Theorem lnopcoi 32206

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 32172	. . 3 ⊢ 𝑆: ℋ⟶ ℋ
3		lnopco.2	. . . 4 ⊢ 𝑇 ∈ LinOp
4	3	lnopfi 32172	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
5	2, 4	hocofi 31969	. 2 ⊢ (𝑆 ∘ 𝑇): ℋ⟶ ℋ
6	3	lnopli 32171	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)))
7	6	fveq2d 6871	. . . . . . 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 7064	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℋ)
10	4	ffvelcdmi 7064	. . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℋ)
11	1	lnopli 32171	. . . . . . . 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 1173	. . . . . . 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 2797	. . . . . 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 1131	. . . . 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 31216	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
16		hvaddcl 31215	. . . . . . 7 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ)
17	15, 16	sylan 589	. . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ)
18	2, 4	hocoi 31967	. . . . . 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 31967	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝑦) = (𝑆‘(𝑇‘𝑦)))
21	20	oveq2d 7412	. . . . . . 7 ⊢ (𝑦 ∈ ℋ → (𝑥 ·ℎ ((𝑆 ∘ 𝑇)‘𝑦)) = (𝑥 ·ℎ (𝑆‘(𝑇‘𝑦))))
22	21	adantl 485	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ ((𝑆 ∘ 𝑇)‘𝑦)) = (𝑥 ·ℎ (𝑆‘(𝑇‘𝑦))))
23	2, 4	hocoi 31967	. . . . . 6 ⊢ (𝑧 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝑧) = (𝑆‘(𝑇‘𝑧)))
24	22, 23	oveqan12d 7415	. . . . 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 2807	. . . 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 1122	. . 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 3207	. 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 32061	. 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 721	1 ⊢ (𝑆 ∘ 𝑇) ∈ LinOp

Syntax hints:   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  ∀wral 3076   ∘ ccom 5651  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396  ℂcc 11071   ℋchba 31122   +_ℎ cva 31123   ·_ℎ csm 31124  LinOpclo 31150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-hilex 31202  ax-hfvadd 31203  ax-hfvmul 31208

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-map 8810  df-lnop 32044

This theorem is referenced by:  lnopco0i  32207  nmopcoi  32298  bdopcoi  32301  nmopcoadj0i  32306