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Theorem idcnop 31501
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 6870 . . 3 ( I ↾ ℋ): ℋ–1-1-onto→ ℋ
2 f1of 6832 . . 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 7172 . . . . . . . . 9 (𝑤 ∈ ℋ → (( I ↾ ℋ)‘𝑤) = 𝑤)
6 fvresi 7172 . . . . . . . . 9 (𝑥 ∈ ℋ → (( I ↾ ℋ)‘𝑥) = 𝑥)
75, 6oveqan12rd 7431 . . . . . . . 8 ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥)) = (𝑤 𝑥))
87fveq2d 6894 . . . . . . 7 ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) = (norm‘(𝑤 𝑥)))
98breq1d 5157 . . . . . 6 ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦 ↔ (norm‘(𝑤 𝑥)) < 𝑦))
109biimprd 247 . . . . 5 ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((norm‘(𝑤 𝑥)) < 𝑦 → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦))
1110ralrimiva 3144 . . . 4 (𝑥 ∈ ℋ → ∀𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑦 → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦))
12 breq2 5151 . . . . 5 (𝑧 = 𝑦 → ((norm‘(𝑤 𝑥)) < 𝑧 ↔ (norm‘(𝑤 𝑥)) < 𝑦))
1312rspceaimv 3616 . . . 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 595 . . 3 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℝ+) → ∃𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦))
1514rgen2 3195 . 2 𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦)
16 elcnop 31377 . 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 707 1 ( I ↾ ℋ) ∈ ContOp
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2104  wral 3059  wrex 3068   class class class wbr 5147   I cid 5572  cres 5677  wf 6538  1-1-ontowf1o 6541  cfv 6542  (class class class)co 7411   < clt 11252  +crp 12978  chba 30439  normcno 30443   cmv 30445  ContOpccop 30466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-hilex 30519
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8824  df-cnop 31360
This theorem is referenced by:  nmcopex  31549  nmcoplb  31550
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