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Theorem nmoeq0 20327
Description: The operator norm is zero only for the zero operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
Hypotheses
Ref Expression
nmo0.1  |-  N  =  ( S normOp T )
nmo0.2  |-  V  =  ( Base `  S
)
nmo0.3  |-  .0.  =  ( 0g `  T )
Assertion
Ref Expression
nmoeq0  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( ( N `
 F )  =  0  <->  F  =  ( V  X.  {  .0.  }
) ) )

Proof of Theorem nmoeq0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . . . . . . . 11  |-  ( ( N `  F )  =  0  ->  ( N `  F )  =  0 )
2 0re 9398 . . . . . . . . . . 11  |-  0  e.  RR
31, 2syl6eqel 2531 . . . . . . . . . 10  |-  ( ( N `  F )  =  0  ->  ( N `  F )  e.  RR )
4 nmo0.1 . . . . . . . . . . . 12  |-  N  =  ( S normOp T )
54isnghm2 20315 . . . . . . . . . . 11  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( F  e.  ( S NGHom  T )  <-> 
( N `  F
)  e.  RR ) )
65biimpar 485 . . . . . . . . . 10  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  e.  RR )  ->  F  e.  ( S NGHom  T ) )
73, 6sylan2 474 . . . . . . . . 9  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  F  e.  ( S NGHom  T ) )
8 nmo0.2 . . . . . . . . . 10  |-  V  =  ( Base `  S
)
9 eqid 2443 . . . . . . . . . 10  |-  ( norm `  S )  =  (
norm `  S )
10 eqid 2443 . . . . . . . . . 10  |-  ( norm `  T )  =  (
norm `  T )
114, 8, 9, 10nmoi 20319 . . . . . . . . 9  |-  ( ( F  e.  ( S NGHom 
T )  /\  x  e.  V )  ->  (
( norm `  T ) `  ( F `  x
) )  <_  (
( N `  F
)  x.  ( (
norm `  S ) `  x ) ) )
127, 11sylan 471 . . . . . . . 8  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( norm `  T ) `  ( F `  x
) )  <_  (
( N `  F
)  x.  ( (
norm `  S ) `  x ) ) )
13 simplr 754 . . . . . . . . . 10  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  ( N `  F )  =  0 )
1413oveq1d 6118 . . . . . . . . 9  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( N `  F
)  x.  ( (
norm `  S ) `  x ) )  =  ( 0  x.  (
( norm `  S ) `  x ) ) )
15 simpl1 991 . . . . . . . . . . . 12  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  S  e. NrmGrp )
168, 9nmcl 20219 . . . . . . . . . . . 12  |-  ( ( S  e. NrmGrp  /\  x  e.  V )  ->  (
( norm `  S ) `  x )  e.  RR )
1715, 16sylan 471 . . . . . . . . . . 11  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( norm `  S ) `  x )  e.  RR )
1817recnd 9424 . . . . . . . . . 10  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( norm `  S ) `  x )  e.  CC )
1918mul02d 9579 . . . . . . . . 9  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
0  x.  ( (
norm `  S ) `  x ) )  =  0 )
2014, 19eqtrd 2475 . . . . . . . 8  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( N `  F
)  x.  ( (
norm `  S ) `  x ) )  =  0 )
2112, 20breqtrd 4328 . . . . . . 7  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( norm `  T ) `  ( F `  x
) )  <_  0
)
22 simpll2 1028 . . . . . . . 8  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  T  e. NrmGrp )
23 simpl3 993 . . . . . . . . . 10  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  F  e.  ( S  GrpHom  T ) )
24 eqid 2443 . . . . . . . . . . 11  |-  ( Base `  T )  =  (
Base `  T )
258, 24ghmf 15763 . . . . . . . . . 10  |-  ( F  e.  ( S  GrpHom  T )  ->  F : V
--> ( Base `  T
) )
2623, 25syl 16 . . . . . . . . 9  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  F : V
--> ( Base `  T
) )
2726ffvelrnda 5855 . . . . . . . 8  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  ( F `  x )  e.  ( Base `  T
) )
2824, 10nmge0 20220 . . . . . . . 8  |-  ( ( T  e. NrmGrp  /\  ( F `  x )  e.  ( Base `  T
) )  ->  0  <_  ( ( norm `  T
) `  ( F `  x ) ) )
2922, 27, 28syl2anc 661 . . . . . . 7  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  0  <_  ( ( norm `  T
) `  ( F `  x ) ) )
3024, 10nmcl 20219 . . . . . . . . 9  |-  ( ( T  e. NrmGrp  /\  ( F `  x )  e.  ( Base `  T
