lnophsi - Hilbert Space Explorer

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

Theorem lnophsi 29778

Description: The sum of two linear operators is linear. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
lnopco.1	⊢ 𝑆 ∈ LinOp
lnopco.2	⊢ 𝑇 ∈ LinOp

**Assertion**
Ref	Expression
lnophsi	⊢ (𝑆 +_op 𝑇) ∈ LinOp

Proof of Theorem lnophsi Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lnopco.1	. . . 4 ⊢ 𝑆 ∈ LinOp
2	1	lnopfi 29746	. . 3 ⊢ 𝑆: ℋ⟶ ℋ
3		lnopco.2	. . . 4 ⊢ 𝑇 ∈ LinOp
4	3	lnopfi 29746	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
5	2, 4	hoaddcli 29545	. 2 ⊢ (𝑆 +_op 𝑇): ℋ⟶ ℋ
6		hvmulcl 28790	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
7	1	lnopaddi 29748	. . . . . . . 8 ⊢ (((𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑆‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑆‘𝑧)))
8	3	lnopaddi 29748	. . . . . . . 8 ⊢ (((𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑇‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑇‘𝑧)))
9	7, 8	oveq12d 7174	. . . . . . 7 ⊢ (((𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑆‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) +_ℎ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧))) = (((𝑆‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑆‘𝑧)) +_ℎ ((𝑇‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑇‘𝑧))))
10	6, 9	sylan 582	. . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑆‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) +_ℎ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧))) = (((𝑆‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑆‘𝑧)) +_ℎ ((𝑇‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑇‘𝑧))))
11	2	ffvelrni 6850	. . . . . . . . 9 ⊢ ((𝑥 ·_ℎ 𝑦) ∈ ℋ → (𝑆‘(𝑥 ·_ℎ 𝑦)) ∈ ℋ)
12	6, 11	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑆‘(𝑥 ·_ℎ 𝑦)) ∈ ℋ)
13	2	ffvelrni 6850	. . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑆‘𝑧) ∈ ℋ)
14	12, 13	anim12i 614	. . . . . . 7 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑆‘(𝑥 ·_ℎ 𝑦)) ∈ ℋ ∧ (𝑆‘𝑧) ∈ ℋ))
15	4	ffvelrni 6850	. . . . . . . . 9 ⊢ ((𝑥 ·_ℎ 𝑦) ∈ ℋ → (𝑇‘(𝑥 ·_ℎ 𝑦)) ∈ ℋ)
16	6, 15	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 ·_ℎ 𝑦)) ∈ ℋ)
17	4	ffvelrni 6850	. . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℋ)
18	16, 17	anim12i 614	. . . . . . 7 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑇‘(𝑥 ·_ℎ 𝑦)) ∈ ℋ ∧ (𝑇‘𝑧) ∈ ℋ))
19		hvadd4 28813	. . . . . . 7 ⊢ ((((𝑆‘(𝑥 ·_ℎ 𝑦)) ∈ ℋ ∧ (𝑆‘𝑧) ∈ ℋ) ∧ ((𝑇‘(𝑥 ·_ℎ 𝑦)) ∈ ℋ ∧ (𝑇‘𝑧) ∈ ℋ)) → (((𝑆‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑆‘𝑧)) +_ℎ ((𝑇‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑇‘𝑧))) = (((𝑆‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑇‘(𝑥 ·_ℎ 𝑦))) +_ℎ ((𝑆‘𝑧) +_ℎ (𝑇‘𝑧))))
20	14, 18, 19	syl2anc 586	. . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (((𝑆‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑆‘𝑧)) +_ℎ ((𝑇‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑇‘𝑧))) = (((𝑆‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑇‘(𝑥 ·_ℎ 𝑦))) +_ℎ ((𝑆‘𝑧) +_ℎ (𝑇‘𝑧))))
21	10, 20	eqtrd 2856	. . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑆‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) +_ℎ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧))) = (((𝑆‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑇‘(𝑥 ·_ℎ 𝑦))) +_ℎ ((𝑆‘𝑧) +_ℎ (𝑇‘𝑧))))
22		hvaddcl 28789	. . . . . . 7 ⊢ (((𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
23	6, 22	sylan 582	. . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
24		hosval 29517	. . . . . . 7 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ) → ((𝑆 +_op 𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑆‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) +_ℎ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧))))
25	2, 4, 24	mp3an12 1447	. . . . . 6 ⊢ (((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ → ((𝑆 +_op 𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑆‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) +_ℎ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧))))
