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Theorem h2hnm 30779

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 6894	. 2 ⊢ (norm_CV‘𝑈) = (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
3		eqid 2728	. . 3 ⊢ (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
4	3	nmcvfval 30410	. 2 ⊢ (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = (2^nd ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
5		opex 5460	. . 3 ⊢ ⟨ +_ℎ , ·_ℎ ⟩ ∈ V
6		h2h.2	. . . . . 6 ⊢ 𝑈 ∈ NrmCVec
7	1, 6	eqeltrri 2826	. . . . 5 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec
8		nvex 30414	. . . . 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 1139	. . 3 ⊢ norm_ℎ ∈ V
11	5, 10	op2nd 7996	. 2 ⊢ (2^nd ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = norm_ℎ
12	2, 4, 11	3eqtrri 2761	1 ⊢ norm_ℎ = (norm_CV‘𝑈)

Colors of variables: wff setvar class

Syntax hints: ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 Vcvv 3470 ⟨cop 4630 ‘cfv 6542 2^nd c2nd 7986 NrmCVeccnv 30387 norm_CVcnmcv 30393 +_ℎ cva 30723 ·_ℎ csm 30724 norm_ℎcno 30726

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6494 df-fun 6544 df-fv 6550 df-oprab 7418 df-2nd 7988 df-vc 30362 df-nv 30395 df-nmcv 30403

This theorem is referenced by: h2hmetdval 30781 hhnm 30974

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