Theorem idcnop 28840

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 6174	. . 3 ⊢ ( I ↾ ℋ): ℋ–1-1-onto→ ℋ
2		f1of 6137	. . 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 6439	. . . . . . . . 9 ⊢ (𝑤 ∈ ℋ → (( I ↾ ℋ)‘𝑤) = 𝑤)
6		fvresi 6439	. . . . . . . . 9 ⊢ (𝑥 ∈ ℋ → (( I ↾ ℋ)‘𝑥) = 𝑥)
7	5, 6	oveqan12rd 6670	. . . . . . . 8 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥)) = (𝑤 −ℎ 𝑥))
8	7	fveq2d 6195	. . . . . . 7 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) = (normℎ‘(𝑤 −ℎ 𝑥)))
9	8	breq1d 4663	. . . . . 6 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦 ↔ (normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦))
10	9	biimprd 238	. . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦))
11	10	ralrimiva 2966	. . . 4 ⊢ (𝑥 ∈ ℋ → ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦))
12		breq2 4657	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 ↔ (normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦))
13	12	imbi1d 331	. . . . . 6 ⊢ (𝑧 = 𝑦 → (((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦) ↔ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦)))
14	13	ralbidv 2986	. . . . 5 ⊢ (𝑧 = 𝑦 → (∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦) ↔ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦)))
15	14	rspcev 3309	. . . 4 ⊢ ((𝑦 ∈ ℝ+ ∧ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦)) → ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦))
16	4, 11, 15	syl2anr 495	. . 3 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℝ+) → ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦))
17	16	rgen2 2975	. 2 ⊢ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦)
18		elcnop 28716	. 2 ⊢ (( I ↾ ℋ) ∈ ContOp ↔ (( I ↾ ℋ): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦)))
19	3, 17, 18	mpbir2an 955	1 ⊢ ( I ↾ ℋ) ∈ ContOp

Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   class class class wbr 4653   I cid 5023   ↾ cres 5116  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   < clt 10074  ℝ⁺crp 11832   ℋchil 27776  norm_ℎcno 27780   −_ℎ cmv 27782  ContOpccop 27803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-hilex 27856

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-cnop 28699

This theorem is referenced by:  nmcopex  28888  nmcoplb  28889