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Theorem dfhnorm2 29525
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.)
Assertion
Ref Expression
dfhnorm2 normβ„Ž = (π‘₯ ∈ β„‹ ↦ (βˆšβ€˜(π‘₯ Β·ih π‘₯)))

Proof of Theorem dfhnorm2
StepHypRef Expression
1 df-hnorm 29371 . 2 normβ„Ž = (π‘₯ ∈ dom dom Β·ih ↦ (βˆšβ€˜(π‘₯ Β·ih π‘₯)))
2 ax-hfi 29482 . . . . . 6 Β·ih :( β„‹ Γ— β„‹)βŸΆβ„‚
32fdmi 6638 . . . . 5 dom Β·ih = ( β„‹ Γ— β„‹)
43dmeqi 5822 . . . 4 dom dom Β·ih = dom ( β„‹ Γ— β„‹)
5 dmxpid 5847 . . . 4 dom ( β„‹ Γ— β„‹) = β„‹
64, 5eqtr2i 2765 . . 3 β„‹ = dom dom Β·ih
76mpteq1i 5177 . 2 (π‘₯ ∈ β„‹ ↦ (βˆšβ€˜(π‘₯ Β·ih π‘₯))) = (π‘₯ ∈ dom dom Β·ih ↦ (βˆšβ€˜(π‘₯ Β·ih π‘₯)))
81, 7eqtr4i 2767 1 normβ„Ž = (π‘₯ ∈ β„‹ ↦ (βˆšβ€˜(π‘₯ Β·ih π‘₯)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539   ↦ cmpt 5164   Γ— cxp 5594  dom cdm 5596  β€˜cfv 6454  (class class class)co 7303  β„‚cc 10911  βˆšcsqrt 14985   β„‹chba 29322   Β·ih csp 29325  normβ„Žcno 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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