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| Description: The value of the norm of a vector in Hilbert space. Definition of norm in [Beran] p. 96. In the literature, the norm of 𝐴 is usually written as "|| 𝐴 ||", but we use function value notation to take advantage of our existing theorems about functions. (Contributed by NM, 29-May-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| normval | ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) = (√‘(𝐴 ·ih 𝐴))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | oveq12 7441 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥 ·ih 𝑥) = (𝐴 ·ih 𝐴)) | |
| 2 | 1 | anidms 566 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ·ih 𝑥) = (𝐴 ·ih 𝐴)) | 
| 3 | 2 | fveq2d 6909 | . 2 ⊢ (𝑥 = 𝐴 → (√‘(𝑥 ·ih 𝑥)) = (√‘(𝐴 ·ih 𝐴))) | 
| 4 | dfhnorm2 31142 | . 2 ⊢ normℎ = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥))) | |
| 5 | fvex 6918 | . 2 ⊢ (√‘(𝐴 ·ih 𝐴)) ∈ V | |
| 6 | 3, 4, 5 | fvmpt 7015 | 1 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) = (√‘(𝐴 ·ih 𝐴))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ‘cfv 6560 (class class class)co 7432 √csqrt 15273 ℋchba 30939 ·ih csp 30942 normℎcno 30943 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-hfi 31099 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-hnorm 30988 | 
| This theorem is referenced by: normge0 31146 normgt0 31147 norm0 31148 normsqi 31152 norm-ii-i 31157 norm-iii-i 31159 bcsiALT 31199 | 
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