Theorem lcomf 20077

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	⊢ · = ( ·𝑠 ‘𝑊)
lcomf.b	⊢ 𝐵 = (Base‘𝑊)
lcomf.w	⊢ (𝜑 → 𝑊 ∈ LMod)
lcomf.g	⊢ (𝜑 → 𝐺:𝐼⟶𝐾)
lcomf.h	⊢ (𝜑 → 𝐻:𝐼⟶𝐵)
lcomf.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)

Assertion

Ref	Expression
lcomf	⊢ (𝜑 → (𝐺 ∘f · 𝐻):𝐼⟶𝐵)

Proof of Theorem lcomf

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lcomf.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ LMod)
2		lcomf.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
3		lcomf.f	. . . . 5 ⊢ 𝐹 = (Scalar‘𝑊)
4		lcomf.s	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
5		lcomf.k	. . . . 5 ⊢ 𝐾 = (Base‘𝐹)
6	2, 3, 4, 5	lmodvscl 20055	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵)
7	6	3expb 1118	. . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵)) → (𝑥 · 𝑦) ∈ 𝐵)
8	1, 7	sylan 579	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵)) → (𝑥 · 𝑦) ∈ 𝐵)
9		lcomf.g	. 2 ⊢ (𝜑 → 𝐺:𝐼⟶𝐾)
10		lcomf.h	. 2 ⊢ (𝜑 → 𝐻:𝐼⟶𝐵)
11		lcomf.i	. 2 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
12		inidm 4149	. 2 ⊢ (𝐼 ∩ 𝐼) = 𝐼
13	8, 9, 10, 11, 11, 12	off 7529	1 ⊢ (𝜑 → (𝐺 ∘f · 𝐻):𝐼⟶𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ∘_f cof 7509  Basecbs 16840  Scalarcsca 16891   ·_𝑠 cvsca 16892  LModclmod 20038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-lmod 20040

This theorem is referenced by:  lcomfsupp  20078  frlmup2  20916  islindf4  20955  fedgmullem2  31613