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Theorem lnof 30787
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 2740 . . . 4 ( +𝑣𝑈) = ( +𝑣𝑈)
4 eqid 2740 . . . 4 ( +𝑣𝑊) = ( +𝑣𝑊)
5 eqid 2740 . . . 4 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
6 eqid 2740 . . . 4 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
7 lnof.7 . . . 4 𝐿 = (𝑈 LnOp 𝑊)
81, 2, 3, 4, 5, 6, 7islno 30785 . . 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 1537  wcel 2108  wral 3067  wf 6569  cfv 6573  (class class class)co 7448  cc 11182  NrmCVeccnv 30616   +𝑣 cpv 30617  BaseSetcba 30618   ·𝑠OLD cns 30619   LnOp clno 30772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-lno 30776
This theorem is referenced by:  lno0  30788  lnocoi  30789  lnoadd  30790  lnosub  30791  lnomul  30792  isblo2  30815  blof  30817  nmlno0lem  30825  nmlnoubi  30828  nmlnogt0  30829  lnon0  30830  isblo3i  30833  blocnilem  30836  blocni  30837  htthlem  30949
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