Theorem lnopcoi 32032

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 31998	. . 3 ⊢ 𝑆: ℋ⟶ ℋ
3		lnopco.2	. . . 4 ⊢ 𝑇 ∈ LinOp
4	3	lnopfi 31998	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
5	2, 4	hocofi 31795	. 2 ⊢ (𝑆 ∘ 𝑇): ℋ⟶ ℋ
6	3	lnopli 31997	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)))
7	6	fveq2d 6911	. . . . . . 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 7103	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℋ)
10	4	ffvelcdmi 7103	. . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℋ)
11	1	lnopli 31997	. . . . . . . 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 1159	. . . . . . 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 2775	. . . . . 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 1117	. . . . 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 31042	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
16		hvaddcl 31041	. . . . . . 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 31793	. . . . . 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 31793	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝑦) = (𝑆‘(𝑇‘𝑦)))
21	20	oveq2d 7447	. . . . . . 7 ⊢ (𝑦 ∈ ℋ → (𝑥 ·ℎ ((𝑆 ∘ 𝑇)‘𝑦)) = (𝑥 ·ℎ (𝑆‘(𝑇‘𝑦))))
22	21	adantl 481	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ ((𝑆 ∘ 𝑇)‘𝑦)) = (𝑥 ·ℎ (𝑆‘(𝑇‘𝑦))))
23	2, 4	hocoi 31793	. . . . . 6 ⊢ (𝑧 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝑧) = (𝑆‘(𝑇‘𝑧)))
24	22, 23	oveqan12d 7450	. . . . 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 2785	. . . 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 1109	. . 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 3202	. 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 31887	. 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 711	1 ⊢ (𝑆 ∘ 𝑇) ∈ LinOp

Syntax hints:   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059   ∘ ccom 5693  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  ℂcc 11151   ℋchba 30948   +_ℎ cva 30949   ·_ℎ csm 30950  LinOpclo 30976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-hilex 31028  ax-hfvadd 31029  ax-hfvmul 31034

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-lnop 31870

This theorem is referenced by:  lnopco0i  32033  nmopcoi  32124  bdopcoi  32127  nmopcoadj0i  32132