Theorem h2hnm 31401

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 6888	. 2 ⊢ (normCV‘𝑈) = (normCV‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
3		eqid 2765	. . 3 ⊢ (normCV‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) = (normCV‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
4	3	nmcvfval 31032	. 2 ⊢ (normCV‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) = (2nd ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
5		opex 5447	. . 3 ⊢ ⟨ +ℎ , ·ℎ ⟩ ∈ V
6		h2h.2	. . . . . 6 ⊢ 𝑈 ∈ NrmCVec
7	1, 6	eqeltrri 2862	. . . . 5 ⊢ ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ ∈ NrmCVec
8		nvex 31036	. . . . 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 8001	. 2 ⊢ (2nd ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) = normℎ
12	2, 4, 11	3eqtrri 2793	1 ⊢ normℎ = (normCV‘𝑈)

This proof depends on syntax axioms:   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ⟨cop 4597  ‘cfv 6540  2^nd c2nd 7991  NrmCVeccnv 31009  norm_CVcnmcv 31015   +_ℎ cva 31345   ·_ℎ csm 31346  norm_ℎcno 31348

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fv 6548  df-oprab 7423  df-2nd 7993  df-vc 30984  df-nv 31017  df-nmcv 31025

This theorem is used by:  h2hmetdval  31403  hhnm  31596