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Theorem h2hnm 28756
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 6676 . 2 (normCV𝑈) = (normCV‘⟨⟨ + , · ⟩, norm⟩)
3 eqid 2824 . . 3 (normCV‘⟨⟨ + , · ⟩, norm⟩) = (normCV‘⟨⟨ + , · ⟩, norm⟩)
43nmcvfval 28387 . 2 (normCV‘⟨⟨ + , · ⟩, norm⟩) = (2nd ‘⟨⟨ + , · ⟩, norm⟩)
5 opex 5359 . . 3 ⟨ + , · ⟩ ∈ V
6 h2h.2 . . . . . 6 𝑈 ∈ NrmCVec
71, 6eqeltrri 2913 . . . . 5 ⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec
8 nvex 28391 . . . . 5 (⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec → ( + ∈ V ∧ · ∈ V ∧ norm ∈ V))
97, 8ax-mp 5 . . . 4 ( + ∈ V ∧ · ∈ V ∧ norm ∈ V)
109simp3i 1137 . . 3 norm ∈ V
115, 10op2nd 7701 . 2 (2nd ‘⟨⟨ + , · ⟩, norm⟩) = norm
122, 4, 113eqtrri 2852 1 norm = (normCV𝑈)
Colors of variables: wff setvar class
Syntax hints:  w3a 1083   = wceq 1536  wcel 2113  Vcvv 3497  cop 4576  cfv 6358  2nd c2nd 7691  NrmCVeccnv 28364  normCVcnmcv 28370   + cva 28700   · csm 28701  normcno 28703
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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-iota 6317  df-fun 6360  df-fv 6366  df-oprab 7163  df-2nd 7693  df-vc 28339  df-nv 28372  df-nmcv 28380
This theorem is referenced by:  h2hmetdval  28758  hhnm  28951
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