idcnop - Hilbert Space Explorer

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Theorem idcnop 31784

Description: The identity function (restricted to Hilbert space) is a continuous operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
idcnop	⊢ ( I ↾ ℋ) ∈ ContOp

Proof of Theorem idcnop Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1oi 6871	. . 3 ⊢ ( I ↾ ℋ): ℋ–1-1-onto→ ℋ
2		f1of 6833	. . 3 ⊢ (( I ↾ ℋ): ℋ–1-1-onto→ ℋ → ( I ↾ ℋ): ℋ⟶ ℋ)
3	1, 2	ax-mp 5	. 2 ⊢ ( I ↾ ℋ): ℋ⟶ ℋ
4		id 22	. . . 4 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℝ⁺)
5		fvresi 7176	. . . . . . . . 9 ⊢ (𝑤 ∈ ℋ → (( I ↾ ℋ)‘𝑤) = 𝑤)
6		fvresi 7176	. . . . . . . . 9 ⊢ (𝑥 ∈ ℋ → (( I ↾ ℋ)‘𝑥) = 𝑥)
7	5, 6	oveqan12rd 7434	. . . . . . . 8 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((( I ↾ ℋ)‘𝑤) −_ℎ (( I ↾ ℋ)‘𝑥)) = (𝑤 −_ℎ 𝑥))
8	7	fveq2d 6895	. . . . . . 7 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (norm_ℎ‘((( I ↾ ℋ)‘𝑤) −_ℎ (( I ↾ ℋ)‘𝑥))) = (norm_ℎ‘(𝑤 −_ℎ 𝑥)))
9	8	breq1d 5152	. . . . . 6 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((norm_ℎ‘((( I ↾ ℋ)‘𝑤) −_ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦 ↔ (norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑦))
10	9	biimprd 247	. . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑦 → (norm_ℎ‘((( I ↾ ℋ)‘𝑤) −_ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦))
11	10	ralrimiva 3142	. . . 4 ⊢ (𝑥 ∈ ℋ → ∀𝑤 ∈ ℋ ((norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑦 → (norm_ℎ‘((( I ↾ ℋ)‘𝑤) −_ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦))
12		breq2 5146	. . . . 5 ⊢ (𝑧 = 𝑦 → ((norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑧 ↔ (norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑦))
13	12	rspceaimv 3614	. . . 4 ⊢ ((𝑦 ∈ ℝ⁺ ∧ ∀𝑤 ∈ ℋ ((norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑦 → (norm_ℎ‘((( I ↾ ℋ)‘𝑤) −_ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦)) → ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℋ ((norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑧 → (norm_ℎ‘((( I ↾ ℋ)‘𝑤) −_ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦))
14	4, 11, 13	syl2anr 596	. . 3 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℝ⁺) → ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℋ ((norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑧 → (norm_ℎ‘((( I ↾ ℋ)‘𝑤) −_ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦))
15	14	rgen2 3193	. 2 ⊢ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℋ ((norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑧 → (norm_ℎ‘((( I ↾ ℋ)‘𝑤) −_ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦)
16		elcnop 31660	. 2 ⊢ (( I ↾ ℋ) ∈ ContOp ↔ (( I ↾ ℋ): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℋ ((norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑧 → (norm_ℎ‘((( I ↾ ℋ)‘𝑤) −_ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦)))
17	3, 15, 16	mpbir2an 710	1 ⊢ ( I ↾ ℋ) ∈ ContOp

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 ∀wral 3057 ∃wrex 3066 class class class wbr 5142 I cid 5569 ↾ cres 5674 ⟶wf 6538 –1-1-onto→wf1o 6541 ‘cfv 6542 (class class class)co 7414 < clt 11272 ℝ⁺crp 13000 ℋchba 30722 norm_ℎcno 30726 −_ℎ cmv 30728 ContOpccop 30749

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-pow 5359 ax-pr 5423 ax-un 7734 ax-hilex 30802

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-sbc 3776 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-map 8840 df-cnop 31643

This theorem is referenced by: nmcopex 31832 nmcoplb 31833

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