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Theorem h2hnm 28753

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 6673	. 2 ⊢ (norm_CV‘𝑈) = (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
3		eqid 2821	. . 3 ⊢ (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
4	3	nmcvfval 28384	. 2 ⊢ (norm_CV‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = (2^nd ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
5		opex 5356	. . 3 ⊢ ⟨ +_ℎ , ·_ℎ ⟩ ∈ V
6		h2h.2	. . . . . 6 ⊢ 𝑈 ∈ NrmCVec
7	1, 6	eqeltrri 2910	. . . . 5 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec
8		nvex 28388	. . . . 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 1137	. . 3 ⊢ norm_ℎ ∈ V
11	5, 10	op2nd 7698	. 2 ⊢ (2^nd ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = norm_ℎ
12	2, 4, 11	3eqtrri 2849	1 ⊢ norm_ℎ = (norm_CV‘𝑈)

Colors of variables: wff setvar class

Syntax hints: ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⟨cop 4573 ‘cfv 6355 2^nd c2nd 7688 NrmCVeccnv 28361 norm_CVcnmcv 28367 +_ℎ cva 28697 ·_ℎ csm 28698 norm_ℎcno 28700

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fv 6363 df-oprab 7160 df-2nd 7690 df-vc 28336 df-nv 28369 df-nmcv 28377

This theorem is referenced by: h2hmetdval 28755 hhnm 28948

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