hosmval - Hilbert Space Explorer

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

Theorem hosmval 31608

Description: Value of the sum of two Hilbert space operators. (Contributed by NM, 9-Nov-2000.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)

**Assertion**
Ref	Expression
hosmval	⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +_op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +_ℎ (𝑇‘𝑥))))

Distinct variable groups: 𝑥,𝑆 𝑥,𝑇

Proof of Theorem hosmval Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-hilex 30872	. . 3 ⊢ ℋ ∈ V
2	1, 1	elmap 8888	. 2 ⊢ (𝑆 ∈ ( ℋ ↑_m ℋ) ↔ 𝑆: ℋ⟶ ℋ)
3	1, 1	elmap 8888	. 2 ⊢ (𝑇 ∈ ( ℋ ↑_m ℋ) ↔ 𝑇: ℋ⟶ ℋ)
4		fveq1 6893	. . . . 5 ⊢ (𝑓 = 𝑆 → (𝑓‘𝑥) = (𝑆‘𝑥))
5	4	oveq1d 7432	. . . 4 ⊢ (𝑓 = 𝑆 → ((𝑓‘𝑥) +_ℎ (𝑔‘𝑥)) = ((𝑆‘𝑥) +_ℎ (𝑔‘𝑥)))
6	5	mpteq2dv 5250	. . 3 ⊢ (𝑓 = 𝑆 → (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) +_ℎ (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +_ℎ (𝑔‘𝑥))))
7		fveq1 6893	. . . . 5 ⊢ (𝑔 = 𝑇 → (𝑔‘𝑥) = (𝑇‘𝑥))
8	7	oveq2d 7433	. . . 4 ⊢ (𝑔 = 𝑇 → ((𝑆‘𝑥) +_ℎ (𝑔‘𝑥)) = ((𝑆‘𝑥) +_ℎ (𝑇‘𝑥)))
9	8	mpteq2dv 5250	. . 3 ⊢ (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +_ℎ (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +_ℎ (𝑇‘𝑥))))
10		df-hosum 31603	. . 3 ⊢ +_op = (𝑓 ∈ ( ℋ ↑_m ℋ), 𝑔 ∈ ( ℋ ↑_m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) +_ℎ (𝑔‘𝑥))))
11	1	mptex 7233	. . 3 ⊢ (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +_ℎ (𝑇‘𝑥))) ∈ V
12	6, 9, 10, 11	ovmpo 7579	. 2 ⊢ ((𝑆 ∈ ( ℋ ↑_m ℋ) ∧ 𝑇 ∈ ( ℋ ↑_m ℋ)) → (𝑆 +_op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +_ℎ (𝑇‘𝑥))))
13	2, 3, 12	syl2anbr 597	1 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +_op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +_ℎ (𝑇‘𝑥))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ↦ cmpt 5231 ⟶wf 6543 ‘cfv 6547 (class class class)co 7417 ↑_m cmap 8843 ℋchba 30792 +_ℎ cva 30793 +_op chos 30811

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-hilex 30872

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 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 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-ov 7420 df-oprab 7421 df-mpo 7422 df-map 8845 df-hosum 31603

This theorem is referenced by: hosval 31613 hoaddcl 31631

W3C validator

	
		OSZAR »