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Theorem h2hnm 30995
Description: The norm function of Hilbert space. (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
h2h.1 𝑈 = ⟨⟨ + , · ⟩, norm
h2h.2 𝑈 ∈ NrmCVec
Assertion
Ref Expression
h2hnm norm = (normCV𝑈)

Proof of Theorem h2hnm
StepHypRef Expression
1 h2h.1 . . 3 𝑈 = ⟨⟨ + , · ⟩, norm
21fveq2i 6909 . 2 (normCV𝑈) = (normCV‘⟨⟨ + , · ⟩, norm⟩)
3 eqid 2737 . . 3 (normCV‘⟨⟨ + , · ⟩, norm⟩) = (normCV‘⟨⟨ + , · ⟩, norm⟩)
43nmcvfval 30626 . 2 (normCV‘⟨⟨ + , · ⟩, norm⟩) = (2nd ‘⟨⟨ + , · ⟩, norm⟩)
5 opex 5469 . . 3 ⟨ + , · ⟩ ∈ V
6 h2h.2 . . . . . 6 𝑈 ∈ NrmCVec
71, 6eqeltrri 2838 . . . . 5 ⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec
8 nvex 30630 . . . . 5 (⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec → ( + ∈ V ∧ · ∈ V ∧ norm ∈ V))
97, 8ax-mp 5 . . . 4 ( + ∈ V ∧ · ∈ V ∧ norm ∈ V)
109simp3i 1142 . . 3 norm ∈ V
115, 10op2nd 8023 . 2 (2nd ‘⟨⟨ + , · ⟩, norm⟩) = norm
122, 4, 113eqtrri 2770 1 norm = (normCV𝑈)
Colors of variables: wff setvar class
Syntax hints:  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632  cfv 6561  2nd c2nd 8013  NrmCVeccnv 30603  normCVcnmcv 30609   + cva 30939   · csm 30940  normcno 30942
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-oprab 7435  df-2nd 8015  df-vc 30578  df-nv 30611  df-nmcv 30619
This theorem is referenced by:  h2hmetdval  30997  hhnm  31190
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