MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nghmfval Structured version   Visualization version   Unicode version

Theorem nghmfval 21739
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 6304 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s normOp t )  =  ( S normOp T ) )
2 nmofval.1 . . . . . 6  |-  N  =  ( S normOp T )
31, 2syl6eqr 2505 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s normOp t )  =  N )
43cnveqd 5013 . . . 4  |-  ( ( s  =  S  /\  t  =  T )  ->  `' ( s normOp t )  =  `' N
)
54imaeq1d 5170 . . 3  |-  ( ( s  =  S  /\  t  =  T )  ->  ( `' ( s
normOp t ) " RR )  =  ( `' N " RR ) )
6 df-nghm 21725 . . 3  |- NGHom  =  ( s  e. NrmGrp ,  t  e. NrmGrp  |->  ( `' ( s
normOp t ) " RR ) )
7 ovex 6323 . . . . . 6  |-  ( S
normOp T )  e.  _V
82, 7eqeltri 2527 . . . . 5  |-  N  e. 
_V
98cnvex 6745 . . . 4  |-  `' N  e.  _V
10 imaexg 6735 . . . 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 21728 . . . . . . . . . 10  |-  normOp  Fn  (NrmGrp  X. NrmGrp
)
15 fndm 5680 . . . . . . . . . 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 2499 . . . . . . 7  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  N  =  (/) )
1918cnveqd 5013 . . . . . 6  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  `' N  =  `' (/) )
20 cnv0 5242 . . . . . 6  |-  `' (/)  =  (/)
2119, 20syl6eq 2503 . . . . 5  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  `' N  =  (/) )
2221imaeq1d 5170 . . . 4  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( `' N " RR )  =  ( (/) " RR ) )
23 0ima 5187 . . . 4  |-  ( (/) " RR )  =  (/)
2422, 23syl6eq 2503 . . 3  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( `' N " RR )  =  (/) )
2513, 24eqtr4d 2490 . 2  |-  ( -.  ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  ( S NGHom  T )  =  ( `' N " RR ) )
2612, 25pm2.61i 168 1  |-  ( S NGHom 
T )  =  ( `' N " RR )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 371    = wceq 1446    e. wcel 1889   _Vcvv 3047   (/)c0 3733    X. cxp 4835   `'ccnv 4836   dom cdm 4837   "cima 4840    Fn wfn 5580  (class class class)co 6295   RRcr 9543  NrmGrpcngp 21604   normOpcnmo 21718   NGHom cnghm 21720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-po 4758  df-so 4759  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6798  df-2nd 6799  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-sup 7961  df-inf 7962  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-ico 11648  df-nmo 21723  df-nghm 21725
This theorem is referenced by:  isnghm  21740
  Copyright terms: Public domain W3C validator