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

Theorem nvz 22146
Description: The norm of a vector is zero iff the vector is zero. First part of Problem 2 of [Kreyszig] p. 64. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvz.1  |-  X  =  ( BaseSet `  U )
nvz.5  |-  Z  =  ( 0vec `  U
)
nvz.6  |-  N  =  ( normCV `  U )
Assertion
Ref Expression
nvz  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  A
)  =  0  <->  A  =  Z ) )

Proof of Theorem nvz
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nvz.1 . . . . . 6  |-  X  =  ( BaseSet `  U )
2 eqid 2435 . . . . . 6  |-  ( +v
`  U )  =  ( +v `  U
)
3 eqid 2435 . . . . . 6  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
4 nvz.5 . . . . . 6  |-  Z  =  ( 0vec `  U
)
5 nvz.6 . . . . . 6  |-  N  =  ( normCV `  U )
61, 2, 3, 4, 5nvi 22081 . . . . 5  |-  ( U  e.  NrmCVec  ->  ( <. ( +v `  U ) ,  ( .s OLD `  U
) >.  e.  CVec OLD  /\  N : X --> RR  /\  A. x  e.  X  ( ( ( N `  x )  =  0  ->  x  =  Z )  /\  A. y  e.  CC  ( N `  ( y ( .s
OLD `  U )
x ) )  =  ( ( abs `  y
)  x.  ( N `
 x ) )  /\  A. y  e.  X  ( N `  ( x ( +v
`  U ) y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) ) ) )
76simp3d 971 . . . 4  |-  ( U  e.  NrmCVec  ->  A. x  e.  X  ( ( ( N `
 x )  =  0  ->  x  =  Z )  /\  A. y  e.  CC  ( N `  ( y
( .s OLD `  U
) x ) )  =  ( ( abs `  y )  x.  ( N `  x )
)  /\  A. y  e.  X  ( N `  ( x ( +v
`  U ) y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) ) )
8 simp1 957 . . . . 5  |-  ( ( ( ( N `  x )  =  0  ->  x  =  Z )  /\  A. y  e.  CC  ( N `  ( y ( .s
OLD `  U )
x ) )  =  ( ( abs `  y
)  x.  ( N `
 x ) )  /\  A. y  e.  X  ( N `  ( x ( +v
`  U ) y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) )  ->  ( ( N `  x )  =  0  ->  x  =  Z ) )
98ralimi 2773 . . . 4  |-  ( A. x  e.  X  (
( ( N `  x )  =  0  ->  x  =  Z )  /\  A. y  e.  CC  ( N `  ( y ( .s
OLD `  U )
x ) )  =  ( ( abs `  y
)  x.  ( N `
 x ) )  /\  A. y  e.  X  ( N `  ( x ( +v
`  U ) y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) )  ->  A. x  e.  X  ( ( N `  x )  =  0  ->  x  =  Z ) )
10 fveq2 5719 . . . . . . 7  |-  ( x  =  A  ->  ( N `  x )  =  ( N `  A ) )
1110eqeq1d 2443 . . . . . 6  |-  ( x  =  A  ->  (
( N `  x
)  =  0  <->  ( N `  A )  =  0 ) )
12 eqeq1 2441 . . . . . 6  |-  ( x  =  A  ->  (
x  =  Z  <->  A  =  Z ) )
1311, 12imbi12d 312 . . . . 5  |-  ( x  =  A  ->  (
( ( N `  x )  =  0  ->  x  =  Z )  <->  ( ( N `
 A )  =  0  ->  A  =  Z ) ) )
1413rspccv 3041 . . . 4  |-  ( A. x  e.  X  (
( N `  x
)  =  0  ->  x  =  Z )  ->  ( A  e.  X  ->  ( ( N `  A )  =  0  ->  A  =  Z ) ) )
157, 9, 143syl 19 . . 3  |-  ( U  e.  NrmCVec  ->  ( A  e.  X  ->  ( ( N `  A )  =  0  ->  A  =  Z ) ) )
1615imp 419 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  A
)  =  0  ->  A  =  Z )
)
17 fveq2 5719 . . . . 5  |-  ( A  =  Z  ->  ( N `  A )  =  ( N `  Z ) )
184, 5nvz0 22145 . . . . 5  |-  ( U  e.  NrmCVec  ->  ( N `  Z )  =  0 )
1917, 18sylan9eqr 2489 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  =  Z )  ->  ( N `  A )  =  0 )
2019ex 424 . . 3  |-  ( U  e.  NrmCVec  ->  ( A  =  Z  ->  ( N `  A )  =  0 ) )
2120adantr 452 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A  =  Z  ->  ( N `  A )  =  0 ) )
2216, 21impbid 184 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  A
)  =  0  <->  A  =  Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   <.cop 3809   class class class wbr 4204   -->wf 5441   ` cfv 5445  (class class class)co 6072   CCcc 8977   RRcr 8978   0cc0 8979    + caddc 8982    x. cmul 8984    <_ cle 9110   abscabs 12027   CVec
OLDcvc 22012   NrmCVeccnv 22051   +vcpv 22052   BaseSetcba 22053   .s
OLDcns 22054   0veccn0v 22055   normCVcnmcv 22057
This theorem is referenced by:  nvgt0  22152  nv1  22153  imsmetlem  22170  ipz  22206  nmlno0lem  22282  nmblolbii  22288  blocnilem  22293  siii  22342  hlipgt0  22404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-sup 7437  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-n0 10211  df-z 10272  df-uz 10478  df-rp 10602  df-seq 11312  df-exp 11371  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-grpo 21767  df-gid 21768  df-ginv 21769  df-ablo 21858  df-vc 22013  df-nv 22059  df-va 22062  df-ba 22063  df-sm 22064  df-0v 22065  df-nmcv 22067
  Copyright terms: Public domain W3C validator