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Theorem h2hnm 31008

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 6923	. 2 ⊢ (norm_CV‘𝑈) = (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
3		eqid 2740	. . 3 ⊢ (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
4	3	nmcvfval 30639	. 2 ⊢ (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = (2^nd ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
5		opex 5484	. . 3 ⊢ ⟨ +_ℎ , ·_ℎ ⟩ ∈ V
6		h2h.2	. . . . . 6 ⊢ 𝑈 ∈ NrmCVec
7	1, 6	eqeltrri 2841	. . . . 5 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec
8		nvex 30643	. . . . 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 1141	. . 3 ⊢ norm_ℎ ∈ V
11	5, 10	op2nd 8039	. 2 ⊢ (2^nd ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = norm_ℎ
12	2, 4, 11	3eqtrri 2773	1 ⊢ norm_ℎ = (norm_CV‘𝑈)

Colors of variables: wff setvar class

Syntax hints: ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 Vcvv 3488 ⟨cop 4654 ‘cfv 6573 2^nd c2nd 8029 NrmCVeccnv 30616 norm_CVcnmcv 30622 +_ℎ cva 30952 ·_ℎ csm 30953 norm_ℎcno 30955

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-iota 6525 df-fun 6575 df-fv 6581 df-oprab 7452 df-2nd 8031 df-vc 30591 df-nv 30624 df-nmcv 30632

This theorem is referenced by: h2hmetdval 31010 hhnm 31203

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