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Mirrors > Home > HSE Home > Th. List > hoscli | Structured version Visualization version GIF version |
Description: Closure of Hilbert space operator sum. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hoeq.1 | ⊢ 𝑆: ℋ⟶ ℋ |
hoeq.2 | ⊢ 𝑇: ℋ⟶ ℋ |
Ref | Expression |
---|---|
hoscli | ⊢ (𝐴 ∈ ℋ → ((𝑆 +op 𝑇)‘𝐴) ∈ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hoeq.1 | . 2 ⊢ 𝑆: ℋ⟶ ℋ | |
2 | hoeq.2 | . 2 ⊢ 𝑇: ℋ⟶ ℋ | |
3 | hoscl 31768 | . 2 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝐴 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝐴) ∈ ℋ) | |
4 | 1, 2, 3 | mpanl12 701 | 1 ⊢ (𝐴 ∈ ℋ → ((𝑆 +op 𝑇)‘𝐴) ∈ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2103 ⟶wf 6568 ‘cfv 6572 (class class class)co 7445 ℋchba 30942 +op chos 30961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-hilex 31022 ax-hfvadd 31023 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-ov 7448 df-oprab 7449 df-mpo 7450 df-map 8882 df-hosum 31753 |
This theorem is referenced by: nmoptrii 32117 |
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