Theorem idcnop 30351

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 6746	. . 3 ⊢ ( I ↾ ℋ): ℋ–1-1-onto→ ℋ
2		f1of 6708	. . 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 7037	. . . . . . . . 9 ⊢ (𝑤 ∈ ℋ → (( I ↾ ℋ)‘𝑤) = 𝑤)
6		fvresi 7037	. . . . . . . . 9 ⊢ (𝑥 ∈ ℋ → (( I ↾ ℋ)‘𝑥) = 𝑥)
7	5, 6	oveqan12rd 7287	. . . . . . . 8 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥)) = (𝑤 −ℎ 𝑥))
8	7	fveq2d 6770	. . . . . . 7 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) = (normℎ‘(𝑤 −ℎ 𝑥)))
9	8	breq1d 5083	. . . . . 6 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦 ↔ (normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦))
10	9	biimprd 247	. . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦))
11	10	ralrimiva 3108	. . . 4 ⊢ (𝑥 ∈ ℋ → ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦))
12		breq2 5077	. . . . 5 ⊢ (𝑧 = 𝑦 → ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 ↔ (normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦))
13	12	rspceaimv 3564	. . . 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))
14	4, 11, 13	syl2anr 597	. . 3 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℝ+) → ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦))
15	14	rgen2 3127	. 2 ⊢ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦)
16		elcnop 30227	. 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)))
17	3, 15, 16	mpbir2an 708	1 ⊢ ( I ↾ ℋ) ∈ ContOp

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   class class class wbr 5073   I cid 5483   ↾ cres 5586  ⟶wf 6422  –1-1-onto→wf1o 6425  ‘cfv 6426  (class class class)co 7267   < clt 11019  ℝ⁺crp 12740   ℋchba 29289  norm_ℎcno 29293   −_ℎ cmv 29295  ContOpccop 29316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350  ax-un 7578  ax-hilex 29369

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-sbc 3716  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-f1 6431  df-fo 6432  df-f1o 6433  df-fv 6434  df-ov 7270  df-oprab 7271  df-mpo 7272  df-map 8604  df-cnop 30210

This theorem is referenced by:  nmcopex  30399  nmcoplb  30400