Theorem lnopmi 32352

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 32321	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
3		homulcl 32111	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
4	2, 3	mpan2 703	. 2 ⊢ (𝐴 ∈ ℂ → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
5		hvmulcl 31365	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
6		hvaddcl 31364	. . . . . . . 8 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ)
7	5, 6	sylan 591	. . . . . . 7 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ)
8		homval 32093	. . . . . . . 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 604	. . . . . 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 7078	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℋ)
13		hvmulcl 31365	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ (𝑇‘𝑦) ∈ ℋ) → (𝑥 ·ℎ (𝑇‘𝑦)) ∈ ℋ)
14	12, 13	sylan2 604	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ (𝑇‘𝑦)) ∈ ℋ)
15	2	ffvelcdmi 7078	. . . . . . . . 9 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℋ)
16		ax-hvdistr1 31360	. . . . . . . . 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 32320	. . . . . . . . . 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 7426	. . . . . . . 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 486	. . . . . . 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 32093	. . . . . . . . . . . . 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 728	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 ·ℎ (𝑇‘𝑦)))
26	25	oveq2d 7426	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ℎ ((𝐴 ·op 𝑇)‘𝑦)) = (𝑥 ·ℎ (𝐴 ·ℎ (𝑇‘𝑦))))
27		hvmulcom 31395	. . . . . . . . . . . 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 2801	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ℎ ((𝐴 ·op 𝑇)‘𝑦)) = (𝐴 ·ℎ (𝑥 ·ℎ (𝑇‘𝑦))))
31		homval 32093	. . . . . . . . . 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 7429	. . . . . . . 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 690	. . . . . . 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 2809	. . . . . 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 2801	. . . . 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 425	. . . 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 3163	. . 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 3206	. 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 32210	. 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 594	1 ⊢ (𝐴 ∈ ℂ → (𝐴 ·op 𝑇) ∈ LinOp)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  ℂcc 11093   ℋchba 31271   +_ℎ cva 31272   ·_ℎ csm 31273   ·_op chot 31291  LinOpclo 31299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-mulcom 11159  ax-hilex 31351  ax-hfvadd 31352  ax-hfvmul 31357  ax-hvmulass 31359  ax-hvdistr1 31360

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-map 8822  df-homul 32083  df-lnop 32193

This theorem is referenced by:  lnophdi  32354  bdophmi  32384