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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 29484 | . . . . . 6 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ) | |
2 | 1 | adantr 481 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℝ) |
3 | normne0 29489 | . . . . . 6 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0ℎ)) | |
4 | 3 | biimpar 478 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ≠ 0) |
5 | 2, 4 | rereccld 11800 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℝ) |
6 | 5 | recnd 11001 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℂ) |
7 | simpl 483 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 𝐴 ∈ ℋ) | |
8 | norm-iii 29499 | . . 3 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = ((abs‘(1 / (normℎ‘𝐴))) · (normℎ‘𝐴))) | |
9 | 6, 7, 8 | syl2anc 584 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = ((abs‘(1 / (normℎ‘𝐴))) · (normℎ‘𝐴))) |
10 | normgt0 29486 | . . . . . 6 ⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ ↔ 0 < (normℎ‘𝐴))) | |
11 | 10 | biimpa 477 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (normℎ‘𝐴)) |
12 | 1re 10973 | . . . . . 6 ⊢ 1 ∈ ℝ | |
13 | 0le1 11496 | . . . . . 6 ⊢ 0 ≤ 1 | |
14 | divge0 11842 | . . . . . 6 ⊢ (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ ((normℎ‘𝐴) ∈ ℝ ∧ 0 < (normℎ‘𝐴))) → 0 ≤ (1 / (normℎ‘𝐴))) | |
15 | 12, 13, 14 | mpanl12 699 | . . . . 5 ⊢ (((normℎ‘𝐴) ∈ ℝ ∧ 0 < (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))) |
16 | 2, 11, 15 | syl2anc 584 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ (1 / (normℎ‘𝐴))) |
17 | 5, 16 | absidd 15132 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (abs‘(1 / (normℎ‘𝐴))) = (1 / (normℎ‘𝐴))) |
18 | 17 | oveq1d 7292 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((abs‘(1 / (normℎ‘𝐴))) · (normℎ‘𝐴)) = ((1 / (normℎ‘𝐴)) · (normℎ‘𝐴))) |
19 | 1 | recnd 11001 | . . . 4 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℂ) |
20 | 19 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℂ) |
21 | 20, 4 | recid2d 11745 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) · (normℎ‘𝐴)) = 1) |
22 | 9, 18, 21 | 3eqtrd 2782 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5076 ‘cfv 6435 (class class class)co 7277 ℂcc 10867 ℝcr 10868 0cc0 10869 1c1 10870 · cmul 10874 < clt 11007 ≤ cle 11008 / cdiv 11630 abscabs 14943 ℋchba 29278 ·ℎ csm 29280 normℎcno 29282 0ℎc0v 29283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 ax-pre-sup 10947 ax-hv0cl 29362 ax-hfvmul 29364 ax-hvmul0 29369 ax-hfi 29438 ax-his1 29441 ax-his3 29443 ax-his4 29444 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-om 7713 df-2nd 7832 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-er 8496 df-en 8732 df-dom 8733 df-sdom 8734 df-sup 9199 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-2 12034 df-3 12035 df-n0 12232 df-z 12318 df-uz 12581 df-rp 12729 df-seq 13720 df-exp 13781 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-hnorm 29327 |
This theorem is referenced by: norm1exi 29609 nmlnop0iALT 30354 nmbdoplbi 30383 nmcoplbi 30387 nmbdfnlbi 30408 nmcfnlbi 30411 branmfn 30464 |
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