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Theorem idcnop 29395
 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.
StepHypRef Expression
1 f1oi 6415 . . 3 ( I ↾ ℋ): ℋ–1-1-onto→ ℋ
2 f1of 6378 . . 3 (( I ↾ ℋ): ℋ–1-1-onto→ ℋ → ( I ↾ ℋ): ℋ⟶ ℋ)
31, 2ax-mp 5 . 2 ( I ↾ ℋ): ℋ⟶ ℋ
4 id 22 . . . 4 (𝑦 ∈ ℝ+𝑦 ∈ ℝ+)
5 fvresi 6691 . . . . . . . . 9 (𝑤 ∈ ℋ → (( I ↾ ℋ)‘𝑤) = 𝑤)
6 fvresi 6691 . . . . . . . . 9 (𝑥 ∈ ℋ → (( I ↾ ℋ)‘𝑥) = 𝑥)
75, 6oveqan12rd 6925 . . . . . . . 8 ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥)) = (𝑤 𝑥))
87fveq2d 6437 . . . . . . 7 ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) = (norm‘(𝑤 𝑥)))
98breq1d 4883 . . . . . 6 ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦 ↔ (norm‘(𝑤 𝑥)) < 𝑦))
109biimprd 240 . . . . 5 ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((norm‘(𝑤 𝑥)) < 𝑦 → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦))
1110ralrimiva 3175 . . . 4 (𝑥 ∈ ℋ → ∀𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑦 → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦))
12 breq2 4877 . . . . 5 (𝑧 = 𝑦 → ((norm‘(𝑤 𝑥)) < 𝑧 ↔ (norm‘(𝑤 𝑥)) < 𝑦))
1312rspceaimv 3534 . . . 4 ((𝑦 ∈ ℝ+ ∧ ∀𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑦 → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦)) → ∃𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦))
144, 11, 13syl2anr 592 . . 3 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℝ+) → ∃𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦))
1514rgen2 3184 . 2 𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦)
16 elcnop 29271 . 2 (( I ↾ ℋ) ∈ ContOp ↔ (( I ↾ ℋ): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦)))
173, 15, 16mpbir2an 704 1 ( I ↾ ℋ) ∈ ContOp
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∈ wcel 2166  ∀wral 3117  ∃wrex 3118   class class class wbr 4873   I cid 5249   ↾ cres 5344  ⟶wf 6119  –1-1-onto→wf1o 6122  ‘cfv 6123  (class class class)co 6905   < clt 10391  ℝ+crp 12112   ℋchba 28331  normℎcno 28335   −ℎ cmv 28337  ContOpccop 28358 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-hilex 28411 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-map 8124  df-cnop 29254 This theorem is referenced by:  nmcopex  29443  nmcoplb  29444
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