lnophsi - Hilbert Space Explorer

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Theorem lnophsi 32076

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 32044	. . 3 ⊢ 𝑆: ℋ⟶ ℋ
3		lnopco.2	. . . 4 ⊢ 𝑇 ∈ LinOp
4	3	lnopfi 32044	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
5	2, 4	hoaddcli 31843	. 2 ⊢ (𝑆 +_op 𝑇): ℋ⟶ ℋ
6		hvmulcl 31088	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
7	1	lnopaddi 32046	. . . . . . . 8 ⊢ (((𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑆‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑆‘𝑧)))
8	3	lnopaddi 32046	. . . . . . . 8 ⊢ (((𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑇‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑇‘𝑧)))
9	7, 8	oveq12d 7376	. . . . . . 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 7028	. . . . . . . . 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 7028	. . . . . . . 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 7028	. . . . . . . . 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 7028	. . . . . . . 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 31111	. . . . . . 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 2771	. . . . 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 31087	. . . . . . 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 31815	. . . . . . 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 7028	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (𝑆‘𝑦) ∈ ℋ)
28	4	ffvelcdmi 7028	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℋ)
29	27, 28	jca 511	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → ((𝑆‘𝑦) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ))
30		ax-hvdistr1 31083	. . . . . . . . 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 31815	. . . . . . . . . 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 7374	. . . . . . . 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 32047	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑆‘(𝑥 ·_ℎ 𝑦)) = (𝑥 ·_ℎ (𝑆‘𝑦)))
38	3	lnopmuli 32047	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 ·_ℎ 𝑦)) = (𝑥 ·_ℎ (𝑇‘𝑦)))
39	37, 38	oveq12d 7376	. . . . . . 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 2781	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ℎ ((𝑆 +_op 𝑇)‘𝑦)) = ((𝑆‘(𝑥 ·_ℎ 𝑦)) +_ℎ (𝑇‘(𝑥 ·_ℎ 𝑦))))
41		hosval 31815	. . . . . . 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 7377	. . . . 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 2781	. . . 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 3128	. . 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 3176	. 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 31933	. 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 1541 ∈ wcel 2113 ∀wral 3051 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ℂcc 11024 ℋchba 30994 +_ℎ cva 30995 ·_ℎ csm 30996 +_op chos 31013 LinOpclo 31022

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-hilex 31074 ax-hfvadd 31075 ax-hvcom 31076 ax-hvass 31077 ax-hv0cl 31078 ax-hvaddid 31079 ax-hfvmul 31080 ax-hvmulid 31081 ax-hvdistr1 31083 ax-hvdistr2 31084 ax-hvmul0 31085

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-po 5532 df-so 5533 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-er 8635 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11168 df-mnf 11169 df-ltxr 11171 df-sub 11366 df-neg 11367 df-hvsub 31046 df-hosum 31805 df-lnop 31916

This theorem is referenced by: lnophdi 32077 bdophsi 32171

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