Theorem lcomf 21003

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 20980	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵)
7	6	3expb 1138	. . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵)) → (𝑥 · 𝑦) ∈ 𝐵)
8	1, 7	sylan 591	. 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 4180	. 2 ⊢ (𝐼 ∩ 𝐼) = 𝐼
13	8, 9, 10, 11, 11, 12	off 7694	1 ⊢ (𝜑 → (𝐺 ∘f · 𝐻):𝐼⟶𝐵)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412   ∘_f cof 7674  Basecbs 17270  Scalarcsca 17314   ·_𝑠 cvsca 17315  LModclmod 20962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7676  df-lmod 20964

This theorem is referenced by:  lcomfsupp  21004  frlmup2  21930  islindf4  21969  fedgmullem2  34001