Theorem lnopcoi 32092

Description: The composition of two linear operators is linear. (Contributed by NM, 8-Mar-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lnopco.1	⊢ 𝑆 ∈ LinOp
lnopco.2	⊢ 𝑇 ∈ LinOp

Assertion

Ref	Expression
lnopcoi	⊢ (𝑆 ∘ 𝑇) ∈ LinOp

Proof of Theorem lnopcoi

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

Step	Hyp	Ref	Expression
1		lnopco.1	. . . 4 ⊢ 𝑆 ∈ LinOp
2	1	lnopfi 32058	. . 3 ⊢ 𝑆: ℋ⟶ ℋ
3		lnopco.2	. . . 4 ⊢ 𝑇 ∈ LinOp
4	3	lnopfi 32058	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
5	2, 4	hocofi 31855	. 2 ⊢ (𝑆 ∘ 𝑇): ℋ⟶ ℋ
6	3	lnopli 32057	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)))
7	6	fveq2d 6831	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘(𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧))) = (𝑆‘((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧))))
8		id 22	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ ℂ)
9	4	ffvelcdmi 7024	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℋ)
10	4	ffvelcdmi 7024	. . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℋ)
11	1	lnopli 32057	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ (𝑇‘𝑦) ∈ ℋ ∧ (𝑇‘𝑧) ∈ ℋ) → (𝑆‘((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧))) = ((𝑥 ·ℎ (𝑆‘(𝑇‘𝑦))) +ℎ (𝑆‘(𝑇‘𝑧))))
12	8, 9, 10, 11	syl3an 1166	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧))) = ((𝑥 ·ℎ (𝑆‘(𝑇‘𝑦))) +ℎ (𝑆‘(𝑇‘𝑧))))
13	7, 12	eqtrd 2774	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘(𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧))) = ((𝑥 ·ℎ (𝑆‘(𝑇‘𝑦))) +ℎ (𝑆‘(𝑇‘𝑧))))
14	13	3expa 1124	. . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝑆‘(𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧))) = ((𝑥 ·ℎ (𝑆‘(𝑇‘𝑦))) +ℎ (𝑆‘(𝑇‘𝑧))))
15		hvmulcl 31102	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
16		hvaddcl 31101	. . . . . . 7 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ)
17	15, 16	sylan 586	. . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ)
18	2, 4	hocoi 31853	. . . . . 6 ⊢ (((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ → ((𝑆 ∘ 𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = (𝑆‘(𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧))))
19	17, 18	syl 17	. . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑆 ∘ 𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = (𝑆‘(𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧))))
20	2, 4	hocoi 31853	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝑦) = (𝑆‘(𝑇‘𝑦)))
21	20	oveq2d 7372	. . . . . . 7 ⊢ (𝑦 ∈ ℋ → (𝑥 ·ℎ ((𝑆 ∘ 𝑇)‘𝑦)) = (𝑥 ·ℎ (𝑆‘(𝑇‘𝑦))))
22	21	adantl 482	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ ((𝑆 ∘ 𝑇)‘𝑦)) = (𝑥 ·ℎ (𝑆‘(𝑇‘𝑦))))
23	2, 4	hocoi 31853	. . . . . 6 ⊢ (𝑧 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝑧) = (𝑆‘(𝑇‘𝑧)))
24	22, 23	oveqan12d 7375	. . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ ((𝑆 ∘ 𝑇)‘𝑦)) +ℎ ((𝑆 ∘ 𝑇)‘𝑧)) = ((𝑥 ·ℎ (𝑆‘(𝑇‘𝑦))) +ℎ (𝑆‘(𝑇‘𝑧))))
25	14, 19, 24	3eqtr4d 2784	. . . 4 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑆 ∘ 𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ ((𝑆 ∘ 𝑇)‘𝑦)) +ℎ ((𝑆 ∘ 𝑇)‘𝑧)))
26	25	3impa 1115	. . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑆 ∘ 𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ ((𝑆 ∘ 𝑇)‘𝑦)) +ℎ ((𝑆 ∘ 𝑇)‘𝑧)))
27	26	rgen3 3184	. 2 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑆 ∘ 𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ ((𝑆 ∘ 𝑇)‘𝑦)) +ℎ ((𝑆 ∘ 𝑇)‘𝑧))
28		ellnop 31947	. 2 ⊢ ((𝑆 ∘ 𝑇) ∈ LinOp ↔ ((𝑆 ∘ 𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑆 ∘ 𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ ((𝑆 ∘ 𝑇)‘𝑦)) +ℎ ((𝑆 ∘ 𝑇)‘𝑧))))
29	5, 27, 28	mpbir2an 717	1 ⊢ (𝑆 ∘ 𝑇) ∈ LinOp

Syntax hints:   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053   ∘ ccom 5622  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  ℂcc 11027   ℋchba 31008   +_ℎ cva 31009   ·_ℎ csm 31010  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-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-hilex 31088  ax-hfvadd 31089  ax-hfvmul 31094

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-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-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-map 8765  df-lnop 31930

This theorem is referenced by:  lnopco0i  32093  nmopcoi  32184  bdopcoi  32187  nmopcoadj0i  32192