Theorem lnoadd 28519

Description: Addition property of a linear operator. (Contributed by NM, 7-Dec-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lnoadd.1	⊢ 𝑋 = (BaseSet‘𝑈)
lnoadd.5	⊢ 𝐺 = ( +𝑣 ‘𝑈)
lnoadd.6	⊢ 𝐻 = ( +𝑣 ‘𝑊)
lnoadd.7	⊢ 𝐿 = (𝑈 LnOp 𝑊)

Assertion

Ref	Expression
lnoadd	⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑇‘(𝐴𝐺𝐵)) = ((𝑇‘𝐴)𝐻(𝑇‘𝐵)))

Proof of Theorem lnoadd

Step	Hyp	Ref	Expression
1		ax-1cn 10581	. . 3 ⊢ 1 ∈ ℂ
2		lnoadd.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
3		eqid 2821	. . . 4 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
4		lnoadd.5	. . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
5		lnoadd.6	. . . 4 ⊢ 𝐻 = ( +𝑣 ‘𝑊)
6		eqid 2821	. . . 4 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
7		eqid 2821	. . . 4 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊)
8		lnoadd.7	. . . 4 ⊢ 𝐿 = (𝑈 LnOp 𝑊)
9	2, 3, 4, 5, 6, 7, 8	lnolin 28515	. . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (1 ∈ ℂ ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑇‘((1( ·𝑠OLD ‘𝑈)𝐴)𝐺𝐵)) = ((1( ·𝑠OLD ‘𝑊)(𝑇‘𝐴))𝐻(𝑇‘𝐵)))
10	1, 9	mp3anr1 1454	. 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑇‘((1( ·𝑠OLD ‘𝑈)𝐴)𝐺𝐵)) = ((1( ·𝑠OLD ‘𝑊)(𝑇‘𝐴))𝐻(𝑇‘𝐵)))
11		simp1 1132	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑈 ∈ NrmCVec)
12		simpl 485	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑋)
13	2, 6	nvsid 28388	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (1( ·𝑠OLD ‘𝑈)𝐴) = 𝐴)
14	11, 12, 13	syl2an 597	. . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (1( ·𝑠OLD ‘𝑈)𝐴) = 𝐴)
15	14	fvoveq1d 7164	. 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑇‘((1( ·𝑠OLD ‘𝑈)𝐴)𝐺𝐵)) = (𝑇‘(𝐴𝐺𝐵)))
16		simpl2 1188	. . . 4 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝑊 ∈ NrmCVec)
17	2, 3, 8	lnof 28516	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:𝑋⟶(BaseSet‘𝑊))
18		ffvelrn 6835	. . . . 5 ⊢ ((𝑇:𝑋⟶(BaseSet‘𝑊) ∧ 𝐴 ∈ 𝑋) → (𝑇‘𝐴) ∈ (BaseSet‘𝑊))
19	17, 12, 18	syl2an 597	. . . 4 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑇‘𝐴) ∈ (BaseSet‘𝑊))
20	3, 7	nvsid 28388	. . . 4 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑇‘𝐴) ∈ (BaseSet‘𝑊)) → (1( ·𝑠OLD ‘𝑊)(𝑇‘𝐴)) = (𝑇‘𝐴))
21	16, 19, 20	syl2anc 586	. . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (1( ·𝑠OLD ‘𝑊)(𝑇‘𝐴)) = (𝑇‘𝐴))
22	21	oveq1d 7157	. 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((1( ·𝑠OLD ‘𝑊)(𝑇‘𝐴))𝐻(𝑇‘𝐵)) = ((𝑇‘𝐴)𝐻(𝑇‘𝐵)))
23	10, 15, 22	3eqtr3d 2864	1 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑇‘(𝐴𝐺𝐵)) = ((𝑇‘𝐴)𝐻(𝑇‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ⟶wf 6337  ‘cfv 6341  (class class class)co 7142  ℂcc 10521  1c1 10524  NrmCVeccnv 28345   +_𝑣 cpv 28346  BaseSetcba 28347   ·_𝑠OLD cns 28348   LnOp clno 28501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5252  ax-pr 5316  ax-un 7447  ax-1cn 10581

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3488  df-sbc 3764  df-csb 3872  df-dif 3927  df-un 3929  df-in 3931  df-ss 3940  df-nul 4280  df-if 4454  df-pw 4527  df-sn 4554  df-pr 4556  df-op 4560  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5446  df-xp 5547  df-rel 5548  df-cnv 5549  df-co 5550  df-dm 5551  df-rn 5552  df-res 5553  df-ima 5554  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-ov 7145  df-oprab 7146  df-mpo 7147  df-1st 7675  df-2nd 7676  df-map 8394  df-vc 28320  df-nv 28353  df-va 28356  df-ba 28357  df-sm 28358  df-0v 28359  df-nmcv 28361  df-lno 28505

This theorem is referenced by: (None)