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Theorem lnoadd 30588
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 11204 . . 3 1 ∈ ℂ
2 lnoadd.1 . . . 4 𝑋 = (BaseSet‘𝑈)
3 eqid 2728 . . . 4 (BaseSet‘𝑊) = (BaseSet‘𝑊)
4 lnoadd.5 . . . 4 𝐺 = ( +𝑣𝑈)
5 lnoadd.6 . . . 4 𝐻 = ( +𝑣𝑊)
6 eqid 2728 . . . 4 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
7 eqid 2728 . . . 4 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
8 lnoadd.7 . . . 4 𝐿 = (𝑈 LnOp 𝑊)
92, 3, 4, 5, 6, 7, 8lnolin 30584 . . 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 1133 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) → 𝑈 ∈ NrmCVec)
12 simpl 481 . . . 4 ((𝐴𝑋𝐵𝑋) → 𝐴𝑋)
132, 6nvsid 30457 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (1( ·𝑠OLD𝑈)𝐴) = 𝐴)
1411, 12, 13syl2an 594 . . 3 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → (1( ·𝑠OLD𝑈)𝐴) = 𝐴)
1514fvoveq1d 7448 . 2 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → (𝑇‘((1( ·𝑠OLD𝑈)𝐴)𝐺𝐵)) = (𝑇‘(𝐴𝐺𝐵)))
16 simpl2 1189 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → 𝑊 ∈ NrmCVec)
172, 3, 8lnof 30585 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) → 𝑇:𝑋⟶(BaseSet‘𝑊))
18 ffvelcdm 7096 . . . . 5 ((𝑇:𝑋⟶(BaseSet‘𝑊) ∧ 𝐴𝑋) → (𝑇𝐴) ∈ (BaseSet‘𝑊))
1917, 12, 18syl2an 594 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → (𝑇𝐴) ∈ (BaseSet‘𝑊))
203, 7nvsid 30457 . . . 4 ((𝑊 ∈ NrmCVec ∧ (𝑇𝐴) ∈ (BaseSet‘𝑊)) → (1( ·𝑠OLD𝑊)(𝑇𝐴)) = (𝑇𝐴))
2116, 19, 20syl2anc 582 . . 3 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → (1( ·𝑠OLD𝑊)(𝑇𝐴)) = (𝑇𝐴))
2221oveq1d 7441 . 2 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → ((1( ·𝑠OLD𝑊)(𝑇𝐴))𝐻(𝑇𝐵)) = ((𝑇𝐴)𝐻(𝑇𝐵)))
2310, 15, 223eqtr3d 2776 1 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → (𝑇‘(𝐴𝐺𝐵)) = ((𝑇𝐴)𝐻(𝑇𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wf 6549  cfv 6553  (class class class)co 7426  cc 11144  1c1 11147  NrmCVeccnv 30414   +𝑣 cpv 30415  BaseSetcba 30416   ·𝑠OLD cns 30417   LnOp clno 30570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-1cn 11204
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-map 8853  df-vc 30389  df-nv 30422  df-va 30425  df-ba 30426  df-sm 30427  df-0v 30428  df-nmcv 30430  df-lno 30574
This theorem is referenced by: (None)
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