Theorem lnophsi 32090

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 32058	. . 3 ⊢ 𝑆: ℋ⟶ ℋ
3		lnopco.2	. . . 4 ⊢ 𝑇 ∈ LinOp
4	3	lnopfi 32058	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
5	2, 4	hoaddcli 31857	. 2 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ
6		hvmulcl 31102	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
7	1	lnopaddi 32060	. . . . . . . 8 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑆‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑆‘𝑧)))
8	3	lnopaddi 32060	. . . . . . . 8 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑇‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑇‘𝑧)))
9	7, 8	oveq12d 7374	. . . . . . 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 586	. . . . . 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 7024	. . . . . . . . 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 7024	. . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑆‘𝑧) ∈ ℋ)
14	12, 13	anim12i 619	. . . . . . 7 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑆‘(𝑥 ·ℎ 𝑦)) ∈ ℋ ∧ (𝑆‘𝑧) ∈ ℋ))
15	4	ffvelcdmi 7024	. . . . . . . . 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 7024	. . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℋ)
18	16, 17	anim12i 619	. . . . . . 7 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑇‘(𝑥 ·ℎ 𝑦)) ∈ ℋ ∧ (𝑇‘𝑧) ∈ ℋ))
19		hvadd4 31125	. . . . . . 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 590	. . . . . 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 2774	. . . . 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 31101	. . . . . . 7 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ)
23	6, 22	sylan 586	. . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ)
24		hosval 31829	. . . . . . 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 1459	. . . . . 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 7024	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (𝑆‘𝑦) ∈ ℋ)
28	4	ffvelcdmi 7024	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℋ)
29	27, 28	jca 516	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → ((𝑆‘𝑦) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ))
30		ax-hvdistr1 31097	. . . . . . . . 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 1126	. . . . . . . 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 599	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ ((𝑆‘𝑦) +ℎ (𝑇‘𝑦))) = ((𝑥 ·ℎ (𝑆‘𝑦)) +ℎ (𝑥 ·ℎ (𝑇‘𝑦))))
33		hosval 31829	. . . . . . . . . 10 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑦) = ((𝑆‘𝑦) +ℎ (𝑇‘𝑦)))
34	2, 4, 33	mp3an12 1459	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑦) = ((𝑆‘𝑦) +ℎ (𝑇‘𝑦)))
35	34	oveq2d 7372	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝑥 ·ℎ ((𝑆 +op 𝑇)‘𝑦)) = (𝑥 ·ℎ ((𝑆‘𝑦) +ℎ (𝑇‘𝑦))))
36	35	adantl 482	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ ((𝑆 +op 𝑇)‘𝑦)) = (𝑥 ·ℎ ((𝑆‘𝑦) +ℎ (𝑇‘𝑦))))
37	1	lnopmuli 32061	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑆‘(𝑥 ·ℎ 𝑦)) = (𝑥 ·ℎ (𝑆‘𝑦)))
38	3	lnopmuli 32061	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 ·ℎ 𝑦)) = (𝑥 ·ℎ (𝑇‘𝑦)))
39	37, 38	oveq12d 7374	. . . . . . 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 2784	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ ((𝑆 +op 𝑇)‘𝑦)) = ((𝑆‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑇‘(𝑥 ·ℎ 𝑦))))
41		hosval 31829	. . . . . . 7 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑧) = ((𝑆‘𝑧) +ℎ (𝑇‘𝑧)))
42	2, 4, 41	mp3an12 1459	. . . . . 6 ⊢ (𝑧 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑧) = ((𝑆‘𝑧) +ℎ (𝑇‘𝑧)))
43	40, 42	oveqan12d 7375	. . . . 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 2784	. . . 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 3131	. . 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 3179	. 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 31947	. 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 717	1 ⊢ (𝑆 +op 𝑇) ∈ LinOp

Syntax hints:   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  ℂcc 11027   ℋchba 31008   +_ℎ cva 31009   ·_ℎ csm 31010   +_op chos 31027  LinOpclo 31036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-hilex 31088  ax-hfvadd 31089  ax-hvcom 31090  ax-hvass 31091  ax-hv0cl 31092  ax-hvaddid 31093  ax-hfvmul 31094  ax-hvmulid 31095  ax-hvdistr1 31097  ax-hvdistr2 31098  ax-hvmul0 31099

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-po 5526  df-so 5527  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-er 8633  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11172  df-mnf 11173  df-ltxr 11175  df-sub 11370  df-neg 11371  df-hvsub 31060  df-hosum 31819  df-lnop 31930

This theorem is referenced by:  lnophdi  32091  bdophsi  32185