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Mirrors > Home > HSE Home > Th. List > dfhnorm2 | Structured version Visualization version GIF version |
Description: Alternate definition of the norm of a vector of Hilbert space. Definition of norm in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dfhnorm2 | ⊢ normℎ = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-hnorm 30148 | . 2 ⊢ normℎ = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥))) | |
2 | ax-hfi 30259 | . . . . . 6 ⊢ ·ih :( ℋ × ℋ)⟶ℂ | |
3 | 2 | fdmi 6717 | . . . . 5 ⊢ dom ·ih = ( ℋ × ℋ) |
4 | 3 | dmeqi 5897 | . . . 4 ⊢ dom dom ·ih = dom ( ℋ × ℋ) |
5 | dmxpid 5922 | . . . 4 ⊢ dom ( ℋ × ℋ) = ℋ | |
6 | 4, 5 | eqtr2i 2761 | . . 3 ⊢ ℋ = dom dom ·ih |
7 | 6 | mpteq1i 5238 | . 2 ⊢ (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥))) = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥))) |
8 | 1, 7 | eqtr4i 2763 | 1 ⊢ normℎ = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ↦ cmpt 5225 × cxp 5668 dom cdm 5670 ‘cfv 6533 (class class class)co 7394 ℂcc 11092 √csqrt 15164 ℋchba 30099 ·ih csp 30102 normℎcno 30103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-sep 5293 ax-nul 5300 ax-pr 5421 ax-hfi 30259 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rab 3433 df-v 3476 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5143 df-opab 5205 df-mpt 5226 df-xp 5676 df-dm 5680 df-fn 6536 df-f 6537 df-hnorm 30148 |
This theorem is referenced by: normf 30303 normval 30304 hilnormi 30343 |
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