Theorem lnocoi 31180

Description: The composition of two linear operators is linear. (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lnocoi.l	⊢ 𝐿 = (𝑈 LnOp 𝑊)
lnocoi.m	⊢ 𝑀 = (𝑊 LnOp 𝑋)
lnocoi.n	⊢ 𝑁 = (𝑈 LnOp 𝑋)
lnocoi.u	⊢ 𝑈 ∈ NrmCVec
lnocoi.w	⊢ 𝑊 ∈ NrmCVec
lnocoi.x	⊢ 𝑋 ∈ NrmCVec
lnocoi.s	⊢ 𝑆 ∈ 𝐿
lnocoi.t	⊢ 𝑇 ∈ 𝑀

Assertion

Ref	Expression
lnocoi	⊢ (𝑇 ∘ 𝑆) ∈ 𝑁

Proof of Theorem lnocoi

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

Step	Hyp	Ref	Expression
1		lnocoi.w	. . . 4 ⊢ 𝑊 ∈ NrmCVec
2		lnocoi.x	. . . 4 ⊢ 𝑋 ∈ NrmCVec
3		lnocoi.t	. . . 4 ⊢ 𝑇 ∈ 𝑀
4		eqid 2765	. . . . 5 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
5		eqid 2765	. . . . 5 ⊢ (BaseSet‘𝑋) = (BaseSet‘𝑋)
6		lnocoi.m	. . . . 5 ⊢ 𝑀 = (𝑊 LnOp 𝑋)
7	4, 5, 6	lnof 31178	. . . 4 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑋 ∈ NrmCVec ∧ 𝑇 ∈ 𝑀) → 𝑇:(BaseSet‘𝑊)⟶(BaseSet‘𝑋))
8	1, 2, 3, 7	mp3an 1490	. . 3 ⊢ 𝑇:(BaseSet‘𝑊)⟶(BaseSet‘𝑋)
9		lnocoi.u	. . . 4 ⊢ 𝑈 ∈ NrmCVec
10		lnocoi.s	. . . 4 ⊢ 𝑆 ∈ 𝐿
11		eqid 2765	. . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
12		lnocoi.l	. . . . 5 ⊢ 𝐿 = (𝑈 LnOp 𝑊)
13	11, 4, 12	lnof 31178	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑆 ∈ 𝐿) → 𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊))
14	9, 1, 10, 13	mp3an 1490	. . 3 ⊢ 𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)
15		fco 6734	. . 3 ⊢ ((𝑇:(BaseSet‘𝑊)⟶(BaseSet‘𝑋) ∧ 𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → (𝑇 ∘ 𝑆):(BaseSet‘𝑈)⟶(BaseSet‘𝑋))
16	8, 14, 15	mp2an 705	. 2 ⊢ (𝑇 ∘ 𝑆):(BaseSet‘𝑈)⟶(BaseSet‘𝑋)
17		eqid 2765	. . . . . . . 8 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
18	11, 17	nvscl 31049	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈)) → (𝑥( ·𝑠OLD ‘𝑈)𝑦) ∈ (BaseSet‘𝑈))
19	9, 18	mp3an1 1477	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈)) → (𝑥( ·𝑠OLD ‘𝑈)𝑦) ∈ (BaseSet‘𝑈))
20		eqid 2765	. . . . . . . 8 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
21	11, 20	nvgcl 31043	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥( ·𝑠OLD ‘𝑈)𝑦) ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧) ∈ (BaseSet‘𝑈))
22	9, 21	mp3an1 1477	. . . . . 6 ⊢ (((𝑥( ·𝑠OLD ‘𝑈)𝑦) ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧) ∈ (BaseSet‘𝑈))
23	19, 22	stoic3 1809	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧) ∈ (BaseSet‘𝑈))
24		fvco3 6985	. . . . 5 ⊢ ((𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ ((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧) ∈ (BaseSet‘𝑈)) → ((𝑇 ∘ 𝑆)‘((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧)) = (𝑇‘(𝑆‘((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧))))
