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Theorem isnghm 18749
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 18748 . . 3  |-  ( S NGHom 
T )  =  ( `' N " RR )
32eleq2i 2499 . 2  |-  ( F  e.  ( S NGHom  T
)  <->  F  e.  ( `' N " RR ) )
4 n0i 3625 . . . 4  |-  ( F  e.  ( `' N " RR )  ->  -.  ( `' N " RR )  =  (/) )
5 nmoffn 18737 . . . . . . . . . . 11  |-  normOp  Fn  (NrmGrp  X. NrmGrp
)
6 fndm 5536 . . . . . . . . . . 11  |-  ( normOp  Fn  (NrmGrp  X. NrmGrp )  ->  dom  normOp  =  (NrmGrp  X. NrmGrp )
)
75, 6ax-mp 8 . . . . . . . . . 10  |-  dom  normOp  =  (NrmGrp  X. NrmGrp )
87ndmov 6223 . . . . . . . . 9  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( S normOp T )  =  (/) )
91, 8syl5eq 2479 . . . . . . . 8  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  N  =  (/) )
109cnveqd 5040 . . . . . . 7  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  `' N  =  `' (/) )
11 cnv0 5267 . . . . . . 7  |-  `' (/)  =  (/)
1210, 11syl6eq 2483 . . . . . 6  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  `' N  =  (/) )
1312imaeq1d 5194 . . . . 5  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( `' N " RR )  =  ( (/) " RR ) )
14 0ima 5214 . . . . 5  |-  ( (/) " RR )  =  (/)
1513, 14syl6eq 2483 . . . 4  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( `' N " RR )  =  (/) )
164, 15nsyl2 121 . . 3  |-  ( F  e.  ( `' N " RR )  ->  ( S  e. NrmGrp  /\  T  e. NrmGrp
) )
171nmof 18745 . . . 4  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  N :
( S  GrpHom  T ) -->
RR* )
18 ffn 5583 . . . 4  |-  ( N : ( S  GrpHom  T ) --> RR*  ->  N  Fn  ( S  GrpHom  T ) )
19 elpreima 5842 . . . 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 1652    e. wcel 1725   (/)c0 3620    X. cxp 4868   `'ccnv 4869   dom cdm 4870   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   RRcr 8981   RR*cxr 9111    GrpHom cghm 14995  NrmGrpcngp 18617   normOpcnmo 18731   NGHom cnghm 18732
This theorem is referenced by:  isnghm2  18750  nghmcl  18753  nmoi  18754  nghmrcl1  18758  nghmrcl2  18759  nghmghm  18760  isnmhm2  18778
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-ico 10914  df-nmo 18734  df-nghm 18735
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