Theorem lnopmi 30362

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 30331	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
3		homulcl 30121	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
4	2, 3	mpan2 688	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
5		hvmulcl 29375	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
6		hvaddcl 29374	. . . . . . . 8 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ)
7	5, 6	sylan 580	. . . . . . 7 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ)
8		homval 30103	. . . . . . . 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 1448	. . . . . . 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 593	. . . . . 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	ffvelrni 6960	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℋ)
13		hvmulcl 29375	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ (𝑇‘𝑦) ∈ ℋ) → (𝑥 ·ℎ (𝑇‘𝑦)) ∈ ℋ)
14	12, 13	sylan2 593	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ (𝑇‘𝑦)) ∈ ℋ)
15	2	ffvelrni 6960	. . . . . . . . 9 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℋ)
16		ax-hvdistr1 29370	. . . . . . . . 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 1159	. . . . . . . 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 1119	. . . . . . 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 30330	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)))
20	19	3expa 1117	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)))
21	20	oveq2d 7291	. . . . . . . 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 30103	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 ·ℎ (𝑇‘𝑦)))
24	2, 23	mp3an2 1448	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 ·ℎ (𝑇‘𝑦)))
25	24	adantrl 713	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 ·ℎ (𝑇‘𝑦)))
26	25	oveq2d 7291	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ℎ ((𝐴 ·op 𝑇)‘𝑦)) = (𝑥 ·ℎ (𝐴 ·ℎ (𝑇‘𝑦))))
27		hvmulcom 29405	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (𝑇‘𝑦) ∈ ℋ) → (𝐴 ·ℎ (𝑥 ·ℎ (𝑇‘𝑦))) = (𝑥 ·ℎ (𝐴 ·ℎ (𝑇‘𝑦))))
28	12, 27	syl3an3 1164	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝐴 ·ℎ (𝑥 ·ℎ (𝑇‘𝑦))) = (𝑥 ·ℎ (𝐴 ·ℎ (𝑇‘𝑦))))
29	28	3expb 1119	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝐴 ·ℎ (𝑥 ·ℎ (𝑇‘𝑦))) = (𝑥 ·ℎ (𝐴 ·ℎ (𝑇‘𝑦))))
30	26, 29	eqtr4d 2781	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ℎ ((𝐴 ·op 𝑇)‘𝑦)) = (𝐴 ·ℎ (𝑥 ·ℎ (𝑇‘𝑦))))
31		homval 30103	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑧 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑧) = (𝐴 ·ℎ (𝑇‘𝑧)))
32	2, 31	mp3an2 1448	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑧) = (𝐴 ·ℎ (𝑇‘𝑧)))
33	30, 32	oveqan12d 7294	. . . . . . . 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 675	. . . . . . 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 2789	. . . . . 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 2781	. . . . 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 3105	. . 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 3122	. 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 30220	. 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 583	1 ⊢ (𝐴 ∈ ℂ → (𝐴 ·op 𝑇) ∈ LinOp)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  ℂcc 10869   ℋchba 29281   +_ℎ cva 29282   ·_ℎ csm 29283   ·_op chot 29301  LinOpclo 29309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-mulcom 10935  ax-hilex 29361  ax-hfvadd 29362  ax-hfvmul 29367  ax-hvmulass 29369  ax-hvdistr1 29370

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-homul 30093  df-lnop 30203

This theorem is referenced by:  lnophdi  30364  bdophmi  30394