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Theorem lcomf 18725
 Description: A linear-combination sum is a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
Hypotheses
Ref Expression
lcomf.f 𝐹 = (Scalar‘𝑊)
lcomf.k 𝐾 = (Base‘𝐹)
lcomf.s · = ( ·𝑠𝑊)
lcomf.b 𝐵 = (Base‘𝑊)
lcomf.w (𝜑𝑊 ∈ LMod)
lcomf.g (𝜑𝐺:𝐼𝐾)
lcomf.h (𝜑𝐻:𝐼𝐵)
lcomf.i (𝜑𝐼𝑉)
Assertion
Ref Expression
lcomf (𝜑 → (𝐺𝑓 · 𝐻):𝐼𝐵)

Proof of Theorem lcomf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lcomf.w . . 3 (𝜑𝑊 ∈ LMod)
2 lcomf.b . . . . 5 𝐵 = (Base‘𝑊)
3 lcomf.f . . . . 5 𝐹 = (Scalar‘𝑊)
4 lcomf.s . . . . 5 · = ( ·𝑠𝑊)
5 lcomf.k . . . . 5 𝐾 = (Base‘𝐹)
62, 3, 4, 5lmodvscl 18703 . . . 4 ((𝑊 ∈ LMod ∧ 𝑥𝐾𝑦𝐵) → (𝑥 · 𝑦) ∈ 𝐵)
763expb 1258 . . 3 ((𝑊 ∈ LMod ∧ (𝑥𝐾𝑦𝐵)) → (𝑥 · 𝑦) ∈ 𝐵)
81, 7sylan 487 . 2 ((𝜑 ∧ (𝑥𝐾𝑦𝐵)) → (𝑥 · 𝑦) ∈ 𝐵)
9 lcomf.g . 2 (𝜑𝐺:𝐼𝐾)
10 lcomf.h . 2 (𝜑𝐻:𝐼𝐵)
11 lcomf.i . 2 (𝜑𝐼𝑉)
12 inidm 3784 . 2 (𝐼𝐼) = 𝐼
138, 9, 10, 11, 11, 12off 6810 1 (𝜑 → (𝐺𝑓 · 𝐻):𝐼𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793  Basecbs 15695  Scalarcsca 15771   ·𝑠 cvsca 15772  LModclmod 18686 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-lmod 18688 This theorem is referenced by:  lcomfsupp  18726  frlmup2  19957  islindf4  19996
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