![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > norm1 | Structured version Visualization version GIF version |
Description: From any nonzero Hilbert space vector, construct a vector whose norm is 1. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
norm1 | ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | normcl 28110 | . . . . . 6 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ) | |
2 | 1 | adantr 480 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℝ) |
3 | normne0 28115 | . . . . . 6 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0ℎ)) | |
4 | 3 | biimpar 501 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ≠ 0) |
5 | 2, 4 | rereccld 10890 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℝ) |
6 | 5 | recnd 10106 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℂ) |
7 | simpl 472 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 𝐴 ∈ ℋ) | |
8 | norm-iii 28125 | . . 3 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = ((abs‘(1 / (normℎ‘𝐴))) · (normℎ‘𝐴))) | |
9 | 6, 7, 8 | syl2anc 694 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = ((abs‘(1 / (normℎ‘𝐴))) · (normℎ‘𝐴))) |
10 | normgt0 28112 | . . . . . 6 ⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ ↔ 0 < (normℎ‘𝐴))) | |
11 | 10 | biimpa 500 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (normℎ‘𝐴)) |
12 | 1re 10077 | . . . . . 6 ⊢ 1 ∈ ℝ | |
13 | 0le1 10589 | . . . . . 6 ⊢ 0 ≤ 1 | |
14 | divge0 10930 | . . . . . 6 ⊢ (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ ((normℎ‘𝐴) ∈ ℝ ∧ 0 < (normℎ‘𝐴))) → 0 ≤ (1 / (normℎ‘𝐴))) | |
15 | 12, 13, 14 | mpanl12 718 | . . . . 5 ⊢ (((normℎ‘𝐴) ∈ ℝ ∧ 0 < (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))) |
16 | 2, 11, 15 | syl2anc 694 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ (1 / (normℎ‘𝐴))) |
17 | 5, 16 | absidd 14205 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (abs‘(1 / (normℎ‘𝐴))) = (1 / (normℎ‘𝐴))) |
18 | 17 | oveq1d 6705 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((abs‘(1 / (normℎ‘𝐴))) · (normℎ‘𝐴)) = ((1 / (normℎ‘𝐴)) · (normℎ‘𝐴))) |
19 | 1 | recnd 10106 | . . . 4 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℂ) |
20 | 19 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℂ) |
21 | 20, 4 | recid2d 10835 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) · (normℎ‘𝐴)) = 1) |
22 | 9, 18, 21 | 3eqtrd 2689 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 class class class wbr 4685 ‘cfv 5926 (class class class)co 6690 ℂcc 9972 ℝcr 9973 0cc0 9974 1c1 9975 · cmul 9979 < clt 10112 ≤ cle 10113 / cdiv 10722 abscabs 14018 ℋchil 27904 ·ℎ csm 27906 normℎcno 27908 0ℎc0v 27909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-hv0cl 27988 ax-hfvmul 27990 ax-hvmul0 27995 ax-hfi 28064 ax-his1 28067 ax-his3 28069 ax-his4 28070 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-sup 8389 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-n0 11331 df-z 11416 df-uz 11726 df-rp 11871 df-seq 12842 df-exp 12901 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-hnorm 27953 |
This theorem is referenced by: norm1exi 28235 nmlnop0iALT 28982 nmbdoplbi 29011 nmcoplbi 29015 nmbdfnlbi 29036 nmcfnlbi 29039 branmfn 29092 |
Copyright terms: Public domain | W3C validator |