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Theorem nv1 22015
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  =  ( .s OLD `  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 957 . . 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 21998 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  e.  RR )
543adant3 977 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  =/= 
Z )  ->  ( N `  A )  e.  RR )
6 nv1.5 . . . . . . 7  |-  Z  =  ( 0vec `  U
)
72, 6, 3nvz 22008 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  A
)  =  0  <->  A  =  Z ) )
87necon3bid 2587 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
( N `  A
)  =/=  0  <->  A  =/=  Z ) )
98biimp3ar 1284 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  =/= 
Z )  ->  ( N `  A )  =/=  0 )
105, 9rereccld 9775 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  =/= 
Z )  ->  (
1  /  ( N `
 A ) )  e.  RR )
112, 6, 3nvgt0 22014 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A  =/=  Z  <->  0  <  ( N `  A ) ) )
1211biimp3a 1283 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  =/= 
Z )  ->  0  <  ( N `  A
) )
13 1re 9025 . . . . 5  |-  1  e.  RR
14 0le1 9485 . . . . 5  |-  0  <_  1
15 divge0 9813 . . . . 5  |-  ( ( ( 1  e.  RR  /\  0  <_  1 )  /\  ( ( N `
 A )  e.  RR  /\  0  < 
( N `  A
) ) )  -> 
0  <_  ( 1  /  ( N `  A ) ) )
1613, 14, 15mpanl12 664 . . . 4  |-  ( ( ( N `  A
)  e.  RR  /\  0  <  ( N `  A ) )  -> 
0  <_  ( 1  /  ( N `  A ) ) )
175, 12, 16syl2anc 643 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  =/= 
Z )  ->  0  <_  ( 1  /  ( N `  A )
) )
18 simp2 958 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  =/= 
Z )  ->  A  e.  X )
19 nv1.4 . . . 4  |-  S  =  ( .s OLD `  U
)
202, 19, 3nvsge0 22002 . . 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 1189 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  =/= 
Z )  ->  ( N `  ( (
1  /  ( N `
 A ) ) S A ) )  =  ( ( 1  /  ( N `  A ) )  x.  ( N `  A
) ) )
224recnd 9049 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  e.  CC )
23223adant3 977 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  =/= 
Z )  ->  ( N `  A )  e.  CC )
2423, 9recid2d 9720 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  =/= 
Z )  ->  (
( 1  /  ( N `  A )
)  x.  ( N `
 A ) )  =  1 )
2521, 24eqtrd 2421 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  =/= 
Z )  ->  ( N `  ( (
1  /  ( N `
 A ) ) S A ) )  =  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2552   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   CCcc 8923   RRcr 8924   0cc0 8925   1c1 8926    x. cmul 8930    < clt 9055    <_ cle 9056    / cdiv 9611   NrmCVeccnv 21913   BaseSetcba 21915   .s
OLDcns 21916   0veccn0v 21917   normCVcnmcv 21919
This theorem is referenced by:  nmlno0lem  22144  nmblolbii  22150
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-n0 10156  df-z 10217  df-uz 10423  df-rp 10547  df-seq 11253  df-exp 11312  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-grpo 21629  df-gid 21630  df-ginv 21631  df-ablo 21720  df-vc 21875  df-nv 21921  df-va 21924  df-ba 21925  df-sm 21926  df-0v 21927  df-nmcv 21929
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