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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 6659 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥 ·ih 𝑥) = (𝐴 ·ih 𝐴)) | |
2 | 1 | anidms 677 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ·ih 𝑥) = (𝐴 ·ih 𝐴)) |
3 | 2 | fveq2d 6195 | . 2 ⊢ (𝑥 = 𝐴 → (√‘(𝑥 ·ih 𝑥)) = (√‘(𝐴 ·ih 𝐴))) |
4 | dfhnorm2 27979 | . 2 ⊢ normℎ = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥))) | |
5 | fvex 6201 | . 2 ⊢ (√‘(𝐴 ·ih 𝐴)) ∈ V | |
6 | 3, 4, 5 | fvmpt 6282 | 1 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) = (√‘(𝐴 ·ih 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 √csqrt 13973 ℋchil 27776 ·ih csp 27779 normℎcno 27780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-hfi 27936 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-hnorm 27825 |
This theorem is referenced by: normge0 27983 normgt0 27984 norm0 27985 normsqi 27989 norm-ii-i 27994 norm-iii-i 27996 bcsiALT 28036 |
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