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Theorem idcnop 31234
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 6872 . . 3 ( I ↾ ℋ): ℋ–1-1-onto→ ℋ
2 f1of 6834 . . 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 7171 . . . . . . . . 9 (𝑤 ∈ ℋ → (( I ↾ ℋ)‘𝑤) = 𝑤)
6 fvresi 7171 . . . . . . . . 9 (𝑥 ∈ ℋ → (( I ↾ ℋ)‘𝑥) = 𝑥)
75, 6oveqan12rd 7429 . . . . . . . 8 ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥)) = (𝑤 𝑥))
87fveq2d 6896 . . . . . . 7 ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) = (norm‘(𝑤 𝑥)))
98breq1d 5159 . . . . . 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 3147 . . . 4 (𝑥 ∈ ℋ → ∀𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑦 → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦))
12 breq2 5153 . . . . 5 (𝑧 = 𝑦 → ((norm‘(𝑤 𝑥)) < 𝑧 ↔ (norm‘(𝑤 𝑥)) < 𝑦))
1312rspceaimv 3618 . . . 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 598 . . 3 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℝ+) → ∃𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦))
1514rgen2 3198 . 2 𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦)
16 elcnop 31110 . 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 710 1 ( I ↾ ℋ) ∈ ContOp
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wral 3062  wrex 3071   class class class wbr 5149   I cid 5574  cres 5679  wf 6540  1-1-ontowf1o 6543  cfv 6544  (class class class)co 7409   < clt 11248  +crp 12974  chba 30172  normcno 30176   cmv 30178  ContOpccop 30199
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-hilex 30252
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-cnop 31093
This theorem is referenced by:  nmcopex  31282  nmcoplb  31283
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