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Theorem nmoeq0 18261
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 19 . . . . . . . . . . 11  |-  ( ( N `  F )  =  0  ->  ( N `  F )  =  0 )
2 0re 8854 . . . . . . . . . . 11  |-  0  e.  RR
31, 2syl6eqel 2384 . . . . . . . . . 10  |-  ( ( N `  F )  =  0  ->  ( N `  F )  e.  RR )
4 nmo0.1 . . . . . . . . . . . 12  |-  N  =  ( S normOp T )
54isnghm2 18249 . . . . . . . . . . 11  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( F  e.  ( S NGHom  T )  <-> 
( N `  F
)  e.  RR ) )
65biimpar 471 . . . . . . . . . 10  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  e.  RR )  ->  F  e.  ( S NGHom  T ) )
73, 6sylan2 460 . . . . . . . . 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 2296 . . . . . . . . . 10  |-  ( norm `  S )  =  (
norm `  S )
10 eqid 2296 . . . . . . . . . 10  |-  ( norm `  T )  =  (
norm `  T )
114, 8, 9, 10nmoi 18253 . . . . . . . . 9  |-  ( ( F  e.  ( S NGHom 
T )  /\  x  e.  V )  ->  (
( norm `  T ) `  ( F `  x
) )  <_  (
( N `  F
)  x.  ( (
norm `  S ) `  x ) ) )
127, 11sylan 457 . . . . . . . 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 731 . . . . . . . . . 10  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  ( N `  F )  =  0 )
1413oveq1d 5889 . . . . . . . . 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 958 . . . . . . . . . . . 12  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  S  e. NrmGrp )
168, 9nmcl 18153 . . . . . . . . . . . 12  |-  ( ( S  e. NrmGrp  /\  x  e.  V )  ->  (
( norm `  S ) `  x )  e.  RR )
1715, 16sylan 457 . . . . . . . . . . 11  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( norm `  S ) `  x )  e.  RR )
1817recnd 8877 . . . . . . . . . 10  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( norm `  S ) `  x )  e.  CC )
1918mul02d 9026 . . . . . . . . 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 2328 . . . . . . . 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 4063 . . . . . . 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 995 . . . . . . . 8  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  T  e. NrmGrp )
23 simpl3 960 . . . . . . . . . 10  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  F  e.  ( S  GrpHom  T ) )
24 eqid 2296 . . . . . . . . . . 11  |-  ( Base `  T )  =  (
Base `  T )
258, 24ghmf 14703 . . . . . . . . . 10  |-  ( F  e.  ( S  GrpHom  T )  ->  F : V
--> ( Base `  T
) )
2623, 25syl 15 . . . . . . . . 9  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  F : V
--> ( Base `  T
) )
27 ffvelrn 5679 . . . . . . . . 9  |-  ( ( F : V --> ( Base `  T )  /\  x  e.  V )  ->  ( F `  x )  e.  ( Base `  T
) )
2826, 27sylan 457 . . . . . . . 8  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  ( F `  x )  e.  ( Base `  T
) )
2924, 10nmge0 18154 . . . . . . . 8  |-  ( ( T  e. NrmGrp  /\  ( F `  x )  e.  ( Base `  T
) )  ->  0  <_  ( ( norm `  T
) `  ( F `  x ) ) )
3022, 28, 29syl2anc 642 . . . . . . 7  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  0  <_  ( ( norm `  T
) `  ( F `  x ) ) )
3124, 10nmcl 18153 . . . . . . . . 9  |-  ( ( T  e. NrmGrp  /\  ( F `  x )  e.  ( Base `  T
) )  ->  (
( norm `  T ) `  ( F `  x
) )  e.  RR )
3222, 28, 31syl2anc 642 . . . . . . . 8  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( norm `  T ) `  ( F `  x
) )  e.  RR )
33 letri3 8923 . . . . . . . 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
) ) ) ) )
3432, 2, 33sylancl 643 . . . . . . 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
) ) ) ) )
3521, 30, 34mpbir2and 888 . . . . . 6  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  (
( norm `  T ) `  ( F `  x
) )  =  0 )
36 nmo0.3 . . . . . . . 8  |-  .0.  =  ( 0g `  T )
3724, 10, 36nmeq0 18155 . . . . . . 7  |-  ( ( T  e. NrmGrp  /\  ( F `  x )  e.  ( Base `  T
) )  ->  (
( ( norm `  T
) `  ( F `  x ) )  =  0  <->  ( F `  x )  =  .0.  ) )
3822, 28, 37syl2anc 642 . . . . . 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.  ) )
3935, 38mpbid 201 . . . . 5  |-  ( ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `
 F )  =  0 )  /\  x  e.  V )  ->  ( F `  x )  =  .0.  )
4039mpteq2dva 4122 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  ( x  e.  V  |->  ( F `
 x ) )  =  ( x  e.  V  |->  .0.  ) )
4126feqmptd 5591 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  F  =  ( x  e.  V  |->  ( F `  x
) ) )
42 fconstmpt 4748 . . . . 5  |-  ( V  X.  {  .0.  }
)  =  ( x  e.  V  |->  .0.  )
4342a1i 10 . . . 4  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  ( V  X.  {  .0.  } )  =  ( x  e.  V  |->  .0.  ) )
4440, 41, 433eqtr4d 2338 . . 3  |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( N `  F )  =  0 )  ->  F  =  ( V  X.  {  .0.  } ) )
4544ex 423 . 2  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( ( N `
 F )  =  0  ->  F  =  ( V  X.  {  .0.  } ) ) )
464, 8, 36nmo0 18260 . . . 4  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp
)  ->  ( N `  ( V  X.  {  .0.  } ) )  =  0 )
47463adant3 975 . . 3  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( N `  ( V  X.  {  .0.  } ) )  =  0 )
48 fveq2 5541 . . . 4  |-  ( F  =  ( V  X.  {  .0.  } )  -> 
( N `  F
)  =  ( N `
 ( V  X.  {  .0.  } ) ) )
4948eqeq1d 2304 . . 3  |-  ( F  =  ( V  X.  {  .0.  } )  -> 
( ( N `  F )  =  0  <-> 
( N `  ( V  X.  {  .0.  }
) )  =  0 ) )
5047, 49syl5ibrcom 213 . 2  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( F  =  ( V  X.  {  .0.  } )  ->  ( N `  F )  =  0 ) )
5145, 50impbid 183 1  |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( ( N `
 F )  =  0  <->  F  =  ( V  X.  {  .0.  }
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {csn 3653   class class class wbr 4039    e. cmpt 4093    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753    x. cmul 8758    <_ cle 8884   Basecbs 13164   0gc0g 13416    GrpHom cghm 14696   normcnm 18115  NrmGrpcngp 18116   normOpcnmo 18230   NGHom cnghm 18231
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ico 10678  df-topgen 13360  df-0g 13420  df-mnd 14383  df-mhm 14431  df-grp 14505  df-ghm 14697  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-xms 17901  df-ms 17902  df-nm 18121  df-ngp 18122  df-nmo 18233  df-nghm 18234
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