25	14, 23, 24	sylancr 599	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑇 ∘ 𝑆)‘((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧)) = (𝑇‘(𝑆‘((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧))))
26		id 23	. . . . . 6 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ ℂ)
27	14	ffvelcdmi 7082	. . . . . 6 ⊢ (𝑦 ∈ (BaseSet‘𝑈) → (𝑆‘𝑦) ∈ (BaseSet‘𝑊))
28	14	ffvelcdmi 7082	. . . . . 6 ⊢ (𝑧 ∈ (BaseSet‘𝑈) → (𝑆‘𝑧) ∈ (BaseSet‘𝑊))
29	1, 2, 3	3pm3.2i 1358	. . . . . . 7 ⊢ (𝑊 ∈ NrmCVec ∧ 𝑋 ∈ NrmCVec ∧ 𝑇 ∈ 𝑀)
30		eqid 2765	. . . . . . . 8 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊)
31		eqid 2765	. . . . . . . 8 ⊢ ( +𝑣 ‘𝑋) = ( +𝑣 ‘𝑋)
32		eqid 2765	. . . . . . . 8 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊)
33		eqid 2765	. . . . . . . 8 ⊢ ( ·𝑠OLD ‘𝑋) = ( ·𝑠OLD ‘𝑋)
34	4, 5, 30, 31, 32, 33, 6	lnolin 31177	. . . . . . 7 ⊢ (((𝑊 ∈ NrmCVec ∧ 𝑋 ∈ NrmCVec ∧ 𝑇 ∈ 𝑀) ∧ (𝑥 ∈ ℂ ∧ (𝑆‘𝑦) ∈ (BaseSet‘𝑊) ∧ (𝑆‘𝑧) ∈ (BaseSet‘𝑊))) → (𝑇‘((𝑥( ·𝑠OLD ‘𝑊)(𝑆‘𝑦))( +𝑣 ‘𝑊)(𝑆‘𝑧))) = ((𝑥( ·𝑠OLD ‘𝑋)(𝑇‘(𝑆‘𝑦)))( +𝑣 ‘𝑋)(𝑇‘(𝑆‘𝑧))))
35	29, 34	mpan 703	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ (𝑆‘𝑦) ∈ (BaseSet‘𝑊) ∧ (𝑆‘𝑧) ∈ (BaseSet‘𝑊)) → (𝑇‘((𝑥( ·𝑠OLD ‘𝑊)(𝑆‘𝑦))( +𝑣 ‘𝑊)(𝑆‘𝑧))) = ((𝑥( ·𝑠OLD ‘𝑋)(𝑇‘(𝑆‘𝑦)))( +𝑣 ‘𝑋)(𝑇‘(𝑆‘𝑧))))
36	26, 27, 28, 35	syl3an 1178	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → (𝑇‘((𝑥( ·𝑠OLD ‘𝑊)(𝑆‘𝑦))( +𝑣 ‘𝑊)(𝑆‘𝑧))) = ((𝑥( ·𝑠OLD ‘𝑋)(𝑇‘(𝑆‘𝑦)))( +𝑣 ‘𝑋)(𝑇‘(𝑆‘𝑧))))
37	9, 1, 10	3pm3.2i 1358	. . . . . . 7 ⊢ (𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑆 ∈ 𝐿)
38	11, 4, 20, 30, 17, 32, 12	lnolin 31177	. . . . . . 7 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑆 ∈ 𝐿) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑆‘((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧)) = ((𝑥( ·𝑠OLD ‘𝑊)(𝑆‘𝑦))( +𝑣 ‘𝑊)(𝑆‘𝑧)))
39	37, 38	mpan 703	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → (𝑆‘((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧)) = ((𝑥( ·𝑠OLD ‘𝑊)(𝑆‘𝑦))( +𝑣 ‘𝑊)(𝑆‘𝑧)))
40	39	fveq2d 6889	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → (𝑇‘(𝑆‘((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧))) = (𝑇‘((𝑥( ·𝑠OLD ‘𝑊)(𝑆‘𝑦))( +𝑣 ‘𝑊)(𝑆‘𝑧))))
41		simp2 1155	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → 𝑦 ∈ (BaseSet‘𝑈))
42		fvco3 6985	. . . . . . . 8 ⊢ ((𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑈)) → ((𝑇 ∘ 𝑆)‘𝑦) = (𝑇‘(𝑆‘𝑦)))
