MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lcomf Structured version   Visualization version   GIF version

Theorem lcomf 19385
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 19363 . . . 4 ((𝑊 ∈ LMod ∧ 𝑥𝐾𝑦𝐵) → (𝑥 · 𝑦) ∈ 𝐵)
763expb 1100 . . 3 ((𝑊 ∈ LMod ∧ (𝑥𝐾𝑦𝐵)) → (𝑥 · 𝑦) ∈ 𝐵)
81, 7sylan 572 . 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 4077 . 2 (𝐼𝐼) = 𝐼
138, 9, 10, 11, 11, 12off 7236 1 (𝜑 → (𝐺𝑓 · 𝐻):𝐼𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2048  wf 6178  cfv 6182  (class class class)co 6970  𝑓 cof 7219  Basecbs 16329  Scalarcsca 16414   ·𝑠 cvsca 16415  LModclmod 19346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pr 5180
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-of 7221  df-lmod 19348
This theorem is referenced by:  lcomfsupp  19386  frlmup2  20635  islindf4  20674  fedgmullem2  30611
  Copyright terms: Public domain W3C validator