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Theorem h2hnm 30779
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 6894 . 2 (normCV𝑈) = (normCV‘⟨⟨ + , · ⟩, norm⟩)
3 eqid 2728 . . 3 (normCV‘⟨⟨ + , · ⟩, norm⟩) = (normCV‘⟨⟨ + , · ⟩, norm⟩)
43nmcvfval 30410 . 2 (normCV‘⟨⟨ + , · ⟩, norm⟩) = (2nd ‘⟨⟨ + , · ⟩, norm⟩)
5 opex 5460 . . 3 ⟨ + , · ⟩ ∈ V
6 h2h.2 . . . . . 6 𝑈 ∈ NrmCVec
71, 6eqeltrri 2826 . . . . 5 ⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec
8 nvex 30414 . . . . 5 (⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec → ( + ∈ V ∧ · ∈ V ∧ norm ∈ V))
97, 8ax-mp 5 . . . 4 ( + ∈ V ∧ · ∈ V ∧ norm ∈ V)
109simp3i 1139 . . 3 norm ∈ V
115, 10op2nd 7996 . 2 (2nd ‘⟨⟨ + , · ⟩, norm⟩) = norm
122, 4, 113eqtrri 2761 1 norm = (normCV𝑈)
Colors of variables: wff setvar class
Syntax hints:  w3a 1085   = wceq 1534  wcel 2099  Vcvv 3470  cop 4630  cfv 6542  2nd c2nd 7986  NrmCVeccnv 30387  normCVcnmcv 30393   + cva 30723   · csm 30724  normcno 30726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fv 6550  df-oprab 7418  df-2nd 7988  df-vc 30362  df-nv 30395  df-nmcv 30403
This theorem is referenced by:  h2hmetdval  30781  hhnm  30974
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