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Theorem isnghm 18621
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 18620 . . 3  |-  ( S NGHom 
T )  =  ( `' N " RR )
32eleq2i 2444 . 2  |-  ( F  e.  ( S NGHom  T
)  <->  F  e.  ( `' N " RR ) )
4 n0i 3569 . . . 4  |-  ( F  e.  ( `' N " RR )  ->  -.  ( `' N " RR )  =  (/) )
5 nmoffn 18609 . . . . . . . . . . 11  |-  normOp  Fn  (NrmGrp  X. NrmGrp
)
6 fndm 5477 . . . . . . . . . . 11  |-  ( normOp  Fn  (NrmGrp  X. NrmGrp )  ->  dom  normOp  =  (NrmGrp  X. NrmGrp )
)
75, 6ax-mp 8 . . . . . . . . . 10  |-  dom  normOp  =  (NrmGrp  X. NrmGrp )
87ndmov 6163 . . . . . . . . 9  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( S normOp T )  =  (/) )
91, 8syl5eq 2424 . . . . . . . 8  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  N  =  (/) )
109cnveqd 4981 . . . . . . 7  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  `' N  =  `' (/) )
11 cnv0 5208 . . . . . . 7  |-  `' (/)  =  (/)
1210, 11syl6eq 2428 . . . . . 6  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  `' N  =  (/) )
1312imaeq1d 5135 . . . . 5  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( `' N " RR )  =  ( (/) " RR ) )
14 0ima 5155 . . . . 5  |-  ( (/) " RR )  =  (/)
1513, 14syl6eq 2428 . . . 4  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( `' N " RR )  =  (/) )
164, 15nsyl2 121 . . 3  |-  ( F  e.  ( `' N " RR )  ->  ( S  e. NrmGrp  /\  T  e. NrmGrp
) )
171nmof 18617 . . . 4  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  N :
( S  GrpHom  T ) -->
RR* )
18 ffn 5524 . . . 4  |-  ( N : ( S  GrpHom  T ) --> RR*  ->  N  Fn  ( S  GrpHom  T ) )
19 elpreima 5782 . . . 4  |-  ( N  Fn  ( S  GrpHom  T )  ->  ( F  e.  ( `' N " RR )  <->  ( F  e.  ( S  GrpHom  T )  /\  ( N `  F )  e.  RR ) ) )
2017, 18, 193syl 19 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( F  e.  ( `' N " RR )  <->  ( F  e.  ( S  GrpHom  T )  /\  ( N `  F )  e.  RR ) ) )
2116, 20biadan2 624 . 2  |-  ( F  e.  ( `' N " RR )  <->  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  /\  ( F  e.  ( S  GrpHom  T )  /\  ( N `  F )  e.  RR ) ) )
223, 21bitri 241 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 set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   (/)c0 3564    X. cxp 4809   `'ccnv 4810   dom cdm 4811   "cima 4814    Fn wfn 5382   -->wf 5383   ` cfv 5387  (class class class)co 6013   RRcr 8915   RR*cxr 9045    GrpHom cghm 14923  NrmGrpcngp 18489   normOpcnmo 18603   NGHom cnghm 18604
This theorem is referenced by:  isnghm2  18622  nghmcl  18625  nmoi  18626  nghmrcl1  18630  nghmrcl2  18631  nghmghm  18632  isnmhm2  18650
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-ico 10847  df-nmo 18606  df-nghm 18607
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