Theorem lnoadd 28164

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 10317	. . 3 ⊢ 1 ∈ ℂ
2		lnoadd.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
3		eqid 2825	. . . 4 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
4		lnoadd.5	. . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
5		lnoadd.6	. . . 4 ⊢ 𝐻 = ( +𝑣 ‘𝑊)
6		eqid 2825	. . . 4 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
7		eqid 2825	. . . 4 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊)
8		lnoadd.7	. . . 4 ⊢ 𝐿 = (𝑈 LnOp 𝑊)
9	2, 3, 4, 5, 6, 7, 8	lnolin 28160	. . 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 1586	. 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 1170	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑈 ∈ NrmCVec)
12		simpl 476	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑋)
13	2, 6	nvsid 28033	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (1( ·𝑠OLD ‘𝑈)𝐴) = 𝐴)
14	11, 12, 13	syl2an 589	. . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (1( ·𝑠OLD ‘𝑈)𝐴) = 𝐴)
15	14	fvoveq1d 6932	. 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑇‘((1( ·𝑠OLD ‘𝑈)𝐴)𝐺𝐵)) = (𝑇‘(𝐴𝐺𝐵)))
16		simpl2 1248	. . . 4 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝑊 ∈ NrmCVec)
17	2, 3, 8	lnof 28161	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:𝑋⟶(BaseSet‘𝑊))
18		ffvelrn 6611	. . . . 5 ⊢ ((𝑇:𝑋⟶(BaseSet‘𝑊) ∧ 𝐴 ∈ 𝑋) → (𝑇‘𝐴) ∈ (BaseSet‘𝑊))
19	17, 12, 18	syl2an 589	. . . 4 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑇‘𝐴) ∈ (BaseSet‘𝑊))
20	3, 7	nvsid 28033	. . . 4 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑇‘𝐴) ∈ (BaseSet‘𝑊)) → (1( ·𝑠OLD ‘𝑊)(𝑇‘𝐴)) = (𝑇‘𝐴))
21	16, 19, 20	syl2anc 579	. . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (1( ·𝑠OLD ‘𝑊)(𝑇‘𝐴)) = (𝑇‘𝐴))
22	21	oveq1d 6925	. 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 2869	1 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑇‘(𝐴𝐺𝐵)) = ((𝑇‘𝐴)𝐻(𝑇‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  ⟶wf 6123  ‘cfv 6127  (class class class)co 6910  ℂcc 10257  1c1 10260  NrmCVeccnv 27990   +_𝑣 cpv 27991  BaseSetcba 27992   ·_𝑠OLD cns 27993   LnOp clno 28146

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-1cn 10317

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  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 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-map 8129  df-vc 27965  df-nv 27998  df-va 28001  df-ba 28002  df-sm 28003  df-0v 28004  df-nmcv 28006  df-lno 28150

This theorem is referenced by: (None)