Theorem idcnop 32494

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 6859	. . 3 ⊢ ( I ↾ ℋ): ℋ–1-1-onto→ ℋ
2		f1of 6820	. . 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 23	. . . 4 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ+)
5		fvresi 7174	. . . . . . . . 9 ⊢ (𝑤 ∈ ℋ → (( I ↾ ℋ)‘𝑤) = 𝑤)
6		fvresi 7174	. . . . . . . . 9 ⊢ (𝑥 ∈ ℋ → (( I ↾ ℋ)‘𝑥) = 𝑥)
7	5, 6	oveqan12rd 7436	. . . . . . . 8 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥)) = (𝑤 −ℎ 𝑥))
8	7	fveq2d 6885	. . . . . . 7 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) = (normℎ‘(𝑤 −ℎ 𝑥)))
9	8	breq1d 5113	. . . . . 6 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦 ↔ (normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦))
10	9	biimprd 251	. . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦))
11	10	ralrimiva 3154	. . . 4 ⊢ (𝑥 ∈ ℋ → ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦))
12		breq2 5107	. . . . 5 ⊢ (𝑧 = 𝑦 → ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 ↔ (normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦))
13	12	rspceaimv 3582	. . . 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 609	. . 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 3202	. 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 32370	. 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 724	1 ⊢ ( I ↾ ℋ) ∈ ContOp

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   class class class wbr 5103   I cid 5549   ↾ cres 5657  ⟶wf 6531  –1-1-onto→wf1o 6534  ‘cfv 6535  (class class class)co 7416   < clt 11292  ℝ⁺crp 13067   ℋchba 31432  norm_ℎcno 31436   −_ℎ cmv 31438  ContOpccop 31459

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-pow 5330  ax-pr 5398  ax-un 7742  ax-hilex 31512

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ov 7419  df-oprab 7420  df-mpo 7421  df-map 8835  df-cnop 32353

This theorem is used by:  nmcopex  32542  nmcoplb  32543