Theorem lnopcoi 30945

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 30911	. . 3 ⊢ 𝑆: ℋ⟶ ℋ
3		lnopco.2	. . . 4 ⊢ 𝑇 ∈ LinOp
4	3	lnopfi 30911	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
5	2, 4	hocofi 30708	. 2 ⊢ (𝑆 ∘ 𝑇): ℋ⟶ ℋ
6	3	lnopli 30910	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)))
7	6	fveq2d 6846	. . . . . . 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 7034	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℋ)
10	4	ffvelcdmi 7034	. . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℋ)
11	1	lnopli 30910	. . . . . . . 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 1160	. . . . . . 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 2776	. . . . . 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 1118	. . . . 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 29955	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
16		hvaddcl 29954	. . . . . . 7 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ)
17	15, 16	sylan 580	. . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ)
18	2, 4	hocoi 30706	. . . . . 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 30706	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝑦) = (𝑆‘(𝑇‘𝑦)))
21	20	oveq2d 7373	. . . . . . 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 30706	. . . . . 6 ⊢ (𝑧 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝑧) = (𝑆‘(𝑇‘𝑧)))
24	22, 23	oveqan12d 7376	. . . . 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 2786	. . . 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 1110	. . 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 3199	. 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 30800	. 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 709	1 ⊢ (𝑆 ∘ 𝑇) ∈ LinOp

Syntax hints:   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064   ∘ ccom 5637  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  ℂcc 11049   ℋchba 29861   +_ℎ cva 29862   ·_ℎ csm 29863  LinOpclo 29889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-hilex 29941  ax-hfvadd 29942  ax-hfvmul 29947

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767  df-lnop 30783

This theorem is referenced by:  lnopco0i  30946  nmopcoi  31037  bdopcoi  31040  nmopcoadj0i  31045