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Theorem h2hnm 30701

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 ⟩, norm_ℎ⟩
h2h.2	⊢ 𝑈 ∈ NrmCVec

**Assertion**
Ref	Expression
h2hnm	⊢ norm_ℎ = (norm_CV‘𝑈)

**Proof of Theorem h2hnm**
Step	Hyp	Ref	Expression
1		h2h.1	. . 3 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
2	1	fveq2i 6885	. 2 ⊢ (norm_CV‘𝑈) = (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
3		eqid 2724	. . 3 ⊢ (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
4	3	nmcvfval 30332	. 2 ⊢ (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = (2^nd ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
5		opex 5455	. . 3 ⊢ ⟨ +_ℎ , ·_ℎ ⟩ ∈ V
6		h2h.2	. . . . . 6 ⊢ 𝑈 ∈ NrmCVec
7	1, 6	eqeltrri 2822	. . . . 5 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec
8		nvex 30336	. . . . 5 ⊢ (⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec → ( +_ℎ ∈ V ∧ ·_ℎ ∈ V ∧ norm_ℎ ∈ V))
9	7, 8	ax-mp 5	. . . 4 ⊢ ( +_ℎ ∈ V ∧ ·_ℎ ∈ V ∧ norm_ℎ ∈ V)
10	9	simp3i 1138	. . 3 ⊢ norm_ℎ ∈ V
11	5, 10	op2nd 7978	. 2 ⊢ (2^nd ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = norm_ℎ
12	2, 4, 11	3eqtrri 2757	1 ⊢ norm_ℎ = (norm_CV‘𝑈)

Colors of variables: wff setvar class

Syntax hints: ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ⟨cop 4627 ‘cfv 6534 2^nd c2nd 7968 NrmCVeccnv 30309 norm_CVcnmcv 30315 +_ℎ cva 30645 ·_ℎ csm 30646 norm_ℎcno 30648

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-un 7719

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-iota 6486 df-fun 6536 df-fv 6542 df-oprab 7406 df-2nd 7970 df-vc 30284 df-nv 30317 df-nmcv 30325

This theorem is referenced by: h2hmetdval 30703 hhnm 30896

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