Theorem lnocoi 28167

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 2825	. . . . 5 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
5		eqid 2825	. . . . 5 ⊢ (BaseSet‘𝑋) = (BaseSet‘𝑋)
6		lnocoi.m	. . . . 5 ⊢ 𝑀 = (𝑊 LnOp 𝑋)
7	4, 5, 6	lnof 28165	. . . 4 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑋 ∈ NrmCVec ∧ 𝑇 ∈ 𝑀) → 𝑇:(BaseSet‘𝑊)⟶(BaseSet‘𝑋))
8	1, 2, 3, 7	mp3an 1591	. . 3 ⊢ 𝑇:(BaseSet‘𝑊)⟶(BaseSet‘𝑋)
9		lnocoi.u	. . . 4 ⊢ 𝑈 ∈ NrmCVec
10		lnocoi.s	. . . 4 ⊢ 𝑆 ∈ 𝐿
11		eqid 2825	. . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
12		lnocoi.l	. . . . 5 ⊢ 𝐿 = (𝑈 LnOp 𝑊)
13	11, 4, 12	lnof 28165	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑆 ∈ 𝐿) → 𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊))
14	9, 1, 10, 13	mp3an 1591	. . 3 ⊢ 𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)
15		fco 6295	. . 3 ⊢ ((𝑇:(BaseSet‘𝑊)⟶(BaseSet‘𝑋) ∧ 𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → (𝑇 ∘ 𝑆):(BaseSet‘𝑈)⟶(BaseSet‘𝑋))
16	8, 14, 15	mp2an 685	. 2 ⊢ (𝑇 ∘ 𝑆):(BaseSet‘𝑈)⟶(BaseSet‘𝑋)
17		eqid 2825	. . . . . . . 8 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
18	11, 17	nvscl 28036	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈)) → (𝑥( ·𝑠OLD ‘𝑈)𝑦) ∈ (BaseSet‘𝑈))
19	9, 18	mp3an1 1578	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈)) → (𝑥( ·𝑠OLD ‘𝑈)𝑦) ∈ (BaseSet‘𝑈))
20		eqid 2825	. . . . . . . 8 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
21	11, 20	nvgcl 28030	. . . . . . 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 1578	. . . . . 6 ⊢ (((𝑥( ·𝑠OLD ‘𝑈)𝑦) ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧) ∈ (BaseSet‘𝑈))
23	19, 22	stoic3 1877	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧) ∈ (BaseSet‘𝑈))
24		fvco3 6522	. . . . 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 583	. . . 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 22	. . . . . 6 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ ℂ)
27	14	ffvelrni 6607	. . . . . 6 ⊢ (𝑦 ∈ (BaseSet‘𝑈) → (𝑆‘𝑦) ∈ (BaseSet‘𝑊))
28	14	ffvelrni 6607	. . . . . 6 ⊢ (𝑧 ∈ (BaseSet‘𝑈) → (𝑆‘𝑧) ∈ (BaseSet‘𝑊))
29	1, 2, 3	3pm3.2i 1444	. . . . . . 7 ⊢ (𝑊 ∈ NrmCVec ∧ 𝑋 ∈ NrmCVec ∧ 𝑇 ∈ 𝑀)
30		eqid 2825	. . . . . . . 8 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊)
31		eqid 2825	. . . . . . . 8 ⊢ ( +𝑣 ‘𝑋) = ( +𝑣 ‘𝑋)
32		eqid 2825	. . . . . . . 8 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊)
33		eqid 2825	. . . . . . . 8 ⊢ ( ·𝑠OLD ‘𝑋) = ( ·𝑠OLD ‘𝑋)
34	4, 5, 30, 31, 32, 33, 6	lnolin 28164	. . . . . . 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 683	. . . . . 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 1205	. . . . 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 1444	. . . . . . 7 ⊢ (𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑆 ∈ 𝐿)
38	11, 4, 20, 30, 17, 32, 12	lnolin 28164	. . . . . . 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 683	. . . . . 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 6437	. . . . 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 1173	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → 𝑦 ∈ (BaseSet‘𝑈))
42		fvco3 6522	. . . . . . . 8 ⊢ ((𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑈)) → ((𝑇 ∘ 𝑆)‘𝑦) = (𝑇‘(𝑆‘𝑦)))
43	14, 41, 42	sylancr 583	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑇 ∘ 𝑆)‘𝑦) = (𝑇‘(𝑆‘𝑦)))
44	43	oveq2d 6921	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → (𝑥( ·𝑠OLD ‘𝑋)((𝑇 ∘ 𝑆)‘𝑦)) = (𝑥( ·𝑠OLD ‘𝑋)(𝑇‘(𝑆‘𝑦))))
45		simp3 1174	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → 𝑧 ∈ (BaseSet‘𝑈))
46		fvco3 6522	. . . . . . 7 ⊢ ((𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑇 ∘ 𝑆)‘𝑧) = (𝑇‘(𝑆‘𝑧)))
47	14, 45, 46	sylancr 583	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑇 ∘ 𝑆)‘𝑧) = (𝑇‘(𝑆‘𝑧)))
48	44, 47	oveq12d 6923	. . . . 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 2872	. . . 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 2864	. . 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 3185	. 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 28163	. . 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 685	. 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 704	1 ⊢ (𝑇 ∘ 𝑆) ∈ 𝑁

Syntax hints:   ↔ wb 198   ∧ wa 386   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166  ∀wral 3117   ∘ ccom 5346  ⟶wf 6119  ‘cfv 6123  (class class class)co 6905  ℂcc 10250  NrmCVeccnv 27994   +_𝑣 cpv 27995  BaseSetcba 27996   ·_𝑠OLD cns 27997   LnOp clno 28150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-map 8124  df-grpo 27903  df-ablo 27955  df-vc 27969  df-nv 28002  df-va 28005  df-ba 28006  df-sm 28007  df-0v 28008  df-nmcv 28010  df-lno 28154

This theorem is referenced by: (None)