Theorem lcomf 20896

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 20873	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵)
7	6	3expb 1121	. . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵)) → (𝑥 · 𝑦) ∈ 𝐵)
8	1, 7	sylan 581	. 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 4167	. 2 ⊢ (𝐼 ∩ 𝐼) = 𝐼
13	8, 9, 10, 11, 11, 12	off 7649	1 ⊢ (𝜑 → (𝐺 ∘f · 𝐻):𝐼⟶𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ∘_f cof 7629  Basecbs 17179  Scalarcsca 17223   ·_𝑠 cvsca 17224  LModclmod 20855

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-lmod 20857

This theorem is referenced by:  lcomfsupp  20897  frlmup2  21779  islindf4  21818  fedgmullem2  33774