lnophsi - Hilbert Space Explorer

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Theorem lnophsi 32033

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 32001	. . 3 ⊢ 𝑆: ℋ⟶ ℋ
3		lnopco.2	. . . 4 ⊢ 𝑇 ∈ LinOp
4	3	lnopfi 32001	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
5	2, 4	hoaddcli 31800	. 2 ⊢ (𝑆 +_op 𝑇): ℋ⟶ ℋ
6		hvmulcl 31045	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
7	1	lnopaddi 32003	. . . . . . . 8 ⊢ (((𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑆‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑆‘𝑧)))
8	3	lnopaddi 32003	. . . . . . . 8 ⊢ (((𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑇‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑇‘𝑧)))
9	7, 8	oveq12d 7466	. . . . . . 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 579	. . . . . 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	ffvelcdmi 7117	. . . . . . . . 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	ffvelcdmi 7117	. . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑆‘𝑧) ∈ ℋ)
14	12, 13	anim12i 612	. . . . . . 7 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑆‘(𝑥 ·_ℎ 𝑦)) ∈ ℋ ∧ (𝑆‘𝑧) ∈ ℋ))
15	4	ffvelcdmi 7117	. . . . . . . . 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	ffvelcdmi 7117	. . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℋ)
18	16, 17	anim12i 612	. . . . . . 7 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑇‘(𝑥 ·_ℎ 𝑦)) ∈ ℋ ∧ (𝑇‘𝑧) ∈ ℋ))
19		hvadd4 31068	. . . . . . 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 583	. . . . . 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 2780	. . . . 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 31044	. . . . . . 7 ⊢ (((𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
23	6, 22	sylan 579	. . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
24		hosval 31772	. . . . . . 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 1451	. . . . . 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	ffvelcdmi 7117	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (𝑆‘𝑦) ∈ ℋ)
28	4	ffvelcdmi 7117	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℋ)
29	27, 28	jca 511	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → ((𝑆‘𝑦) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ))
30		ax-hvdistr1 31040	. . . . . . . . 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 1120	. . . . . . . 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 592	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ℎ ((𝑆‘𝑦) +_ℎ (𝑇‘𝑦))) = ((𝑥 ·_ℎ (𝑆‘𝑦)) +_ℎ (𝑥 ·_ℎ (𝑇‘𝑦))))
33		hosval 31772	. . . . . . . . . 10 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑆 +_op 𝑇)‘𝑦) = ((𝑆‘𝑦) +_ℎ (𝑇‘𝑦)))
34	2, 4, 33	mp3an12 1451	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → ((𝑆 +_op 𝑇)‘𝑦) = ((𝑆‘𝑦) +_ℎ (𝑇‘𝑦)))
35	34	oveq2d 7464	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝑥 ·_ℎ ((𝑆 +_op 𝑇)‘𝑦)) = (𝑥 ·_ℎ ((𝑆‘𝑦) +_ℎ (𝑇‘𝑦))))
36	35	adantl 481	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ℎ ((𝑆 +_op 𝑇)‘𝑦)) = (𝑥 ·_ℎ ((𝑆‘𝑦) +_ℎ (𝑇‘𝑦))))
37	1	lnopmuli 32004	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑆‘(𝑥 ·_ℎ 𝑦)) = (𝑥 ·_ℎ (𝑆‘𝑦)))
38	3	lnopmuli 32004	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 ·_ℎ 𝑦)) = (𝑥 ·_ℎ (𝑇‘𝑦)))
39	37, 38	oveq12d 7466	. . . . . . 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 2790	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ℎ ((𝑆 +_op 𝑇)‘𝑦)) = ((𝑆‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑇‘(𝑥 ·_ℎ 𝑦))))
41		hosval 31772	. . . . . . 7 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑆 +_op 𝑇)‘𝑧) = ((𝑆‘𝑧) +_ℎ (𝑇‘𝑧)))
42	2, 4, 41	mp3an12 1451	. . . . . 6 ⊢ (𝑧 ∈ ℋ → ((𝑆 +_op 𝑇)‘𝑧) = ((𝑆‘𝑧) +_ℎ (𝑇‘𝑧)))
43	40, 42	oveqan12d 7467	. . . . 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 2790	. . . 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 3152	. . 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 3205	. 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 31890	. 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 710	1 ⊢ (𝑆 +_op 𝑇) ∈ LinOp

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℋchba 30951 +_ℎ cva 30952 ·_ℎ csm 30953 +_op chos 30970 LinOpclo 30979

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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-hilex 31031 ax-hfvadd 31032 ax-hvcom 31033 ax-hvass 31034 ax-hv0cl 31035 ax-hvaddid 31036 ax-hfvmul 31037 ax-hvmulid 31038 ax-hvdistr1 31040 ax-hvdistr2 31041 ax-hvmul0 31042

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-ltxr 11329 df-sub 11522 df-neg 11523 df-hvsub 31003 df-hosum 31762 df-lnop 31873

This theorem is referenced by: lnophdi 32034 bdophsi 32128

W3C validator