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Theorem lnoadd 28529
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 10589 . . 3 1 ∈ ℂ
2 lnoadd.1 . . . 4 𝑋 = (BaseSet‘𝑈)
3 eqid 2821 . . . 4 (BaseSet‘𝑊) = (BaseSet‘𝑊)
4 lnoadd.5 . . . 4 𝐺 = ( +𝑣𝑈)
5 lnoadd.6 . . . 4 𝐻 = ( +𝑣𝑊)
6 eqid 2821 . . . 4 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
7 eqid 2821 . . . 4 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
8 lnoadd.7 . . . 4 𝐿 = (𝑈 LnOp 𝑊)
92, 3, 4, 5, 6, 7, 8lnolin 28525 . . 3 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (1 ∈ ℂ ∧ 𝐴𝑋𝐵𝑋)) → (𝑇‘((1( ·𝑠OLD𝑈)𝐴)𝐺𝐵)) = ((1( ·𝑠OLD𝑊)(𝑇𝐴))𝐻(𝑇𝐵)))
101, 9mp3anr1 1454 . 2 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → (𝑇‘((1( ·𝑠OLD𝑈)𝐴)𝐺𝐵)) = ((1( ·𝑠OLD𝑊)(𝑇𝐴))𝐻(𝑇𝐵)))
11 simp1 1132 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) → 𝑈 ∈ NrmCVec)
12 simpl 485 . . . 4 ((𝐴𝑋𝐵𝑋) → 𝐴𝑋)
132, 6nvsid 28398 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (1( ·𝑠OLD𝑈)𝐴) = 𝐴)
1411, 12, 13syl2an 597 . . 3 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → (1( ·𝑠OLD𝑈)𝐴) = 𝐴)
1514fvoveq1d 7172 . 2 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → (𝑇‘((1( ·𝑠OLD𝑈)𝐴)𝐺𝐵)) = (𝑇‘(𝐴𝐺𝐵)))
16 simpl2 1188 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → 𝑊 ∈ NrmCVec)
172, 3, 8lnof 28526 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) → 𝑇:𝑋⟶(BaseSet‘𝑊))
18 ffvelrn 6843 . . . . 5 ((𝑇:𝑋⟶(BaseSet‘𝑊) ∧ 𝐴𝑋) → (𝑇𝐴) ∈ (BaseSet‘𝑊))
1917, 12, 18syl2an 597 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → (𝑇𝐴) ∈ (BaseSet‘𝑊))
203, 7nvsid 28398 . . . 4 ((𝑊 ∈ NrmCVec ∧ (𝑇𝐴) ∈ (BaseSet‘𝑊)) → (1( ·𝑠OLD𝑊)(𝑇𝐴)) = (𝑇𝐴))
2116, 19, 20syl2anc 586 . . 3 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → (1( ·𝑠OLD𝑊)(𝑇𝐴)) = (𝑇𝐴))
2221oveq1d 7165 . 2 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → ((1( ·𝑠OLD𝑊)(𝑇𝐴))𝐻(𝑇𝐵)) = ((𝑇𝐴)𝐻(𝑇𝐵)))
2310, 15, 223eqtr3d 2864 1 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → (𝑇‘(𝐴𝐺𝐵)) = ((𝑇𝐴)𝐻(𝑇𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083   = wceq 1533  wcel 2110  wf 6345  cfv 6349  (class class class)co 7150  cc 10529  1c1 10532  NrmCVeccnv 28355   +𝑣 cpv 28356  BaseSetcba 28357   ·𝑠OLD cns 28358   LnOp clno 28511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-1cn 10589
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  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 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-map 8402  df-vc 28330  df-nv 28363  df-va 28366  df-ba 28367  df-sm 28368  df-0v 28369  df-nmcv 28371  df-lno 28515
This theorem is referenced by: (None)
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