26	23, 25	syl 17	. . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑆 +_op 𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑆‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) +_ℎ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧))))
27	2	ffvelrni 6850	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (𝑆‘𝑦) ∈ ℋ)
28	4	ffvelrni 6850	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℋ)
29	27, 28	jca 514	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → ((𝑆‘𝑦) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ))
30		ax-hvdistr1 28785	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ (𝑆‘𝑦) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → (𝑥 ·_ℎ ((𝑆‘𝑦) +_ℎ (𝑇‘𝑦))) = ((𝑥 ·_ℎ (𝑆‘𝑦)) +_ℎ (𝑥 ·_ℎ (𝑇‘𝑦))))
31	30	3expb 1116	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ ((𝑆‘𝑦) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ)) → (𝑥 ·_ℎ ((𝑆‘𝑦) +_ℎ (𝑇‘𝑦))) = ((𝑥 ·_ℎ (𝑆‘𝑦)) +_ℎ (𝑥 ·_ℎ (𝑇‘𝑦))))
32	29, 31	sylan2 594	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ℎ ((𝑆‘𝑦) +_ℎ (𝑇‘𝑦))) = ((𝑥 ·_ℎ (𝑆‘𝑦)) +_ℎ (𝑥 ·_ℎ (𝑇‘𝑦))))
33		hosval 29517	. . . . . . . . . 10 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑆 +_op 𝑇)‘𝑦) = ((𝑆‘𝑦) +_ℎ (𝑇‘𝑦)))
34	2, 4, 33	mp3an12 1447	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → ((𝑆 +_op 𝑇)‘𝑦) = ((𝑆‘𝑦) +_ℎ (𝑇‘𝑦)))
35	34	oveq2d 7172	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝑥 ·_ℎ ((𝑆 +_op 𝑇)‘𝑦)) = (𝑥 ·_ℎ ((𝑆‘𝑦) +_ℎ (𝑇‘𝑦))))
36	35	adantl 484	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ℎ ((𝑆 +_op 𝑇)‘𝑦)) = (𝑥 ·_ℎ ((𝑆‘𝑦) +_ℎ (𝑇‘𝑦))))
37	1	lnopmuli 29749	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑆‘(𝑥 ·_ℎ 𝑦)) = (𝑥 ·_ℎ (𝑆‘𝑦)))
38	3	lnopmuli 29749	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 ·_ℎ 𝑦)) = (𝑥 ·_ℎ (𝑇‘𝑦)))
39	37, 38	oveq12d 7174	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → ((𝑆‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑇‘(𝑥 ·_ℎ 𝑦))) = ((𝑥 ·_ℎ (𝑆‘𝑦)) +_ℎ (𝑥 ·_ℎ (𝑇‘𝑦))))
40	32, 36, 39	3eqtr4d 2866	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ℎ ((𝑆 +_op 𝑇)‘𝑦)) = ((𝑆‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑇‘(𝑥 ·_ℎ 𝑦))))
41		hosval 29517	. . . . . . 7 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑆 +_op 𝑇)‘𝑧) = ((𝑆‘𝑧) +_ℎ (𝑇‘𝑧)))
42	2, 4, 41	mp3an12 1447	. . . . . 6 ⊢ (𝑧 ∈ ℋ → ((𝑆 +_op 𝑇)‘𝑧) = ((𝑆‘𝑧) +_ℎ (𝑇‘𝑧)))
43	40, 42	oveqan12d 7175	. . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ ((𝑆 +_op 𝑇)‘𝑦)) +_ℎ ((𝑆 +_op 𝑇)‘𝑧)) = (((𝑆‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑇‘(𝑥 ·_ℎ 𝑦))) +_ℎ ((𝑆‘𝑧) +_ℎ (𝑇‘𝑧))))
44	21, 26, 43	3eqtr4d 2866	. . . 4 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑆 +_op 𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ ((𝑆 +_op 𝑇)‘𝑦)) +_ℎ ((𝑆 +_op 𝑇)‘𝑧)))
45	44	ralrimiva 3182	. . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → ∀𝑧 ∈ ℋ ((𝑆 +_op 𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ ((𝑆 +_op 𝑇)‘𝑦)) +_ℎ ((𝑆 +_op 𝑇)‘𝑧)))
46	45	rgen2 3203	. 2 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑆 +_op 𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ ((𝑆 +_op 𝑇)‘𝑦)) +_ℎ ((𝑆 +_op 𝑇)‘𝑧))
47		ellnop 29635	. 2 ⊢ ((𝑆 +_op 𝑇) ∈ LinOp ↔ ((𝑆 +_op 𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑆 +_op 𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ ((𝑆 +_op 𝑇)‘𝑦)) +_ℎ ((𝑆 +_op 𝑇)‘𝑧))))
48	5, 46, 47	mpbir2an 709	1 ⊢ (𝑆 +_op 𝑇) ∈ LinOp

Colors of variables: wff setvar class

Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℋchba 28696 +_ℎ cva 28697 ·_ℎ csm 28698 +_op chos 28715 LinOpclo 28724

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-hilex 28776 ax-hfvadd 28777 ax-hvcom 28778 ax-hvass 28779 ax-hv0cl 28780 ax-hvaddid 28781 ax-hfvmul 28782 ax-hvmulid 28783 ax-hvdistr1 28785 ax-hvdistr2 28786 ax-hvmul0 28787

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-sub 10872 df-neg 10873 df-hvsub 28748 df-hosum 29507 df-lnop 29618

This theorem is referenced by: lnophdi 29779 bdophsi 29873

W3C validator