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Theorem h2hnm 31578

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 6888	. 2 ⊢ (norm_CV‘𝑈) = (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
3		eqid 2761	. . 3 ⊢ (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
4	3	nmcvfval 31209	. 2 ⊢ (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = (2^nd ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
5		opex 5432	. . 3 ⊢ ⟨ +_ℎ , ·_ℎ ⟩ ∈ V
6		h2h.2	. . . . . 6 ⊢ 𝑈 ∈ NrmCVec
7	1, 6	eqeltrri 2858	. . . . 5 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec
8		nvex 31213	. . . . 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 1159	. . 3 ⊢ norm_ℎ ∈ V
11	5, 10	op2nd 8010	. 2 ⊢ (2^nd ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = norm_ℎ
12	2, 4, 11	3eqtrri 2789	1 ⊢ norm_ℎ = (norm_CV‘𝑈)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 Vcvv 3451 ⟨cop 4590 ‘cfv 6538 2^nd c2nd 8000 NrmCVeccnv 31186 norm_CVcnmcv 31192 +_ℎ cva 31522 ·_ℎ csm 31523 norm_ℎcno 31525

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pr 5391 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-iota 6494 df-fun 6540 df-fv 6546 df-oprab 7424 df-2nd 8002 df-vc 31161 df-nv 31194 df-nmcv 31202

This theorem is used by: h2hmetdval 31580 hhnm 31773

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