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Mirrors > Home > HSE Home > Th. List > idcnop | Structured version Visualization version GIF version |
Description: The identity function (restricted to Hilbert space) is a continuous operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
idcnop | ⊢ ( I ↾ ℋ) ∈ ContOp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1oi 6654 | . . 3 ⊢ ( I ↾ ℋ): ℋ–1-1-onto→ ℋ | |
2 | f1of 6617 | . . 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 22 | . . . 4 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ+) | |
5 | fvresi 6937 | . . . . . . . . 9 ⊢ (𝑤 ∈ ℋ → (( I ↾ ℋ)‘𝑤) = 𝑤) | |
6 | fvresi 6937 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℋ → (( I ↾ ℋ)‘𝑥) = 𝑥) | |
7 | 5, 6 | oveqan12rd 7178 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥)) = (𝑤 −ℎ 𝑥)) |
8 | 7 | fveq2d 6676 | . . . . . . 7 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) = (normℎ‘(𝑤 −ℎ 𝑥))) |
9 | 8 | breq1d 5078 | . . . . . 6 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦 ↔ (normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦)) |
10 | 9 | biimprd 250 | . . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦)) |
11 | 10 | ralrimiva 3184 | . . . 4 ⊢ (𝑥 ∈ ℋ → ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 → (normℎ‘((( I ↾ ℋ)‘𝑤) −ℎ (( I ↾ ℋ)‘𝑥))) < 𝑦)) |
12 | breq2 5072 | . . . . 5 ⊢ (𝑧 = 𝑦 → ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 ↔ (normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦)) | |
13 | 12 | rspceaimv 3630 | . . . 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 598 | . . 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 3205 | . 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 29636 | . 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 709 | 1 ⊢ ( I ↾ ℋ) ∈ ContOp |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 class class class wbr 5068 I cid 5461 ↾ cres 5559 ⟶wf 6353 –1-1-onto→wf1o 6356 ‘cfv 6357 (class class class)co 7158 < clt 10677 ℝ+crp 12392 ℋchba 28698 normℎcno 28702 −ℎ cmv 28704 ContOpccop 28725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-hilex 28778 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 df-cnop 29619 |
This theorem is referenced by: nmcopex 29808 nmcoplb 29809 |
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