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Theorem nv1 24201
Description: From any nonzero vector, construct a vector whose norm is one. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nv1.1  |-  X  =  ( BaseSet `  U )
nv1.4  |-  S  =  ( .sOLD `  U )
nv1.5  |-  Z  =  ( 0vec `  U
)
nv1.6  |-  N  =  ( normCV `  U )
Assertion
Ref Expression
nv1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  =/= 
Z )  ->  ( N `  ( (
1  /  ( N `
 A ) ) S A ) )  =  1 )

Proof of Theorem nv1
StepHypRef Expression
1 simp1 988 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  =/= 
Z )  ->  U  e.  NrmCVec )
2 nv1.1 . . . . . 6  |-  X  =  ( BaseSet `  U )
3 nv1.6 . . . . . 6  |-  N  =  ( normCV `  U )
42, 3nvcl 24184 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  e.  RR )
543adant3 1008 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  =/= 
Z )  ->  ( N `  A )  e.  RR )
6 nv1.5 . . . . . . 7  |-  Z  =  ( 0vec `  U
)
72, 6, 3nvz 24194 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  A
)  =  0  <->  A  =  Z ) )
87necon3bid 2706 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  A
)  =/=  0  <->  A  =/=  Z ) )
98biimp3ar 1320 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  =/= 
Z )  ->  ( N `  A )  =/=  0 )
105, 9rereccld 10261 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  =/= 
Z )  ->  (
1  /  ( N `
 A ) )  e.  RR )
112, 6, 3nvgt0 24200 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A  =/=  Z  <->  0  <  ( N `  A ) ) )
1211biimp3a 1319 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  =/= 
Z )  ->  0  <  ( N `  A
) )
13 1re 9488 . . . . 5  |-  1  e.  RR
14 0le1 9966 . . . . 5  |-  0  <_  1
15 divge0 10301 . . . . 5  |-  ( ( ( 1  e.  RR  /\  0  <_  1 )  /\  ( ( N `
 A )  e.  RR  /\  0  < 
( N `  A
) ) )  -> 
0  <_  ( 1  /  ( N `  A ) ) )
1613, 14, 15mpanl12 682 . . . 4  |-  ( ( ( N `  A
)  e.  RR  /\  0  <  ( N `  A ) )  -> 
0  <_  ( 1  /  ( N `  A ) ) )
175, 12, 16syl2anc 661 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  =/= 
Z )  ->  0  <_  ( 1  /  ( N `  A )
) )
18 simp2 989 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  =/= 
Z )  ->  A  e.  X )
19 nv1.4 . . . 4  |-  S  =  ( .sOLD `  U )
202, 19, 3nvsge0 24188 . . 3  |-  ( ( U  e.  NrmCVec  /\  (
( 1  /  ( N `  A )
)  e.  RR  /\  0  <_  ( 1  / 
( N `  A
) ) )  /\  A  e.  X )  ->  ( N `  (
( 1  /  ( N `  A )
) S A ) )  =  ( ( 1  /  ( N `
 A ) )  x.  ( N `  A ) ) )
211, 10, 17, 18, 20syl121anc 1224 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  =/= 
Z )  ->  ( N `  ( (
1  /  ( N `
 A ) ) S A ) )  =  ( ( 1  /  ( N `  A ) )  x.  ( N `  A
) ) )
224recnd 9515 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  e.  CC )
23223adant3 1008 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  =/= 
Z )  ->  ( N `  A )  e.  CC )
2423, 9recid2d 10206 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  =/= 
Z )  ->  (
( 1  /  ( N `  A )
)  x.  ( N `
 A ) )  =  1 )
2521, 24eqtrd 2492 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  =/= 
Z )  ->  ( N `  ( (
1  /  ( N `
 A ) ) S A ) )  =  1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   CCcc 9383   RRcr 9384   0cc0 9385   1c1 9386    x. cmul 9390    < clt 9521    <_ cle 9522    / cdiv 10096   NrmCVeccnv 24099   BaseSetcba 24101   .sOLDcns 24102   0veccn0v 24103   normCVcnmcv 24105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-pre-sup 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-sup 7794  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-div 10097  df-nn 10426  df-2 10483  df-3 10484  df-n0 10683  df-z 10750  df-uz 10965  df-rp 11095  df-seq 11910  df-exp 11969  df-cj 12692  df-re 12693  df-im 12694  df-sqr 12828  df-abs 12829  df-grpo 23815  df-gid 23816  df-ginv 23817  df-ablo 23906  df-vc 24061  df-nv 24107  df-va 24110  df-ba 24111  df-sm 24112  df-0v 24113  df-nmcv 24115
This theorem is referenced by:  nmlno0lem  24330  nmblolbii  24336
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