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Theorem idcnop 28689
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 6131 . . 3 ( I ↾ ℋ): ℋ–1-1-onto→ ℋ
2 f1of 6094 . . 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 6393 . . . . . . . . 9 (𝑤 ∈ ℋ → (( I ↾ ℋ)‘𝑤) = 𝑤)
6 fvresi 6393 . . . . . . . . 9 (𝑥 ∈ ℋ → (( I ↾ ℋ)‘𝑥) = 𝑥)
75, 6oveqan12rd 6624 . . . . . . . 8 ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥)) = (𝑤 𝑥))
87fveq2d 6152 . . . . . . 7 ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) = (norm‘(𝑤 𝑥)))
98breq1d 4623 . . . . . 6 ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦 ↔ (norm‘(𝑤 𝑥)) < 𝑦))
109biimprd 238 . . . . 5 ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((norm‘(𝑤 𝑥)) < 𝑦 → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦))
1110ralrimiva 2960 . . . 4 (𝑥 ∈ ℋ → ∀𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑦 → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦))
12 breq2 4617 . . . . . . 7 (𝑧 = 𝑦 → ((norm‘(𝑤 𝑥)) < 𝑧 ↔ (norm‘(𝑤 𝑥)) < 𝑦))
1312imbi1d 331 . . . . . 6 (𝑧 = 𝑦 → (((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦) ↔ ((norm‘(𝑤 𝑥)) < 𝑦 → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦)))
1413ralbidv 2980 . . . . 5 (𝑧 = 𝑦 → (∀𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦) ↔ ∀𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑦 → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦)))
1514rspcev 3295 . . . 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))
164, 11, 15syl2anr 495 . . 3 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℝ+) → ∃𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦))
1716rgen2 2969 . 2 𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((( I ↾ ℋ)‘𝑤) − (( I ↾ ℋ)‘𝑥))) < 𝑦)
18 elcnop 28565 . 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)))
193, 17, 18mpbir2an 954 1 ( I ↾ ℋ) ∈ ContOp
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wral 2907  wrex 2908   class class class wbr 4613   I cid 4984  cres 5076  wf 5843  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604   < clt 10018  +crp 11776  chil 27625  normcno 27629   cmv 27631  ContOpccop 27652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-hilex 27705
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-cnop 28548
This theorem is referenced by:  nmcopex  28737  nmcoplb  28738
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