )  ->  ( S  e. NrmGrp  /\  T  e. NrmGrp
) )
171nmof 21334 . . . 4  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  N :
( S  GrpHom  T ) -->
RR* )
18 ffn 5656 . . . 4  |-  ( N : ( S  GrpHom  T ) --> RR*  ->  N  Fn  ( S  GrpHom  T ) )
19 elpreima 5926 . . . 4  |-  ( N  Fn  ( S  GrpHom  T )  ->  ( F  e.  ( `' N " RR )  <->  ( F  e.  ( S  GrpHom  T )  /\  ( N `  F )  e.  RR ) ) )
2017, 18, 193syl 20 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( F  e.  ( `' N " RR )  <->  ( F  e.  ( S  GrpHom  T )  /\  ( N `  F )  e.  RR ) ) )
2116, 20biadan2 640 . 2  |-  ( F  e.  ( `' N " RR )  <->  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  /\  ( F  e.  ( S  GrpHom  T )  /\  ( N `  F )  e.  RR ) ) )
223, 21bitri 249 1  |-  ( F  e.  ( S NGHom  T
)  <->  ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( F  e.  ( S  GrpHom  T )  /\  ( N `  F )  e.  RR ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1836   (/)c0 3728    X. cxp 4928   `'ccnv 4929   dom cdm 4930   "cima 4933    Fn wfn 5508   -->wf 5509   ` cfv 5513  (class class class)co 6218   RRcr 9424   RR*cxr 9560    GrpHom cghm 16404  NrmGrpcngp 21206   normOpcnmo 21320   NGHom cnghm 21321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-cnex 9481  ax-resscn 9482  ax-1cn 9483  ax-icn 9484  ax-addcl 9485  ax-addrcl 9486  ax-mulcl 9487  ax-mulrcl 9488  ax-mulcom 9489  ax-addass 9490  ax-mulass 9491  ax-distr 9492  ax-i2m1 9493  ax-1ne0 9494  ax-1rid 9495  ax-rnegex 9496  ax-rrecex 9497  ax-cnre 9498  ax-pre-lttri 9499  ax-pre-lttrn 9500  ax-pre-ltadd 9501  ax-pre-mulgt0 9502  ax-pre-sup 9503
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rmo 2754  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-po 4731  df-so 4732  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-1st 6721  df-2nd 6722  df-er 7251  df-en 7458  df-dom 7459  df-sdom 7460  df-sup 7838  df-pnf 9563  df-mnf 9564  df-xr 9565  df-ltxr 9566  df-le 9567  df-sub 9742  df-neg 9743  df-ico 11478  df-nmo 21323  df-nghm 21324
This theorem is referenced by:  isnghm2  21339  nghmcl  21342  nmoi  21343  nghmrcl1  21347  nghmrcl2  21348  nghmghm  21349  isnmhm2  21367
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