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Theorem lnof 28524
 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 2819 . . . 4 ( +𝑣𝑈) = ( +𝑣𝑈)
4 eqid 2819 . . . 4 ( +𝑣𝑊) = ( +𝑣𝑊)
5 eqid 2819 . . . 4 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
6 eqid 2819 . . . 4 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
7 lnof.7 . . . 4 𝐿 = (𝑈 LnOp 𝑊)
81, 2, 3, 4, 5, 6, 7islno 28522 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇𝐿 ↔ (𝑇:𝑋𝑌 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑇‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧)) = ((𝑥( ·𝑠OLD𝑊)(𝑇𝑦))( +𝑣𝑊)(𝑇𝑧)))))
98simprbda 501 . 2 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑇𝐿) → 𝑇:𝑋𝑌)
1093impa 1105 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) → 𝑇:𝑋𝑌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1082   = wceq 1531   ∈ wcel 2108  ∀wral 3136  ⟶wf 6344  ‘cfv 6348  (class class class)co 7148  ℂcc 10527  NrmCVeccnv 28353   +𝑣 cpv 28354  BaseSetcba 28355   ·𝑠OLD cns 28356   LnOp clno 28509 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-map 8400  df-lno 28513 This theorem is referenced by:  lno0  28525  lnocoi  28526  lnoadd  28527  lnosub  28528  lnomul  28529  isblo2  28552  blof  28554  nmlno0lem  28562  nmlnoubi  28565  nmlnogt0  28566  lnon0  28567  isblo3i  28570  blocnilem  28573  blocni  28574  htthlem  28686
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