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	⊢ · = ( ·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 19090	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵)
7	6	3expb 1113	. . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵)) → (𝑥 · 𝑦) ∈ 𝐵)
8	1, 7	sylan 569	. 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 3971	. 2 ⊢ (𝐼 ∩ 𝐼) = 𝐼
13	8, 9, 10, 11, 11, 12	off 7063	1 ⊢ (𝜑 → (𝐺 ∘𝑓 · 𝐻):𝐼⟶𝐵)

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