Theorem idcnop 32052

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 6818	. . 3 ⊢ ( I ↾ ℋ): ℋ–1-1-onto→ ℋ
2		f1of 6780	. . 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 7128	. . . . . . . . 9 ⊢ (𝑤 ∈ ℋ → (( I ↾ ℋ)‘𝑤) = 𝑤)
6		fvresi 7128	. . . . . . . . 9 ⊢ (𝑥 ∈ ℋ → (( I ↾ ℋ)‘𝑥) = 𝑥)
7	5, 6	oveqan12rd 7387	. . . . . . . 8 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥)) = (𝑤 −ℎ 𝑥))
8	7	fveq2d 6844	. . . . . . 7 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) = (normℎ‘(𝑤 −ℎ 𝑥)))
9	8	breq1d 5095	. . . . . 6 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦 ↔ (normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦))
10	9	biimprd 248	. . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦))
11	10	ralrimiva 3129	. . . 4 ⊢ (𝑥 ∈ ℋ → ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦))
12		breq2 5089	. . . . 5 ⊢ (𝑧 = 𝑦 → ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 ↔ (normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦))
13	12	rspceaimv 3570	. . . 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 598	. . 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 3177	. 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 31928	. 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 712	1 ⊢ ( I ↾ ℋ) ∈ ContOp

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061   class class class wbr 5085   I cid 5525   ↾ cres 5633  ⟶wf 6494  –1-1-onto→wf1o 6497  ‘cfv 6498  (class class class)co 7367   < clt 11179  ℝ⁺crp 12942   ℋchba 30990  norm_ℎcno 30994   −_ℎ cmv 30996  ContOpccop 31017

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-hilex 31070

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-map 8775  df-cnop 31911

This theorem is referenced by:  nmcopex  32100  nmcoplb  32101