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Mirrors > Home > HSE Home > Th. List > lnopf | Structured version Visualization version GIF version |
Description: A linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lnopf | ⊢ (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ellnop 29952 | . 2 ⊢ (𝑇 ∈ LinOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)))) | |
2 | 1 | simplbi 501 | 1 ⊢ (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2111 ∀wral 3062 ⟶wf 6385 ‘cfv 6389 (class class class)co 7222 ℂcc 10740 ℋchba 29013 +ℎ cva 29014 ·ℎ csm 29015 LinOpclo 29041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5201 ax-nul 5208 ax-pow 5267 ax-pr 5331 ax-un 7532 ax-hilex 29093 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3417 df-sbc 3704 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-nul 4247 df-if 4449 df-pw 4524 df-sn 4551 df-pr 4553 df-op 4557 df-uni 4829 df-br 5063 df-opab 5125 df-id 5464 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 df-dm 5570 df-rn 5571 df-iota 6347 df-fun 6391 df-fn 6392 df-f 6393 df-fv 6397 df-ov 7225 df-oprab 7226 df-mpo 7227 df-map 8519 df-lnop 29935 |
This theorem is referenced by: bdopf 29956 elbdop2 29965 unopadj2 30032 lnop0 30060 lnopmul 30061 lnopfi 30063 homco2 30071 nmopun 30108 cnlnadjeui 30171 cnlnssadj 30174 |
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