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Theorem isnghm 18248
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 18247 . . 3  |-  ( S NGHom 
T )  =  ( `' N " RR )
32eleq2i 2360 . 2  |-  ( F  e.  ( S NGHom  T
)  <->  F  e.  ( `' N " RR ) )
4 n0i 3473 . . . 4  |-  ( F  e.  ( `' N " RR )  ->  -.  ( `' N " RR )  =  (/) )
5 nmoffn 18236 . . . . . . . . . . 11  |-  normOp  Fn  (NrmGrp  X. NrmGrp
)
6 fndm 5359 . . . . . . . . . . 11  |-  ( normOp  Fn  (NrmGrp  X. NrmGrp )  ->  dom  normOp  =  (NrmGrp  X. NrmGrp )
)
75, 6ax-mp 8 . . . . . . . . . 10  |-  dom  normOp  =  (NrmGrp  X. NrmGrp )
87ndmov 6020 . . . . . . . . 9  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( S normOp T )  =  (/) )
91, 8syl5eq 2340 . . . . . . . 8  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  N  =  (/) )
109cnveqd 4873 . . . . . . 7  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  `' N  =  `' (/) )
11 cnv0 5100 . . . . . . 7  |-  `' (/)  =  (/)
1210, 11syl6eq 2344 . . . . . 6  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  `' N  =  (/) )
1312imaeq1d 5027 . . . . 5  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( `' N " RR )  =  ( (/) " RR ) )
14 0ima 5047 . . . . 5  |-  ( (/) " RR )  =  (/)
1513, 14syl6eq 2344 . . . 4  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( `' N " RR )  =  (/) )
164, 15nsyl2 119 . . 3  |-  ( F  e.  ( `' N " RR )  ->  ( S  e. NrmGrp  /\  T  e. NrmGrp
) )
171nmof 18244 . . . 4  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  N :
( S  GrpHom  T ) -->
RR* )
18 ffn 5405 . . . 4  |-  ( N : ( S  GrpHom  T ) --> RR*  ->  N  Fn  ( S  GrpHom  T ) )
19 elpreima 5661 . . . 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 623 . 2  |-  ( F  e.  ( `' N " RR )  <->  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  /\  ( F  e.  ( S  GrpHom  T )  /\  ( N `  F )  e.  RR ) ) )
223, 21bitri 240 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   (/)c0 3468    X. cxp 4703   `'ccnv 4704   dom cdm 4705   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   RRcr 8752   RR*cxr 8882    GrpHom cghm 14696  NrmGrpcngp 18116   normOpcnmo 18230   NGHom cnghm 18231
This theorem is referenced by:  isnghm2  18249  nghmcl  18252  nmoi  18253  nghmrcl1  18257  nghmrcl2  18258  nghmghm  18259  isnmhm2  18277
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-ico 10678  df-nmo 18233  df-nghm 18234
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