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Theorem nvz 24057
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 2443 . . . . . 6  |-  ( +v
`  U )  =  ( +v `  U
)
3 eqid 2443 . . . . . 6  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
4 nvz.5 . . . . . 6  |-  Z  =  ( 0vec `  U
)
5 nvz.6 . . . . . 6  |-  N  =  ( normCV `  U )
61, 2, 3, 4, 5nvi 23992 . . . . 5  |-  ( U  e.  NrmCVec  ->  ( <. ( +v `  U ) ,  ( .sOLD `  U ) >.  e.  CVecOLD 
/\  N : X --> RR  /\  A. x  e.  X  ( ( ( N `  x )  =  0  ->  x  =  Z )  /\  A. y  e.  CC  ( N `  ( y
( .sOLD `  U ) x ) )  =  ( ( abs `  y )  x.  ( N `  x ) )  /\  A. y  e.  X  ( N `  ( x ( +v `  U
) y ) )  <_  ( ( N `
 x )  +  ( N `  y
) ) ) ) )
76simp3d 1002 . . . 4  |-  ( U  e.  NrmCVec  ->  A. x  e.  X  ( ( ( N `
 x )  =  0  ->  x  =  Z )  /\  A. y  e.  CC  ( N `  ( y
( .sOLD `  U ) x ) )  =  ( ( abs `  y )  x.  ( N `  x ) )  /\  A. y  e.  X  ( N `  ( x ( +v `  U
) y ) )  <_  ( ( N `
 x )  +  ( N `  y
) ) ) )
8 simp1 988 . . . . 5  |-  ( ( ( ( N `  x )  =  0  ->  x  =  Z )  /\  A. y  e.  CC  ( N `  ( y ( .sOLD `  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 2791 . . . 4  |-  ( A. x  e.  X  (
( ( N `  x )  =  0  ->  x  =  Z )  /\  A. y  e.  CC  ( N `  ( y ( .sOLD `  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 5691 . . . . . . 7  |-  ( x  =  A  ->  ( N `  x )  =  ( N `  A ) )
1110eqeq1d 2451 . . . . . 6  |-  ( x  =  A  ->  (
( N `  x
)  =  0  <->  ( N `  A )  =  0 ) )
12 eqeq1 2449 . . . . . 6  |-  ( x  =  A  ->  (
x  =  Z  <->  A  =  Z ) )
1311, 12imbi12d 320 . . . . 5  |-  ( x  =  A  ->  (
( ( N `  x )  =  0  ->  x  =  Z )  <->  ( ( N `
 A )  =  0  ->  A  =  Z ) ) )
1413rspccv 3070 . . . 4  |-  ( A. x  e.  X  (
( N `  x
)  =  0  ->  x  =  Z )  ->  ( A  e.  X  ->  ( ( N `  A )  =  0  ->  A  =  Z ) ) )
157, 9, 143syl 20 . . 3  |-  ( U  e.  NrmCVec  ->  ( A  e.  X  ->  ( ( N `  A )  =  0  ->  A  =  Z ) ) )
1615imp 429 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  A
)  =  0  ->  A  =  Z )
)
17 fveq2 5691 . . . . 5  |-  ( A  =  Z  ->  ( N `  A )  =  ( N `  Z ) )
184, 5nvz0 24056 . . . . 5  |-  ( U  e.  NrmCVec  ->  ( N `  Z )  =  0 )
1917, 18sylan9eqr 2497 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  =  Z )  ->  ( N `  A )  =  0 )
2019ex 434 . . 3  |-  ( U  e.  NrmCVec  ->  ( A  =  Z  ->  ( N `  A )  =  0 ) )
2120adantr 465 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A  =  Z  ->  ( N `  A )  =  0 ) )
2216, 21impbid 191 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  A
)  =  0  <->  A  =  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2715   <.cop 3883   class class class wbr 4292   -->wf 5414   ` cfv 5418  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282    + caddc 9285    x. cmul 9287    <_ cle 9419   abscabs 12723   CVecOLDcvc 23923   NrmCVeccnv 23962   +vcpv 23963   BaseSetcba 23964   .sOLDcns 23965   0veccn0v 23966   normCVcnmcv 23968
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-grpo 23678  df-gid 23679  df-ginv 23680  df-ablo 23769  df-vc 23924  df-nv 23970  df-va 23973  df-ba 23974  df-sm 23975  df-0v 23976  df-nmcv 23978
This theorem is referenced by:  nvgt0  24063  nv1  24064  imsmetlem  24081  ipz  24117  nmlno0lem  24193  nmblolbii  24199  blocnilem  24204  siii  24253  hlipgt0  24315
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