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Theorem lcomf 19112
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 19090 . . . 4 ((𝑊 ∈ LMod ∧ 𝑥𝐾𝑦𝐵) → (𝑥 · 𝑦) ∈ 𝐵)
763expb 1113 . . 3 ((𝑊 ∈ LMod ∧ (𝑥𝐾𝑦𝐵)) → (𝑥 · 𝑦) ∈ 𝐵)
81, 7sylan 569 . 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 3971 . 2 (𝐼𝐼) = 𝐼
138, 9, 10, 11, 11, 12off 7063 1 (𝜑 → (𝐺𝑓 · 𝐻):𝐼𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wf 6026  cfv 6030  (class class class)co 6796  𝑓 cof 7046  Basecbs 16064  Scalarcsca 16152   ·𝑠 cvsca 16153  LModclmod 19073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-of 7048  df-lmod 19075
This theorem is referenced by:  lcomfsupp  19113  frlmup2  20355  islindf4  20394
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