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Theorem nghmfval 20300
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
nghmfval  |-  ( S NGHom 
T )  =  ( `' N " RR )

Proof of Theorem nghmfval
Dummy variables  s 
t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq12 6099 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s normOp t )  =  ( S normOp T ) )
2 nmofval.1 . . . . . 6  |-  N  =  ( S normOp T )
31, 2syl6eqr 2492 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s normOp t )  =  N )
43cnveqd 5014 . . . 4  |-  ( ( s  =  S  /\  t  =  T )  ->  `' ( s normOp t )  =  `' N
)
54imaeq1d 5167 . . 3  |-  ( ( s  =  S  /\  t  =  T )  ->  ( `' ( s
normOp t ) " RR )  =  ( `' N " RR ) )
6 df-nghm 20287 . . 3  |- NGHom  =  ( s  e. NrmGrp ,  t  e. NrmGrp  |->  ( `' ( s
normOp t ) " RR ) )
7 ovex 6115 . . . . . 6  |-  ( S
normOp T )  e.  _V
82, 7eqeltri 2512 . . . . 5  |-  N  e. 
_V
98cnvex 6524 . . . 4  |-  `' N  e.  _V
10 imaexg 6514 . . . 4  |-  ( `' N  e.  _V  ->  ( `' N " RR )  e.  _V )
119, 10ax-mp 5 . . 3  |-  ( `' N " RR )  e.  _V
125, 6, 11ovmpt2a 6220 . 2  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( S NGHom  T )  =  ( `' N " RR ) )
136mpt2ndm0 6738 . . 3  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( S NGHom  T )  =  (/) )
14 nmoffn 20289 . . . . . . . . . 10  |-  normOp  Fn  (NrmGrp  X. NrmGrp
)
15 fndm 5509 . . . . . . . . . 10  |-  ( normOp  Fn  (NrmGrp  X. NrmGrp )  ->  dom  normOp  =  (NrmGrp  X. NrmGrp )
)
1614, 15ax-mp 5 . . . . . . . . 9  |-  dom  normOp  =  (NrmGrp  X. NrmGrp )
1716ndmov 6246 . . . . . . . 8  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( S normOp T )  =  (/) )
182, 17syl5eq 2486 . . . . . . 7  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  N  =  (/) )
1918cnveqd 5014 . . . . . 6  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  `' N  =  `' (/) )
20 cnv0 5239 . . . . . 6  |-  `' (/)  =  (/)
2119, 20syl6eq 2490 . . . . 5  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  `' N  =  (/) )
2221imaeq1d 5167 . . . 4  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( `' N " RR )  =  ( (/) " RR ) )
23 0ima 5184 . . . 4  |-  ( (/) " RR )  =  (/)
2422, 23syl6eq 2490 . . 3  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( `' N " RR )  =  (/) )
2513, 24eqtr4d 2477 . 2  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( S NGHom  T )  =  ( `' N " RR ) )
2612, 25pm2.61i 164 1  |-  ( S NGHom 
T )  =  ( `' N " RR )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2971   (/)c0 3636    X. cxp 4837   `'ccnv 4838   dom cdm 4839   "cima 4842    Fn wfn 5412  (class class class)co 6090   RRcr 9280  NrmGrpcngp 20169   normOpcnmo 20283   NGHom cnghm 20284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-po 4640  df-so 4641  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-sup 7690  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-ico 11305  df-nmo 20286  df-nghm 20287
This theorem is referenced by:  isnghm  20301
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