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Theorem nghmfval 21355
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 6305 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s normOp t )  =  ( S normOp T ) )
2 nmofval.1 . . . . . 6  |-  N  =  ( S normOp T )
31, 2syl6eqr 2516 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s normOp t )  =  N )
43cnveqd 5188 . . . 4  |-  ( ( s  =  S  /\  t  =  T )  ->  `' ( s normOp t )  =  `' N
)
54imaeq1d 5346 . . 3  |-  ( ( s  =  S  /\  t  =  T )  ->  ( `' ( s
normOp t ) " RR )  =  ( `' N " RR ) )
6 df-nghm 21342 . . 3  |- NGHom  =  ( s  e. NrmGrp ,  t  e. NrmGrp  |->  ( `' ( s
normOp t ) " RR ) )
7 ovex 6324 . . . . . 6  |-  ( S
normOp T )  e.  _V
82, 7eqeltri 2541 . . . . 5  |-  N  e. 
_V
98cnvex 6746 . . . 4  |-  `' N  e.  _V
10 imaexg 6736 . . . 4  |-  ( `' N  e.  _V  ->  ( `' N " RR )  e.  _V )
119, 10ax-mp 5 . . 3  |-  ( `' N " RR )  e.  _V
125, 6, 11ovmpt2a 6432 . 2  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( S NGHom  T )  =  ( `' N " RR ) )
136mpt2ndm0 6515 . . 3  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( S NGHom  T )  =  (/) )
14 nmoffn 21344 . . . . . . . . . 10  |-  normOp  Fn  (NrmGrp  X. NrmGrp
)
15 fndm 5686 . . . . . . . . . 10  |-  ( normOp  Fn  (NrmGrp  X. NrmGrp )  ->  dom  normOp  =  (NrmGrp  X. NrmGrp )
)
1614, 15ax-mp 5 . . . . . . . . 9  |-  dom  normOp  =  (NrmGrp  X. NrmGrp )
1716ndmov 6458 . . . . . . . 8  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( S normOp T )  =  (/) )
182, 17syl5eq 2510 . . . . . . 7  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  N  =  (/) )
1918cnveqd 5188 . . . . . 6  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  `' N  =  `' (/) )
20 cnv0 5416 . . . . . 6  |-  `' (/)  =  (/)
2119, 20syl6eq 2514 . . . . 5  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  `' N  =  (/) )
2221imaeq1d 5346 . . . 4  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( `' N " RR )  =  ( (/) " RR ) )
23 0ima 5363 . . . 4  |-  ( (/) " RR )  =  (/)
2422, 23syl6eq 2514 . . 3  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( `' N " RR )  =  (/) )
2513, 24eqtr4d 2501 . 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 1395    e. wcel 1819   _Vcvv 3109   (/)c0 3793    X. cxp 5006   `'ccnv 5007   dom cdm 5008   "cima 5011    Fn wfn 5589  (class class class)co 6296   RRcr 9508  NrmGrpcngp 21224   normOpcnmo 21338   NGHom cnghm 21339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-ico 11560  df-nmo 21341  df-nghm 21342
This theorem is referenced by:  isnghm  21356
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