Theorem lnopcoi 31939

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 31905	. . 3 ⊢ 𝑆: ℋ⟶ ℋ
3		lnopco.2	. . . 4 ⊢ 𝑇 ∈ LinOp
4	3	lnopfi 31905	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
5	2, 4	hocofi 31702	. 2 ⊢ (𝑆 ∘ 𝑇): ℋ⟶ ℋ
6	3	lnopli 31904	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)))
7	6	fveq2d 6865	. . . . . . 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 7058	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℋ)
10	4	ffvelcdmi 7058	. . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℋ)
11	1	lnopli 31904	. . . . . . . 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 2765	. . . . . 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 30949	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
16		hvaddcl 30948	. . . . . . 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 31700	. . . . . 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 31700	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝑦) = (𝑆‘(𝑇‘𝑦)))
21	20	oveq2d 7406	. . . . . . 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 31700	. . . . . 6 ⊢ (𝑧 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝑧) = (𝑆‘(𝑇‘𝑧)))
24	22, 23	oveqan12d 7409	. . . . 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 2775	. . . 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 3183	. 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 31794	. 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 1540   ∈ wcel 2109  ∀wral 3045   ∘ ccom 5645  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  ℂcc 11073   ℋchba 30855   +_ℎ cva 30856   ·_ℎ csm 30857  LinOpclo 30883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-hilex 30935  ax-hfvadd 30936  ax-hfvmul 30941

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8804  df-lnop 31777

This theorem is referenced by:  lnopco0i  31940  nmopcoi  32031  bdopcoi  32034  nmopcoadj0i  32039