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Theorem nghmfval 18231
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 5867 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s normOp t )  =  ( S normOp T ) )
2 nmofval.1 . . . . . 6  |-  N  =  ( S normOp T )
31, 2syl6eqr 2333 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s normOp t )  =  N )
43cnveqd 4857 . . . 4  |-  ( ( s  =  S  /\  t  =  T )  ->  `' ( s normOp t )  =  `' N
)
54imaeq1d 5011 . . 3  |-  ( ( s  =  S  /\  t  =  T )  ->  ( `' ( s
normOp t ) " RR )  =  ( `' N " RR ) )
6 df-nghm 18218 . . 3  |- NGHom  =  ( s  e. NrmGrp ,  t  e. NrmGrp  |->  ( `' ( s
normOp t ) " RR ) )
7 ovex 5883 . . . . . 6  |-  ( S
normOp T )  e.  _V
82, 7eqeltri 2353 . . . . 5  |-  N  e. 
_V
98cnvex 5209 . . . 4  |-  `' N  e.  _V
10 imaexg 5026 . . . 4  |-  ( `' N  e.  _V  ->  ( `' N " RR )  e.  _V )
119, 10ax-mp 8 . . 3  |-  ( `' N " RR )  e.  _V
125, 6, 11ovmpt2a 5978 . 2  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( S NGHom  T )  =  ( `' N " RR ) )
13 ovex 5883 . . . . . . 7  |-  ( s
normOp t )  e.  _V
1413cnvex 5209 . . . . . 6  |-  `' ( s normOp t )  e. 
_V
15 imaexg 5026 . . . . . 6  |-  ( `' ( s normOp t )  e.  _V  ->  ( `' ( s normOp t ) " RR )  e.  _V )
1614, 15ax-mp 8 . . . . 5  |-  ( `' ( s normOp t )
" RR )  e. 
_V
176, 16dmmpt2 6194 . . . 4  |-  dom NGHom  =  (NrmGrp  X. NrmGrp )
1817ndmov 6004 . . 3  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( S NGHom  T )  =  (/) )
19 nmoffn 18220 . . . . . . . . . 10  |-  normOp  Fn  (NrmGrp  X. NrmGrp
)
20 fndm 5343 . . . . . . . . . 10  |-  ( normOp  Fn  (NrmGrp  X. NrmGrp )  ->  dom  normOp  =  (NrmGrp  X. NrmGrp )
)
2119, 20ax-mp 8 . . . . . . . . 9  |-  dom  normOp  =  (NrmGrp  X. NrmGrp )
2221ndmov 6004 . . . . . . . 8  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( S normOp T )  =  (/) )
232, 22syl5eq 2327 . . . . . . 7  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  N  =  (/) )
2423cnveqd 4857 . . . . . 6  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  `' N  =  `' (/) )
25 cnv0 5084 . . . . . 6  |-  `' (/)  =  (/)
2624, 25syl6eq 2331 . . . . 5  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  `' N  =  (/) )
2726imaeq1d 5011 . . . 4  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( `' N " RR )  =  ( (/) " RR ) )
28 0ima 5031 . . . 4  |-  ( (/) " RR )  =  (/)
2927, 28syl6eq 2331 . . 3  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( `' N " RR )  =  (/) )
3018, 29eqtr4d 2318 . 2  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( S NGHom  T )  =  ( `' N " RR ) )
3112, 30pm2.61i 156 1  |-  ( S NGHom 
T )  =  ( `' N " RR )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455    X. cxp 4687   `'ccnv 4688   dom cdm 4689   "cima 4692    Fn wfn 5250  (class class class)co 5858   RRcr 8736  NrmGrpcngp 18100   normOpcnmo 18214   NGHom cnghm 18215
This theorem is referenced by:  isnghm  18232
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-ico 10662  df-nmo 18217  df-nghm 18218
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