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Theorem nvz 22111
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 2404 . . . . . 6  |-  ( +v
`  U )  =  ( +v `  U
)
3 eqid 2404 . . . . . 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 22046 . . . . 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 2741 . . . 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 5687 . . . . . . 7  |-  ( x  =  A  ->  ( N `  x )  =  ( N `  A ) )
1110eqeq1d 2412 . . . . . 6  |-  ( x  =  A  ->  (
( N `  x
)  =  0  <->  ( N `  A )  =  0 ) )
12 eqeq1 2410 . . . . . 6  |-  ( x  =  A  ->  (
x  =  Z  <->  A  =  Z ) )
1311, 12imbi12d 312 . . . . 5  |-  ( x  =  A  ->  (
( ( N `  x )  =  0  ->  x  =  Z )  <->  ( ( N `
 A )  =  0  ->  A  =  Z ) ) )
1413rspccv 3009 . . . 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 5687 . . . . 5  |-  ( A  =  Z  ->  ( N `  A )  =  ( N `  Z ) )
184, 5nvz0 22110 . . . . 5  |-  ( U  e.  NrmCVec  ->  ( N `  Z )  =  0 )
1917, 18sylan9eqr 2458 . . . 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 1649    e. wcel 1721   A.wral 2666   <.cop 3777   class class class wbr 4172   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946    + caddc 8949    x. cmul 8951    <_ cle 9077   abscabs 11994   CVec
OLDcvc 21977   NrmCVeccnv 22016   +vcpv 22017   BaseSetcba 22018   .s
OLDcns 22019   0veccn0v 22020   normCVcnmcv 22022
This theorem is referenced by:  nvgt0  22117  nv1  22118  imsmetlem  22135  ipz  22171  nmlno0lem  22247  nmblolbii  22253  blocnilem  22258  siii  22307  hlipgt0  22369
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-grpo 21732  df-gid 21733  df-ginv 21734  df-ablo 21823  df-vc 21978  df-nv 22024  df-va 22027  df-ba 22028  df-sm 22029  df-0v 22030  df-nmcv 22032
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