hoscl - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > hoscl		Structured version Visualization version GIF version

Theorem hoscl 30107

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 30102	. . 3 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑆 +_op 𝑇)‘𝐴) = ((𝑆‘𝐴) +_ℎ (𝑇‘𝐴)))
2	1	3expa 1117	. 2 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝐴 ∈ ℋ) → ((𝑆 +_op 𝑇)‘𝐴) = ((𝑆‘𝐴) +_ℎ (𝑇‘𝐴)))
3		ffvelrn 6959	. . . . 5 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → (𝑆‘𝐴) ∈ ℋ)
4		ffvelrn 6959	. . . . 5 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → (𝑇‘𝐴) ∈ ℋ)
5	3, 4	anim12i 613	. . . 4 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ)) → ((𝑆‘𝐴) ∈ ℋ ∧ (𝑇‘𝐴) ∈ ℋ))
6	5	anandirs 676	. . 3 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝐴 ∈ ℋ) → ((𝑆‘𝐴) ∈ ℋ ∧ (𝑇‘𝐴) ∈ ℋ))
7		hvaddcl 29374	. . 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 2839	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 396 = wceq 1539 ∈ wcel 2106 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ℋchba 29281 +_ℎ cva 29282 +_op chos 29300

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-hilex 29361 ax-hfvadd 29362

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-map 8617 df-hosum 30092

This theorem is referenced by: hoscli 30124