) )  ->  (
( norm `  T ) `  ( F `  x
) )  e.  RR )
3122, 27, 30syl2anc 661 . . . . . . . 8  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( norm `  T ) `  ( F `  x
) )  e.  RR )
32 letri3 9472 . . . . . . . 8  |-  ( ( ( ( norm `  T
) `  ( F `  x ) )  e.  RR  /\  0  e.  RR )  ->  (
( ( norm `  T
) `  ( F `  x ) )  =  0  <->  ( ( (
norm `  T ) `  ( F `  x
) )  <_  0  /\  0  <_  ( (
norm `  T ) `  ( F `  x
) ) ) ) )
3331, 2, 32sylancl 662 . . . . . . 7  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( ( norm `  T
) `  ( F `  x ) )  =  0  <->  ( ( (
norm `  T ) `  ( F `  x
) )  <_  0  /\  0  <_  ( (
norm `  T ) `  ( F `  x
) ) ) ) )
3421, 29, 33mpbir2and 913 . . . . . 6  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( norm `  T ) `  ( F `  x
) )  =  0 )
35 nmo0.3 . . . . . . . 8  |-  .0.  =  ( 0g `  T )
3624, 10, 35nmeq0 20221 . . . . . . 7  |-  ( ( T  e. NrmGrp  /\  ( F `  x )  e.  ( Base `  T
) )  ->  (
( ( norm `  T
) `  ( F `  x ) )  =  0  <->  ( F `  x )  =  .0.  ) )
3722, 27, 36syl2anc 661 . . . . . 6  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( ( norm `  T
) `  ( F `  x ) )  =  0  <->  ( F `  x )  =  .0.  ) )
3834, 37mpbid 210 . . . . 5  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  ( F `  x )  =  .0.  )
3938mpteq2dva 4390 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  ( x  e.  V  |->  ( F `
 x ) )  =  ( x  e.  V  |->  .0.  ) )
4026feqmptd 5756 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  F  =  ( x  e.  V  |->  ( F `  x
) ) )
41 fconstmpt 4894 . . . . 5  |-  ( V  X.  {  .0.  }
)  =  ( x  e.  V  |->  .0.  )
4241a1i 11 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  ( V  X.  {  .0.  } )  =  ( x  e.  V  |->  .0.  ) )
4339, 40, 423eqtr4d 2485 . . 3  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  F  =  ( V  X.  {  .0.  } ) )
4443ex 434 . 2  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( ( N `
 F )  =  0  ->  F  =  ( V  X.  {  .0.  } ) ) )
454, 8, 35nmo0 20326 . . . 4  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( N `  ( V  X.  {  .0.  } ) )  =  0 )
46453adant3 1008 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( N `  ( V  X.  {  .0.  } ) )  =  0 )
47 fveq2 5703 . . . 4  |-  ( F  =  ( V  X.  {  .0.  } )  -> 
( N `  F
)  =  ( N `
 ( V  X.  {  .0.  } ) ) )
4847eqeq1d 2451 . . 3  |-  ( F  =  ( V  X.  {  .0.  } )  -> 
( ( N `  F )  =  0  <-> 
( N `  ( V  X.  {  .0.  }
) )  =  0 ) )
4946, 48syl5ibrcom 222 . 2  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( F  =  ( V  X.  {  .0.  } )  ->  ( N `  F )  =  0 ) )
5044, 49impbid 191 1  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( ( N `
 F )  =  0  <->  F  =  ( V  X.  {  .0.  }
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   {csn 3889   class class class wbr 4304    e. cmpt 4362    X. cxp 4850   -->wf 5426   ` cfv 5430  (class class class)co 6103   RRcr 9293   0cc0 9294    x. cmul 9299    <_ cle 9431   Basecbs 14186   0gc0g 14390    GrpHom cghm 15756   normcnm 20181  NrmGrpcngp 20182   normOpcnmo 20296   NGHom cnghm 20297
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-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-er 7113  df-map 7228  df-en 7323  df-dom 7324  df-sdom 7325  df-sup 7703  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-n0 10592  df-z 10659  df-uz 10874  df-q 10966  df-rp 11004  df-xneg 11101  df-xadd 11102  df-xmul 11103  df-ico 11318  df-0g 14392  df-topgen 14394  df-mnd 15427  df-mhm 15476  df-grp 15557  df-ghm 15757  df-psmet 17821  df-xmet 17822  df-met 17823  df-bl 17824  df-mopn 17825  df-top 18515  df-bases 18517  df-topon 18518  df-topsp 18519  df-xms 19907  df-ms 19908  df-nm 20187  df-ngp 20188  df-nmo 20299  df-nghm 20300
This theorem is referenced by: (None)
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