Theorem h2hnm 31593

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 6899	. 2 ⊢ (normCV‘𝑈) = (normCV‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
3		eqid 2763	. . 3 ⊢ (normCV‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) = (normCV‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
4	3	nmcvfval 31224	. 2 ⊢ (normCV‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) = (2nd ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
5		opex 5436	. . 3 ⊢ ⟨ +ℎ , ·ℎ ⟩ ∈ V
6		h2h.2	. . . . . 6 ⊢ 𝑈 ∈ NrmCVec
7	1, 6	eqeltrri 2861	. . . . 5 ⊢ ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ ∈ NrmCVec
8		nvex 31228	. . . . 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 8020	. 2 ⊢ (2nd ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) = normℎ
12	2, 4, 11	3eqtrri 2791	1 ⊢ normℎ = (normCV‘𝑈)

This proof depends on syntax axioms:   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  Vcvv 3454  ⟨cop 4838  ‘cfv 6547  2^nd c2nd 8010  NrmCVeccnv 31201  norm_CVcnmcv 31207   +_ℎ cva 31537   ·_ℎ csm 31538  norm_ℎcno 31540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6501  df-fun 6549  df-fv 6555  df-oprab 7434  df-2nd 8012  df-vc 31176  df-nv 31209  df-nmcv 31217

This theorem is used by:  h2hmetdval  31595  hhnm  31788