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Mirrors > Home > HSE Home > Th. List > normval | Structured version Visualization version GIF version |
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 6919 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥 ·ih 𝑥) = (𝐴 ·ih 𝐴)) | |
2 | 1 | anidms 562 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ·ih 𝑥) = (𝐴 ·ih 𝐴)) |
3 | 2 | fveq2d 6441 | . 2 ⊢ (𝑥 = 𝐴 → (√‘(𝑥 ·ih 𝑥)) = (√‘(𝐴 ·ih 𝐴))) |
4 | dfhnorm2 28530 | . 2 ⊢ normℎ = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥))) | |
5 | fvex 6450 | . 2 ⊢ (√‘(𝐴 ·ih 𝐴)) ∈ V | |
6 | 3, 4, 5 | fvmpt 6533 | 1 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) = (√‘(𝐴 ·ih 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ‘cfv 6127 (class class class)co 6910 √csqrt 14357 ℋchba 28327 ·ih csp 28330 normℎcno 28331 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 ax-hfi 28487 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-fv 6135 df-ov 6913 df-hnorm 28376 |
This theorem is referenced by: normge0 28534 normgt0 28535 norm0 28536 normsqi 28540 norm-ii-i 28545 norm-iii-i 28547 bcsiALT 28587 |
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