lnophsi - Hilbert Space Explorer

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Theorem lnophsi 31945

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 31913	. . 3 ⊢ 𝑆: ℋ⟶ ℋ
3		lnopco.2	. . . 4 ⊢ 𝑇 ∈ LinOp
4	3	lnopfi 31913	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
5	2, 4	hoaddcli 31712	. 2 ⊢ (𝑆 +_op 𝑇): ℋ⟶ ℋ
6		hvmulcl 30957	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
7	1	lnopaddi 31915	. . . . . . . 8 ⊢ (((𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑆‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑆‘𝑧)))
8	3	lnopaddi 31915	. . . . . . . 8 ⊢ (((𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑇‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑇‘𝑧)))
9	7, 8	oveq12d 7367	. . . . . . 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 7017	. . . . . . . . 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 7017	. . . . . . . 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 7017	. . . . . . . . 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 7017	. . . . . . . 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 30980	. . . . . . 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 30956	. . . . . . 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 31684	. . . . . . 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 7017	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (𝑆‘𝑦) ∈ ℋ)
28	4	ffvelcdmi 7017	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℋ)
29	27, 28	jca 511	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → ((𝑆‘𝑦) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ))
30		ax-hvdistr1 30952	. . . . . . . . 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 31684	. . . . . . . . . 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 7365	. . . . . . . 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 31916	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑆‘(𝑥 ·_ℎ 𝑦)) = (𝑥 ·_ℎ (𝑆‘𝑦)))
38	3	lnopmuli 31916	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 ·_ℎ 𝑦)) = (𝑥 ·_ℎ (𝑇‘𝑦)))
39	37, 38	oveq12d 7367	. . . . . . 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 31684	. . . . . . 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 7368	. . . . 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 3121	. . 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 3169	. 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 31802	. 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 6478 ‘cfv 6482 (class class class)co 7349 ℂcc 11007 ℋchba 30863 +_ℎ cva 30864 ·_ℎ csm 30865 +_op chos 30882 LinOpclo 30891

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 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-hilex 30943 ax-hfvadd 30944 ax-hvcom 30945 ax-hvass 30946 ax-hv0cl 30947 ax-hvaddid 30948 ax-hfvmul 30949 ax-hvmulid 30950 ax-hvdistr1 30952 ax-hvdistr2 30953 ax-hvmul0 30954

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 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-po 5527 df-so 5528 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-er 8625 df-map 8755 df-en 8873 df-dom 8874 df-sdom 8875 df-pnf 11151 df-mnf 11152 df-ltxr 11154 df-sub 11349 df-neg 11350 df-hvsub 30915 df-hosum 31674 df-lnop 31785

This theorem is referenced by: lnophdi 31946 bdophsi 32040

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