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Theorem lnof 30774
Description: A linear operator is a mapping. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 18-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
lnof.1 𝑋 = (BaseSet‘𝑈)
lnof.2 𝑌 = (BaseSet‘𝑊)
lnof.7 𝐿 = (𝑈 LnOp 𝑊)
Assertion
Ref Expression
lnof ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) → 𝑇:𝑋𝑌)

Proof of Theorem lnof
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lnof.1 . . . 4 𝑋 = (BaseSet‘𝑈)
2 lnof.2 . . . 4 𝑌 = (BaseSet‘𝑊)
3 eqid 2737 . . . 4 ( +𝑣𝑈) = ( +𝑣𝑈)
4 eqid 2737 . . . 4 ( +𝑣𝑊) = ( +𝑣𝑊)
5 eqid 2737 . . . 4 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
6 eqid 2737 . . . 4 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
7 lnof.7 . . . 4 𝐿 = (𝑈 LnOp 𝑊)
81, 2, 3, 4, 5, 6, 7islno 30772 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇𝐿 ↔ (𝑇:𝑋𝑌 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑇‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧)) = ((𝑥( ·𝑠OLD𝑊)(𝑇𝑦))( +𝑣𝑊)(𝑇𝑧)))))
98simprbda 498 . 2 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑇𝐿) → 𝑇:𝑋𝑌)
1093impa 1110 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) → 𝑇:𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wf 6557  cfv 6561  (class class class)co 7431  cc 11153  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-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
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-rab 3437  df-v 3482  df-sbc 3789  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-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-lno 30763
This theorem is referenced by:  lno0  30775  lnocoi  30776  lnoadd  30777  lnosub  30778  lnomul  30779  isblo2  30802  blof  30804  nmlno0lem  30812  nmlnoubi  30815  nmlnogt0  30816  lnon0  30817  isblo3i  30820  blocnilem  30823  blocni  30824  htthlem  30936
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