Theorem lnopmi 31936

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 31905	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
3		homulcl 31695	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
4	2, 3	mpan2 691	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
5		hvmulcl 30949	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
6		hvaddcl 30948	. . . . . . . 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 31677	. . . . . . . 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 1451	. . . . . . 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	ffvelcdmi 7058	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℋ)
13		hvmulcl 30949	. . . . . . . . . 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	ffvelcdmi 7058	. . . . . . . . 9 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℋ)
16		ax-hvdistr1 30944	. . . . . . . . 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 1160	. . . . . . . 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 1120	. . . . . . 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 31904	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)))
20	19	3expa 1118	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)))
21	20	oveq2d 7406	. . . . . . . 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 481	. . . . . . 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 31677	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 ·ℎ (𝑇‘𝑦)))
24	2, 23	mp3an2 1451	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 ·ℎ (𝑇‘𝑦)))
25	24	adantrl 716	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 ·ℎ (𝑇‘𝑦)))
26	25	oveq2d 7406	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ℎ ((𝐴 ·op 𝑇)‘𝑦)) = (𝑥 ·ℎ (𝐴 ·ℎ (𝑇‘𝑦))))
27		hvmulcom 30979	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (𝑇‘𝑦) ∈ ℋ) → (𝐴 ·ℎ (𝑥 ·ℎ (𝑇‘𝑦))) = (𝑥 ·ℎ (𝐴 ·ℎ (𝑇‘𝑦))))
28	12, 27	syl3an3 1165	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝐴 ·ℎ (𝑥 ·ℎ (𝑇‘𝑦))) = (𝑥 ·ℎ (𝐴 ·ℎ (𝑇‘𝑦))))
29	28	3expb 1120	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝐴 ·ℎ (𝑥 ·ℎ (𝑇‘𝑦))) = (𝑥 ·ℎ (𝐴 ·ℎ (𝑇‘𝑦))))
30	26, 29	eqtr4d 2768	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ℎ ((𝐴 ·op 𝑇)‘𝑦)) = (𝐴 ·ℎ (𝑥 ·ℎ (𝑇‘𝑦))))
31		homval 31677	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑧 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑧) = (𝐴 ·ℎ (𝑇‘𝑧)))
32	2, 31	mp3an2 1451	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑧) = (𝐴 ·ℎ (𝑇‘𝑧)))
33	30, 32	oveqan12d 7409	. . . . . . . 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 678	. . . . . . 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 2776	. . . . . 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 2768	. . . . 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 420	. . . 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 3132	. . 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 3179	. 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 31794	. 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 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  ℂcc 11073   ℋchba 30855   +_ℎ cva 30856   ·_ℎ csm 30857   ·_op chot 30875  LinOpclo 30883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-mulcom 11139  ax-hilex 30935  ax-hfvadd 30936  ax-hfvmul 30941  ax-hvmulass 30943  ax-hvdistr1 30944

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8804  df-homul 31667  df-lnop 31777

This theorem is referenced by:  lnophdi  31938  bdophmi  31968