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Theorem lnoadd 30777
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 = ( +𝑣𝑈)
lnoadd.6 𝐻 = ( +𝑣𝑊)
lnoadd.7 𝐿 = (𝑈 LnOp 𝑊)
Assertion
Ref Expression
lnoadd (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → (𝑇‘(𝐴𝐺𝐵)) = ((𝑇𝐴)𝐻(𝑇𝐵)))

Proof of Theorem lnoadd
StepHypRef Expression
1 ax-1cn 11213 . . 3 1 ∈ ℂ
2 lnoadd.1 . . . 4 𝑋 = (BaseSet‘𝑈)
3 eqid 2737 . . . 4 (BaseSet‘𝑊) = (BaseSet‘𝑊)
4 lnoadd.5 . . . 4 𝐺 = ( +𝑣𝑈)
5 lnoadd.6 . . . 4 𝐻 = ( +𝑣𝑊)
6 eqid 2737 . . . 4 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
7 eqid 2737 . . . 4 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
8 lnoadd.7 . . . 4 𝐿 = (𝑈 LnOp 𝑊)
92, 3, 4, 5, 6, 7, 8lnolin 30773 . . 3 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (1 ∈ ℂ ∧ 𝐴𝑋𝐵𝑋)) → (𝑇‘((1( ·𝑠OLD𝑈)𝐴)𝐺𝐵)) = ((1( ·𝑠OLD𝑊)(𝑇𝐴))𝐻(𝑇𝐵)))
101, 9mp3anr1 1460 . 2 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → (𝑇‘((1( ·𝑠OLD𝑈)𝐴)𝐺𝐵)) = ((1( ·𝑠OLD𝑊)(𝑇𝐴))𝐻(𝑇𝐵)))
11 simp1 1137 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) → 𝑈 ∈ NrmCVec)
12 simpl 482 . . . 4 ((𝐴𝑋𝐵𝑋) → 𝐴𝑋)
132, 6nvsid 30646 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (1( ·𝑠OLD𝑈)𝐴) = 𝐴)
1411, 12, 13syl2an 596 . . 3 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → (1( ·𝑠OLD𝑈)𝐴) = 𝐴)
1514fvoveq1d 7453 . 2 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → (𝑇‘((1( ·𝑠OLD𝑈)𝐴)𝐺𝐵)) = (𝑇‘(𝐴𝐺𝐵)))
16 simpl2 1193 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → 𝑊 ∈ NrmCVec)
172, 3, 8lnof 30774 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) → 𝑇:𝑋⟶(BaseSet‘𝑊))
18 ffvelcdm 7101 . . . . 5 ((𝑇:𝑋⟶(BaseSet‘𝑊) ∧ 𝐴𝑋) → (𝑇𝐴) ∈ (BaseSet‘𝑊))
1917, 12, 18syl2an 596 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → (𝑇𝐴) ∈ (BaseSet‘𝑊))
203, 7nvsid 30646 . . . 4 ((𝑊 ∈ NrmCVec ∧ (𝑇𝐴) ∈ (BaseSet‘𝑊)) → (1( ·𝑠OLD𝑊)(𝑇𝐴)) = (𝑇𝐴))
2116, 19, 20syl2anc 584 . . 3 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → (1( ·𝑠OLD𝑊)(𝑇𝐴)) = (𝑇𝐴))
2221oveq1d 7446 . 2 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → ((1( ·𝑠OLD𝑊)(𝑇𝐴))𝐻(𝑇𝐵)) = ((𝑇𝐴)𝐻(𝑇𝐵)))
2310, 15, 223eqtr3d 2785 1 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → (𝑇‘(𝐴𝐺𝐵)) = ((𝑇𝐴)𝐻(𝑇𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wf 6557  cfv 6561  (class class class)co 7431  cc 11153  1c1 11156  NrmCVeccnv 30603   +𝑣 cpv 30604  BaseSetcba 30605   ·𝑠OLD cns 30606   LnOp clno 30759
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-1cn 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-vc 30578  df-nv 30611  df-va 30614  df-ba 30615  df-sm 30616  df-0v 30617  df-nmcv 30619  df-lno 30763
This theorem is referenced by: (None)
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