Theorem h2hnm 31328

Description: The norm function of Hilbert space. (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
h2h.1	⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
h2h.2	⊢ 𝑈 ∈ NrmCVec

Assertion

Ref	Expression
h2hnm	⊢ normℎ = (normCV‘𝑈)

Proof of Theorem h2hnm

Step	Hyp	Ref	Expression
1		h2h.1	. . 3 ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
2	1	fveq2i 6884	. 2 ⊢ (normCV‘𝑈) = (normCV‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
3		eqid 2763	. . 3 ⊢ (normCV‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) = (normCV‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
4	3	nmcvfval 30959	. 2 ⊢ (normCV‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) = (2nd ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
5		opex 5445	. . 3 ⊢ ⟨ +ℎ , ·ℎ ⟩ ∈ V
6		h2h.2	. . . . . 6 ⊢ 𝑈 ∈ NrmCVec
7	1, 6	eqeltrri 2860	. . . . 5 ⊢ ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ ∈ NrmCVec
8		nvex 30963	. . . . 5 ⊢ (⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ ∈ NrmCVec → ( +ℎ ∈ V ∧ ·ℎ ∈ V ∧ normℎ ∈ V))
9	7, 8	ax-mp 5	. . . 4 ⊢ ( +ℎ ∈ V ∧ ·ℎ ∈ V ∧ normℎ ∈ V)
10	9	simp3i 1159	. . 3 ⊢ normℎ ∈ V
11	5, 10	op2nd 7991	. 2 ⊢ (2nd ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) = normℎ
12	2, 4, 11	3eqtrri 2791	1 ⊢ normℎ = (normCV‘𝑈)

Syntax hints:   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ⟨cop 4595  ‘cfv 6536  2^nd c2nd 7981  NrmCVeccnv 30936  norm_CVcnmcv 30942   +_ℎ cva 31272   ·_ℎ csm 31273  norm_ℎcno 31275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fv 6544  df-oprab 7414  df-2nd 7983  df-vc 30911  df-nv 30944  df-nmcv 30952

This theorem is referenced by:  h2hmetdval  31330  hhnm  31523