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Theorem nvz 21235
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 2283 . . . . . 6  |-  ( +v
`  U )  =  ( +v `  U
)
3 eqid 2283 . . . . . 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 21170 . . . . 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 969 . . . 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 955 . . . . 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 2618 . . . 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 5525 . . . . . . 7  |-  ( x  =  A  ->  ( N `  x )  =  ( N `  A ) )
1110eqeq1d 2291 . . . . . 6  |-  ( x  =  A  ->  (
( N `  x
)  =  0  <->  ( N `  A )  =  0 ) )
12 eqeq1 2289 . . . . . 6  |-  ( x  =  A  ->  (
x  =  Z  <->  A  =  Z ) )
1311, 12imbi12d 311 . . . . 5  |-  ( x  =  A  ->  (
( ( N `  x )  =  0  ->  x  =  Z )  <->  ( ( N `
 A )  =  0  ->  A  =  Z ) ) )
1413rspccv 2881 . . . 4  |-  ( A. x  e.  X  (
( N `  x
)  =  0  ->  x  =  Z )  ->  ( A  e.  X  ->  ( ( N `  A )  =  0  ->  A  =  Z ) ) )
157, 9, 143syl 18 . . 3  |-  ( U  e.  NrmCVec  ->  ( A  e.  X  ->  ( ( N `  A )  =  0  ->  A  =  Z ) ) )
1615imp 418 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  A
)  =  0  ->  A  =  Z )
)
17 fveq2 5525 . . . . 5  |-  ( A  =  Z  ->  ( N `  A )  =  ( N `  Z ) )
184, 5nvz0 21234 . . . . 5  |-  ( U  e.  NrmCVec  ->  ( N `  Z )  =  0 )
1917, 18sylan9eqr 2337 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  =  Z )  ->  ( N `  A )  =  0 )
2019ex 423 . . 3  |-  ( U  e.  NrmCVec  ->  ( A  =  Z  ->  ( N `  A )  =  0 ) )
2120adantr 451 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A  =  Z  ->  ( N `  A )  =  0 ) )
2216, 21impbid 183 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  A
)  =  0  <->  A  =  Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   <.cop 3643   class class class wbr 4023   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    x. cmul 8742    <_ cle 8868   abscabs 11719   CVec
OLDcvc 21101   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   .s
OLDcns 21143   0veccn0v 21144   normCVcnmcv 21146
This theorem is referenced by:  nvgt0  21241  nv1  21242  imsmetlem  21259  ipz  21295  nmlno0lem  21371  nmblolbii  21377  blocnilem  21382  siii  21431  hlipgt0  21493
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-grpo 20858  df-gid 20859  df-ginv 20860  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-nmcv 21156
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