Theorem lnopmi 32603

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 32572	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
3		homulcl 32362	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
4	2, 3	mpan2 704	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
5		hvmulcl 31616	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
6		hvaddcl 31615	. . . . . . . 8 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ)
7	5, 6	sylan 592	. . . . . . 7 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ)
8		homval 32344	. . . . . . . 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 1478	. . . . . . 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 605	. . . . . 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 23	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ)
12	2	ffvelcdmi 7083	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℋ)
13		hvmulcl 31616	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ (𝑇‘𝑦) ∈ ℋ) → (𝑥 ·ℎ (𝑇‘𝑦)) ∈ ℋ)
14	12, 13	sylan2 605	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ (𝑇‘𝑦)) ∈ ℋ)
15	2	ffvelcdmi 7083	. . . . . . . . 9 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℋ)
16		ax-hvdistr1 31611	. . . . . . . . 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 1178	. . . . . . . 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 1138	. . . . . . 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 32571	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)))
20	19	3expa 1136	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)))
21	20	oveq2d 7436	. . . . . . . 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 487	. . . . . . 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 32344	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 ·ℎ (𝑇‘𝑦)))
24	2, 23	mp3an2 1478	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 ·ℎ (𝑇‘𝑦)))
25	24	adantrl 729	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 ·ℎ (𝑇‘𝑦)))
26	25	oveq2d 7436	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ℎ ((𝐴 ·op 𝑇)‘𝑦)) = (𝑥 ·ℎ (𝐴 ·ℎ (𝑇‘𝑦))))
27		hvmulcom 31646	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (𝑇‘𝑦) ∈ ℋ) → (𝐴 ·ℎ (𝑥 ·ℎ (𝑇‘𝑦))) = (𝑥 ·ℎ (𝐴 ·ℎ (𝑇‘𝑦))))
28	12, 27	syl3an3 1183	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝐴 ·ℎ (𝑥 ·ℎ (𝑇‘𝑦))) = (𝑥 ·ℎ (𝐴 ·ℎ (𝑇‘𝑦))))
29	28	3expb 1138	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝐴 ·ℎ (𝑥 ·ℎ (𝑇‘𝑦))) = (𝑥 ·ℎ (𝐴 ·ℎ (𝑇‘𝑦))))
30	26, 29	eqtr4d 2798	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ℎ ((𝐴 ·op 𝑇)‘𝑦)) = (𝐴 ·ℎ (𝑥 ·ℎ (𝑇‘𝑦))))
31		homval 32344	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑧 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑧) = (𝐴 ·ℎ (𝑇‘𝑧)))
32	2, 31	mp3an2 1478	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑧) = (𝐴 ·ℎ (𝑇‘𝑧)))
33	30, 32	oveqan12d 7439	. . . . . . . 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 691	. . . . . . 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 2806	. . . . . 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 2798	. . . . 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 426	. . . 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 3160	. . 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 3203	. 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 32461	. 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 595	1 ⊢ (𝐴 ∈ ℂ → (𝐴 ·op 𝑇) ∈ LinOp)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  ℂcc 11199   ℋchba 31522   +_ℎ cva 31523   ·_ℎ csm 31524   ·_op chot 31542  LinOpclo 31550

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-mulcom 11265  ax-hilex 31602  ax-hfvadd 31603  ax-hfvmul 31608  ax-hvmulass 31610  ax-hvdistr1 31611

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8850  df-homul 32334  df-lnop 32444

This theorem is used by:  lnophdi  32605  bdophmi  32635