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Theorem lcomf 19670
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 19648 . . . 4 ((𝑊 ∈ LMod ∧ 𝑥𝐾𝑦𝐵) → (𝑥 · 𝑦) ∈ 𝐵)
763expb 1117 . . 3 ((𝑊 ∈ LMod ∧ (𝑥𝐾𝑦𝐵)) → (𝑥 · 𝑦) ∈ 𝐵)
81, 7sylan 583 . 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 4148 . 2 (𝐼𝐼) = 𝐼
138, 9, 10, 11, 11, 12off 7408 1 (𝜑 → (𝐺f · 𝐻):𝐼𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wf 6324  cfv 6328  (class class class)co 7139  f cof 7391  Basecbs 16479  Scalarcsca 16564   ·𝑠 cvsca 16565  LModclmod 19631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-lmod 19633
This theorem is referenced by:  lcomfsupp  19671  frlmup2  20492  islindf4  20531  fedgmullem2  31118
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