43	14, 41, 42	sylancr 599	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑇 ∘ 𝑆)‘𝑦) = (𝑇‘(𝑆‘𝑦)))
44	43	oveq2d 7435	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → (𝑥( ·𝑠OLD ‘𝑋)((𝑇 ∘ 𝑆)‘𝑦)) = (𝑥( ·𝑠OLD ‘𝑋)(𝑇‘(𝑆‘𝑦))))
45		simp3 1156	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → 𝑧 ∈ (BaseSet‘𝑈))
46		fvco3 6985	. . . . . . 7 ⊢ ((𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑇 ∘ 𝑆)‘𝑧) = (𝑇‘(𝑆‘𝑧)))
47	14, 45, 46	sylancr 599	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑇 ∘ 𝑆)‘𝑧) = (𝑇‘(𝑆‘𝑧)))
48	44, 47	oveq12d 7437	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑥( ·𝑠OLD ‘𝑋)((𝑇 ∘ 𝑆)‘𝑦))( +𝑣 ‘𝑋)((𝑇 ∘ 𝑆)‘𝑧)) = ((𝑥( ·𝑠OLD ‘𝑋)(𝑇‘(𝑆‘𝑦)))( +𝑣 ‘𝑋)(𝑇‘(𝑆‘𝑧))))
49	36, 40, 48	3eqtr4rd 2811	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑥( ·𝑠OLD ‘𝑋)((𝑇 ∘ 𝑆)‘𝑦))( +𝑣 ‘𝑋)((𝑇 ∘ 𝑆)‘𝑧)) = (𝑇‘(𝑆‘((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧))))
50	25, 49	eqtr4d 2803	. . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑇 ∘ 𝑆)‘((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧)) = ((𝑥( ·𝑠OLD ‘𝑋)((𝑇 ∘ 𝑆)‘𝑦))( +𝑣 ‘𝑋)((𝑇 ∘ 𝑆)‘𝑧)))
51	50	rgen3 3212	. 2 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (BaseSet‘𝑈)∀𝑧 ∈ (BaseSet‘𝑈)((𝑇 ∘ 𝑆)‘((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧)) = ((𝑥( ·𝑠OLD ‘𝑋)((𝑇 ∘ 𝑆)‘𝑦))( +𝑣 ‘𝑋)((𝑇 ∘ 𝑆)‘𝑧))
52		lnocoi.n	. . . 4 ⊢ 𝑁 = (𝑈 LnOp 𝑋)
53	11, 5, 20, 31, 17, 33, 52	islno 31176	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑋 ∈ NrmCVec) → ((𝑇 ∘ 𝑆) ∈ 𝑁 ↔ ((𝑇 ∘ 𝑆):(BaseSet‘𝑈)⟶(BaseSet‘𝑋) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (BaseSet‘𝑈)∀𝑧 ∈ (BaseSet‘𝑈)((𝑇 ∘ 𝑆)‘((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧)) = ((𝑥( ·𝑠OLD ‘𝑋)((𝑇 ∘ 𝑆)‘𝑦))( +𝑣 ‘𝑋)((𝑇 ∘ 𝑆)‘𝑧)))))
54	9, 2, 53	mp2an 705	. 2 ⊢ ((𝑇 ∘ 𝑆) ∈ 𝑁 ↔ ((𝑇 ∘ 𝑆):(BaseSet‘𝑈)⟶(BaseSet‘𝑋) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (BaseSet‘𝑈)∀𝑧 ∈ (BaseSet‘𝑈)((𝑇 ∘ 𝑆)‘((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧)) = ((𝑥( ·𝑠OLD ‘𝑋)((𝑇 ∘ 𝑆)‘𝑦))( +𝑣 ‘𝑋)((𝑇 ∘ 𝑆)‘𝑧))))
55	16, 51, 54	mpbir2an 724	1 ⊢ (𝑇 ∘ 𝑆) ∈ 𝑁

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081   ∘ ccom 5667  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419  ℂcc 11113  NrmCVeccnv 31007   +_𝑣 cpv 31008  BaseSetcba 31009   ·_𝑠OLD cns 31010   LnOp clno 31163

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-map 8832  df-grpo 30916  df-ablo 30968  df-vc 30982  df-nv 31015  df-va 31018  df-ba 31019  df-sm 31020  df-0v 31021  df-nmcv 31023  df-lno 31167

This theorem is used by: (None)