hoscl - Hilbert Space Explorer

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Theorem hoscl 31263

Description: Closure of the sum of two Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)

**Assertion**
Ref	Expression
hoscl	⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝐴 ∈ ℋ) → ((𝑆 +_op 𝑇)‘𝐴) ∈ ℋ)

**Proof of Theorem hoscl**
Step	Hyp	Ref	Expression
1		hosval 31258	. . 3 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑆 +_op 𝑇)‘𝐴) = ((𝑆‘𝐴) +_ℎ (𝑇‘𝐴)))
2	1	3expa 1116	. 2 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝐴 ∈ ℋ) → ((𝑆 +_op 𝑇)‘𝐴) = ((𝑆‘𝐴) +_ℎ (𝑇‘𝐴)))
3		ffvelcdm 7084	. . . . 5 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → (𝑆‘𝐴) ∈ ℋ)
4		ffvelcdm 7084	. . . . 5 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → (𝑇‘𝐴) ∈ ℋ)
5	3, 4	anim12i 611	. . . 4 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ)) → ((𝑆‘𝐴) ∈ ℋ ∧ (𝑇‘𝐴) ∈ ℋ))
6	5	anandirs 675	. . 3 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝐴 ∈ ℋ) → ((𝑆‘𝐴) ∈ ℋ ∧ (𝑇‘𝐴) ∈ ℋ))
7		hvaddcl 30530	. . 3 ⊢ (((𝑆‘𝐴) ∈ ℋ ∧ (𝑇‘𝐴) ∈ ℋ) → ((𝑆‘𝐴) +_ℎ (𝑇‘𝐴)) ∈ ℋ)
8	6, 7	syl 17	. 2 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝐴 ∈ ℋ) → ((𝑆‘𝐴) +_ℎ (𝑇‘𝐴)) ∈ ℋ)
9	2, 8	eqeltrd 2831	1 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝐴 ∈ ℋ) → ((𝑆 +_op 𝑇)‘𝐴) ∈ ℋ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ⟶wf 6540 ‘cfv 6544 (class class class)co 7413 ℋchba 30437 +_ℎ cva 30438 +_op chos 30456

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-hilex 30517 ax-hfvadd 30518

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7416 df-oprab 7417 df-mpo 7418 df-map 8826 df-hosum 31248

This theorem is referenced by: hoscli 31280