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Theorem norm1 25831
Description: From any nonzero Hilbert space vector, construct a vector whose norm is 1. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
norm1  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  =  1 )

Proof of Theorem norm1
StepHypRef Expression
1 normcl 25706 . . . . . 6  |-  ( A  e.  ~H  ->  ( normh `  A )  e.  RR )
21adantr 465 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  A )  e.  RR )
3 normne0 25711 . . . . . 6  |-  ( A  e.  ~H  ->  (
( normh `  A )  =/=  0  <->  A  =/=  0h )
)
43biimpar 485 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  A )  =/=  0 )
52, 4rereccld 10362 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( 1  /  ( normh `  A ) )  e.  RR )
65recnd 9613 . . 3  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( 1  /  ( normh `  A ) )  e.  CC )
7 simpl 457 . . 3  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  ->  A  e.  ~H )
8 norm-iii 25721 . . 3  |-  ( ( ( 1  /  ( normh `  A ) )  e.  CC  /\  A  e.  ~H )  ->  ( normh `  ( ( 1  /  ( normh `  A
) )  .h  A
) )  =  ( ( abs `  (
1  /  ( normh `  A ) ) )  x.  ( normh `  A
) ) )
96, 7, 8syl2anc 661 . 2  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  =  ( ( abs `  (
1  /  ( normh `  A ) ) )  x.  ( normh `  A
) ) )
10 normgt0 25708 . . . . . 6  |-  ( A  e.  ~H  ->  ( A  =/=  0h  <->  0  <  (
normh `  A ) ) )
1110biimpa 484 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( normh `  A ) )
12 1re 9586 . . . . . 6  |-  1  e.  RR
13 0le1 10067 . . . . . 6  |-  0  <_  1
14 divge0 10402 . . . . . 6  |-  ( ( ( 1  e.  RR  /\  0  <_  1 )  /\  ( ( normh `  A )  e.  RR  /\  0  <  ( normh `  A ) ) )  ->  0  <_  (
1  /  ( normh `  A ) ) )
1512, 13, 14mpanl12 682 . . . . 5  |-  ( ( ( normh `  A )  e.  RR  /\  0  < 
( normh `  A )
)  ->  0  <_  ( 1  /  ( normh `  A ) ) )
162, 11, 15syl2anc 661 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <_  ( 1  /  ( normh `  A
) ) )
175, 16absidd 13205 . . 3  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( abs `  (
1  /  ( normh `  A ) ) )  =  ( 1  / 
( normh `  A )
) )
1817oveq1d 6292 . 2  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( abs `  (
1  /  ( normh `  A ) ) )  x.  ( normh `  A
) )  =  ( ( 1  /  ( normh `  A ) )  x.  ( normh `  A
) ) )
191recnd 9613 . . . 4  |-  ( A  e.  ~H  ->  ( normh `  A )  e.  CC )
2019adantr 465 . . 3  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  A )  e.  CC )
2120, 4recid2d 10307 . 2  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( 1  / 
( normh `  A )
)  x.  ( normh `  A ) )  =  1 )
229, 18, 213eqtrd 2507 1  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  =  1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2657   class class class wbr 4442   ` cfv 5581  (class class class)co 6277   CCcc 9481   RRcr 9482   0cc0 9483   1c1 9484    x. cmul 9488    < clt 9619    <_ cle 9620    / cdiv 10197   abscabs 13019   ~Hchil 25500    .h csm 25502   normhcno 25504   0hc0v 25505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561  ax-hv0cl 25584  ax-hfvmul 25586  ax-hvmul0 25591  ax-hfi 25660  ax-his1 25663  ax-his3 25665  ax-his4 25666
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-2nd 6777  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-sup 7892  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-n0 10787  df-z 10856  df-uz 11074  df-rp 11212  df-seq 12066  df-exp 12125  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-hnorm 25549
This theorem is referenced by:  norm1exi  25832  nmlnop0iALT  26578  nmbdoplbi  26607  nmcoplbi  26611  nmbdfnlbi  26632  nmcfnlbi  26635  branmfn  26688
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