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Theorem isnghm 21726
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 21725 . . 3  |-  ( S NGHom 
T )  =  ( `' N " RR )
32eleq2i 2499 . 2  |-  ( F  e.  ( S NGHom  T
)  <->  F  e.  ( `' N " RR ) )
4 n0i 3766 . . . 4  |-  ( F  e.  ( `' N " RR )  ->  -.  ( `' N " RR )  =  (/) )
5 nmoffn 21714 . . . . . . . . . . 11  |-  normOp  Fn  (NrmGrp  X. NrmGrp
)
6 fndm 5693 . . . . . . . . . . 11  |-  ( normOp  Fn  (NrmGrp  X. NrmGrp )  ->  dom  normOp  =  (NrmGrp  X. NrmGrp )
)
75, 6ax-mp 5 . . . . . . . . . 10  |-  dom  normOp  =  (NrmGrp  X. NrmGrp )
87ndmov 6467 . . . . . . . . 9  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( S normOp T )  =  (/) )
91, 8syl5eq 2475 . . . . . . . 8  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  N  =  (/) )
109cnveqd 5029 . . . . . . 7  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  `' N  =  `' (/) )
11 cnv0 5258 . . . . . . 7  |-  `' (/)  =  (/)
1210, 11syl6eq 2479 . . . . . 6  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  `' N  =  (/) )
1312imaeq1d 5186 . . . . 5  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( `' N " RR )  =  ( (/) " RR ) )
14 0ima 5203 . . . . 5  |-  ( (/) " RR )  =  (/)
1513, 14syl6eq 2479 . . . 4  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( `' N " RR )  =  (/) )
164, 15nsyl2 130 . . 3  |-  ( F  e.  ( `' N " RR )  ->  ( S  e. NrmGrp  /\  T  e. NrmGrp
) )
171nmof 21722 . . . 4  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  N :
( S  GrpHom  T ) -->
RR* )
18 ffn 5746 . . . 4  |-  ( N : ( S  GrpHom  T ) --> RR*  ->  N  Fn  ( S  GrpHom  T ) )
19 elpreima 6017 . . . 4  |-  ( N  Fn  ( S  GrpHom  T )  ->  ( F  e.  ( `' N " RR )  <->  ( F  e.  ( S  GrpHom  T )  /\  ( N `  F )  e.  RR ) ) )
2017, 18, 193syl 18 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( F  e.  ( `' N " RR )  <->  ( F  e.  ( S  GrpHom  T )  /\  ( N `  F )  e.  RR ) ) )
2116, 20biadan2 646 . 2  |-  ( F  e.  ( `' N " RR )  <->  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  /\  ( F  e.  ( S  GrpHom  T )  /\  ( N `  F )  e.  RR ) ) )
223, 21bitri 252 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 187    /\ wa 370    = wceq 1437    e. wcel 1872   (/)c0 3761    X. cxp 4851   `'ccnv 4852   dom cdm 4853   "cima 4856    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305   RRcr 9545   RR*cxr 9681    GrpHom cghm 16879  NrmGrpcngp 21590   normOpcnmo 21704   NGHom cnghm 21706
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  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  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-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-po 4774  df-so 4775  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-sup 7965  df-inf 7966  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-ico 11648  df-nmo 21709  df-nghm 21711
This theorem is referenced by:  isnghm2  21727  nghmcl  21730  nmoi  21731  nghmrcl1  21751  nghmrcl2  21752  nghmghm  21753  isnmhm2  21771
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