Theorem h2hnm 31179

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 6870	. 2 ⊢ (normCV‘𝑈) = (normCV‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
3		eqid 2762	. . 3 ⊢ (normCV‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) = (normCV‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
4	3	nmcvfval 30810	. 2 ⊢ (normCV‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) = (2nd ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
5		opex 5431	. . 3 ⊢ ⟨ +ℎ , ·ℎ ⟩ ∈ V
6		h2h.2	. . . . . 6 ⊢ 𝑈 ∈ NrmCVec
7	1, 6	eqeltrri 2859	. . . . 5 ⊢ ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ ∈ NrmCVec
8		nvex 30814	. . . . 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 1154	. . 3 ⊢ normℎ ∈ V
11	5, 10	op2nd 7979	. 2 ⊢ (2nd ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) = normℎ
12	2, 4, 11	3eqtrri 2790	1 ⊢ normℎ = (normCV‘𝑈)

Syntax hints:   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  Vcvv 3454  ⟨cop 4588  ‘cfv 6521  2^nd c2nd 7969  NrmCVeccnv 30787  norm_CVcnmcv 30793   +_ℎ cva 31123   ·_ℎ csm 31124  norm_ℎcno 31126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-iota 6477  df-fun 6523  df-fv 6529  df-oprab 7400  df-2nd 7971  df-vc 30762  df-nv 30795  df-nmcv 30803

This theorem is referenced by:  h2hmetdval  31181  hhnm  31374