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Theorem h2hnm 27803
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 6181 . 2 (normCV𝑈) = (normCV‘⟨⟨ + , · ⟩, norm⟩)
3 eqid 2620 . . 3 (normCV‘⟨⟨ + , · ⟩, norm⟩) = (normCV‘⟨⟨ + , · ⟩, norm⟩)
43nmcvfval 27432 . 2 (normCV‘⟨⟨ + , · ⟩, norm⟩) = (2nd ‘⟨⟨ + , · ⟩, norm⟩)
5 opex 4923 . . 3 ⟨ + , · ⟩ ∈ V
6 h2h.2 . . . . . 6 𝑈 ∈ NrmCVec
71, 6eqeltrri 2696 . . . . 5 ⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec
8 nvex 27436 . . . . 5 (⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec → ( + ∈ V ∧ · ∈ V ∧ norm ∈ V))
97, 8ax-mp 5 . . . 4 ( + ∈ V ∧ · ∈ V ∧ norm ∈ V)
109simp3i 1070 . . 3 norm ∈ V
115, 10op2nd 7162 . 2 (2nd ‘⟨⟨ + , · ⟩, norm⟩) = norm
122, 4, 113eqtrri 2647 1 norm = (normCV𝑈)
Colors of variables: wff setvar class
Syntax hints:  w3a 1036   = wceq 1481  wcel 1988  Vcvv 3195  cop 4174  cfv 5876  2nd c2nd 7152  NrmCVeccnv 27409  normCVcnmcv 27415   + cva 27747   · csm 27748  normcno 27750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fv 5884  df-oprab 6639  df-2nd 7154  df-vc 27384  df-nv 27417  df-nmcv 27425
This theorem is referenced by:  h2hmetdval  27805  hhnm  27998
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