Theorem h2hnm 27217

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 6106	. 2 ⊢ (normCV‘𝑈) = (normCV‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
3		eqid 2610	. . 3 ⊢ (normCV‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) = (normCV‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
4	3	nmcvfval 26846	. 2 ⊢ (normCV‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) = (2nd ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
5		opex 4859	. . 3 ⊢ ⟨ +ℎ , ·ℎ ⟩ ∈ V
6		h2h.2	. . . . . 6 ⊢ 𝑈 ∈ NrmCVec
7	1, 6	eqeltrri 2685	. . . . 5 ⊢ ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ ∈ NrmCVec
8		nvex 26850	. . . . 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 1065	. . 3 ⊢ normℎ ∈ V
11	5, 10	op2nd 7068	. 2 ⊢ (2nd ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) = normℎ
12	2, 4, 11	3eqtrri 2637	1 ⊢ normℎ = (normCV‘𝑈)

Syntax hints:   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131  ‘cfv 5804  2^nd c2nd 7058  NrmCVeccnv 26823  norm_CVcnmcv 26829   +_ℎ cva 27161   ·_ℎ csm 27162  norm_ℎcno 27164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-oprab 6553  df-2nd 7060  df-vc 26798  df-nv 26831  df-nmcv 26839

This theorem is referenced by:  h2hmetdval  27219  hhnm  27412