lnophsi - Hilbert Space Explorer

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Theorem lnophsi 31930

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 31898	. . 3 ⊢ 𝑆: ℋ⟶ ℋ
3		lnopco.2	. . . 4 ⊢ 𝑇 ∈ LinOp
4	3	lnopfi 31898	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
5	2, 4	hoaddcli 31697	. 2 ⊢ (𝑆 +_op 𝑇): ℋ⟶ ℋ
6		hvmulcl 30942	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
7	1	lnopaddi 31900	. . . . . . . 8 ⊢ (((𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑆‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑆‘𝑧)))
8	3	lnopaddi 31900	. . . . . . . 8 ⊢ (((𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑇‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑇‘𝑧)))
9	7, 8	oveq12d 7405	. . . . . . 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 580	. . . . . 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 7055	. . . . . . . . 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 7055	. . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑆‘𝑧) ∈ ℋ)
14	12, 13	anim12i 613	. . . . . . 7 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑆‘(𝑥 ·_ℎ 𝑦)) ∈ ℋ ∧ (𝑆‘𝑧) ∈ ℋ))
15	4	ffvelcdmi 7055	. . . . . . . . 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 7055	. . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℋ)
18	16, 17	anim12i 613	. . . . . . 7 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑇‘(𝑥 ·_ℎ 𝑦)) ∈ ℋ ∧ (𝑇‘𝑧) ∈ ℋ))
19		hvadd4 30965	. . . . . . 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 584	. . . . . 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 2764	. . . . 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 30941	. . . . . . 7 ⊢ (((𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
23	6, 22	sylan 580	. . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
24		hosval 31669	. . . . . . 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 1453	. . . . . 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 7055	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (𝑆‘𝑦) ∈ ℋ)
28	4	ffvelcdmi 7055	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℋ)
29	27, 28	jca 511	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → ((𝑆‘𝑦) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ))
30		ax-hvdistr1 30937	. . . . . . . . 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 593	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ℎ ((𝑆‘𝑦) +_ℎ (𝑇‘𝑦))) = ((𝑥 ·_ℎ (𝑆‘𝑦)) +_ℎ (𝑥 ·_ℎ (𝑇‘𝑦))))
33		hosval 31669	. . . . . . . . . 10 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑆 +_op 𝑇)‘𝑦) = ((𝑆‘𝑦) +_ℎ (𝑇‘𝑦)))
34	2, 4, 33	mp3an12 1453	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → ((𝑆 +_op 𝑇)‘𝑦) = ((𝑆‘𝑦) +_ℎ (𝑇‘𝑦)))
35	34	oveq2d 7403	. . . . . . . 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 31901	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑆‘(𝑥 ·_ℎ 𝑦)) = (𝑥 ·_ℎ (𝑆‘𝑦)))
38	3	lnopmuli 31901	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 ·_ℎ 𝑦)) = (𝑥 ·_ℎ (𝑇‘𝑦)))
39	37, 38	oveq12d 7405	. . . . . . 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 2774	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ℎ ((𝑆 +_op 𝑇)‘𝑦)) = ((𝑆‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑇‘(𝑥 ·_ℎ 𝑦))))
41		hosval 31669	. . . . . . 7 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑆 +_op 𝑇)‘𝑧) = ((𝑆‘𝑧) +_ℎ (𝑇‘𝑧)))
42	2, 4, 41	mp3an12 1453	. . . . . 6 ⊢ (𝑧 ∈ ℋ → ((𝑆 +_op 𝑇)‘𝑧) = ((𝑆‘𝑧) +_ℎ (𝑇‘𝑧)))
43	40, 42	oveqan12d 7406	. . . . 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 2774	. . . 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 3125	. . 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 3177	. 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 31787	. 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 711	1 ⊢ (𝑆 +_op 𝑇) ∈ LinOp

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℋchba 30848 +_ℎ cva 30849 ·_ℎ csm 30850 +_op chos 30867 LinOpclo 30876

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 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-hilex 30928 ax-hfvadd 30929 ax-hvcom 30930 ax-hvass 30931 ax-hv0cl 30932 ax-hvaddid 30933 ax-hfvmul 30934 ax-hvmulid 30935 ax-hvdistr1 30937 ax-hvdistr2 30938 ax-hvmul0 30939

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-po 5546 df-so 5547 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-ltxr 11213 df-sub 11407 df-neg 11408 df-hvsub 30900 df-hosum 31659 df-lnop 31770

This theorem is referenced by: lnophdi 31931 bdophsi 32025

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