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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 29293 | . 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 493 | 1 ⊢ (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ∀wral 3090 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 ℂcc 10272 ℋchba 28352 +ℎ cva 28353 ·ℎ csm 28354 LinOpclo 28380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-hilex 28432 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-map 8144 df-lnop 29276 |
This theorem is referenced by: bdopf 29297 elbdop2 29306 unopadj2 29373 lnop0 29401 lnopmul 29402 lnopfi 29404 homco2 29412 nmopun 29449 cnlnadjeui 29512 cnlnssadj 29515 |
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