Theorem lnopmi 32089

Description: The scalar product of a linear operator is a linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
lnopm.1	⊢ 𝑇 ∈ LinOp

Assertion

Ref	Expression
lnopmi	⊢ (𝐴 ∈ ℂ → (𝐴 ·op 𝑇) ∈ LinOp)

Proof of Theorem lnopmi

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lnopm.1	. . . 4 ⊢ 𝑇 ∈ LinOp
2	1	lnopfi 32058	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
3		homulcl 31848	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
4	2, 3	mpan2 697	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
5		hvmulcl 31102	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
6		hvaddcl 31101	. . . . . . . 8 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ)
7	5, 6	sylan 586	. . . . . . 7 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ)
8		homval 31830	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ) → ((𝐴 ·op 𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = (𝐴 ·ℎ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧))))
9	2, 8	mp3an2 1457	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ) → ((𝐴 ·op 𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = (𝐴 ·ℎ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧))))
10	7, 9	sylan2 599	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((𝐴 ·op 𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = (𝐴 ·ℎ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧))))
11		id 22	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ)
12	2	ffvelcdmi 7024	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℋ)
13		hvmulcl 31102	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ (𝑇‘𝑦) ∈ ℋ) → (𝑥 ·ℎ (𝑇‘𝑦)) ∈ ℋ)
14	12, 13	sylan2 599	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ (𝑇‘𝑦)) ∈ ℋ)
15	2	ffvelcdmi 7024	. . . . . . . . 9 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℋ)
16		ax-hvdistr1 31097	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ·ℎ (𝑇‘𝑦)) ∈ ℋ ∧ (𝑇‘𝑧) ∈ ℋ) → (𝐴 ·ℎ ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧))) = ((𝐴 ·ℎ (𝑥 ·ℎ (𝑇‘𝑦))) +ℎ (𝐴 ·ℎ (𝑇‘𝑧))))
17	11, 14, 15, 16	syl3an 1166	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝐴 ·ℎ ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧))) = ((𝐴 ·ℎ (𝑥 ·ℎ (𝑇‘𝑦))) +ℎ (𝐴 ·ℎ (𝑇‘𝑧))))
18	17	3expb 1126	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝐴 ·ℎ ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧))) = ((𝐴 ·ℎ (𝑥 ·ℎ (𝑇‘𝑦))) +ℎ (𝐴 ·ℎ (𝑇‘𝑧))))
19	1	lnopli 32057	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)))
20	19	3expa 1124	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)))
21	20	oveq2d 7372	. . . . . . . 8 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝐴 ·ℎ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧))) = (𝐴 ·ℎ ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧))))
22	21	adantl 482	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝐴 ·ℎ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧))) = (𝐴 ·ℎ ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧))))
23		homval 31830	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 ·ℎ (𝑇‘𝑦)))
24	2, 23	mp3an2 1457	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 ·ℎ (𝑇‘𝑦)))
25	24	adantrl 722	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 ·ℎ (𝑇‘𝑦)))
26	25	oveq2d 7372	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ℎ ((𝐴 ·op 𝑇)‘𝑦)) = (𝑥 ·ℎ (𝐴 ·ℎ (𝑇‘𝑦))))
27		hvmulcom 31132	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (𝑇‘𝑦) ∈ ℋ) → (𝐴 ·ℎ (𝑥 ·ℎ (𝑇‘𝑦))) = (𝑥 ·ℎ (𝐴 ·ℎ (𝑇‘𝑦))))
28	12, 27	syl3an3 1171	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝐴 ·ℎ (𝑥 ·ℎ (𝑇‘𝑦))) = (𝑥 ·ℎ (𝐴 ·ℎ (𝑇‘𝑦))))
29	28	3expb 1126	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝐴 ·ℎ (𝑥 ·ℎ (𝑇‘𝑦))) = (𝑥 ·ℎ (𝐴 ·ℎ (𝑇‘𝑦))))
30	26, 29	eqtr4d 2777	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ℎ ((𝐴 ·op 𝑇)‘𝑦)) = (𝐴 ·ℎ (𝑥 ·ℎ (𝑇‘𝑦))))
31		homval 31830	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑧 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑧) = (𝐴 ·ℎ (𝑇‘𝑧)))
32	2, 31	mp3an2 1457	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑧) = (𝐴 ·ℎ (𝑇‘𝑧)))
33	30, 32	oveqan12d 7375	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ (𝐴 ∈ ℂ ∧ 𝑧 ∈ ℋ)) → ((𝑥 ·ℎ ((𝐴 ·op 𝑇)‘𝑦)) +ℎ ((𝐴 ·op 𝑇)‘𝑧)) = ((𝐴 ·ℎ (𝑥 ·ℎ (𝑇‘𝑦))) +ℎ (𝐴 ·ℎ (𝑇‘𝑧))))
34	33	anandis 684	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((𝑥 ·ℎ ((𝐴 ·op 𝑇)‘𝑦)) +ℎ ((𝐴 ·op 𝑇)‘𝑧)) = ((𝐴 ·ℎ (𝑥 ·ℎ (𝑇‘𝑦))) +ℎ (𝐴 ·ℎ (𝑇‘𝑧))))
35	18, 22, 34	3eqtr4rd 2785	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((𝑥 ·ℎ ((𝐴 ·op 𝑇)‘𝑦)) +ℎ ((𝐴 ·op 𝑇)‘𝑧)) = (𝐴 ·ℎ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧))))
36	10, 35	eqtr4d 2777	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((𝐴 ·op 𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ ((𝐴 ·op 𝑇)‘𝑦)) +ℎ ((𝐴 ·op 𝑇)‘𝑧)))
37	36	exp32 421	. . . 4 ⊢ (𝐴 ∈ ℂ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑧 ∈ ℋ → ((𝐴 ·op 𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ ((𝐴 ·op 𝑇)‘𝑦)) +ℎ ((𝐴 ·op 𝑇)‘𝑧)))))
38	37	ralrimdv 3137	. . 3 ⊢ (𝐴 ∈ ℂ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → ∀𝑧 ∈ ℋ ((𝐴 ·op 𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ ((𝐴 ·op 𝑇)‘𝑦)) +ℎ ((𝐴 ·op 𝑇)‘𝑧))))
39	38	ralrimivv 3180	. 2 ⊢ (𝐴 ∈ ℂ → ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ ((𝐴 ·op 𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ ((𝐴 ·op 𝑇)‘𝑦)) +ℎ ((𝐴 ·op 𝑇)‘𝑧)))
40		ellnop 31947	. 2 ⊢ ((𝐴 ·op 𝑇) ∈ LinOp ↔ ((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ ((𝐴 ·op 𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ ((𝐴 ·op 𝑇)‘𝑦)) +ℎ ((𝐴 ·op 𝑇)‘𝑧))))
41	4, 39, 40	sylanbrc 589	1 ⊢ (𝐴 ∈ ℂ → (𝐴 ·op 𝑇) ∈ LinOp)

Syntax hints:   → wi 4   ∧ 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 chot 31028  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-mulcom 11093  ax-hilex 31088  ax-hfvadd 31089  ax-hfvmul 31094  ax-hvmulass 31096  ax-hvdistr1 31097

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  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-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-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-ov 7359  df-oprab 7360  df-mpo 7361  df-map 8765  df-homul 31820  df-lnop 31930

This theorem is referenced by:  lnophdi  32091  bdophmi  32121