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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 | β’ normβ = (π₯ β β β¦ (ββ(π₯ Β·ih π₯))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-hnorm 29371 | . 2 β’ normβ = (π₯ β dom dom Β·ih β¦ (ββ(π₯ Β·ih π₯))) | |
2 | ax-hfi 29482 | . . . . . 6 β’ Β·ih :( β Γ β)βΆβ | |
3 | 2 | fdmi 6638 | . . . . 5 β’ dom Β·ih = ( β Γ β) |
4 | 3 | dmeqi 5822 | . . . 4 β’ dom dom Β·ih = dom ( β Γ β) |
5 | dmxpid 5847 | . . . 4 β’ dom ( β Γ β) = β | |
6 | 4, 5 | eqtr2i 2765 | . . 3 β’ β = dom dom Β·ih |
7 | 6 | mpteq1i 5177 | . 2 β’ (π₯ β β β¦ (ββ(π₯ Β·ih π₯))) = (π₯ β dom dom Β·ih β¦ (ββ(π₯ Β·ih π₯))) |
8 | 1, 7 | eqtr4i 2767 | 1 β’ normβ = (π₯ β β β¦ (ββ(π₯ Β·ih π₯))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 β¦ cmpt 5164 Γ cxp 5594 dom cdm 5596 βcfv 6454 (class class class)co 7303 βcc 10911 βcsqrt 14985 βchba 29322 Β·ih csp 29325 normβcno 29326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pr 5361 ax-hfi 29482 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rab 3287 df-v 3439 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4566 df-pr 4568 df-op 4572 df-br 5082 df-opab 5144 df-mpt 5165 df-xp 5602 df-dm 5606 df-fn 6457 df-f 6458 df-hnorm 29371 |
This theorem is referenced by: normf 29526 normval 29527 hilnormi 29566 |
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