Theorem scaffng 14115

Description: The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypotheses

Ref	Expression
scaffval.b	⊢ 𝐵 = (Base‘𝑊)
scaffval.f	⊢ 𝐹 = (Scalar‘𝑊)
scaffval.k	⊢ 𝐾 = (Base‘𝐹)
scaffval.a	⊢ ∙ = ( ·sf ‘𝑊)

Assertion

Ref	Expression
scaffng	⊢ (𝑊 ∈ 𝑉 → ∙ Fn (𝐾 × 𝐵))

Proof of Theorem scaffng

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

Step	Hyp	Ref	Expression
1		vex 2776	. . . . . 6 ⊢ 𝑥 ∈ V
2		vscaslid 13039	. . . . . . 7 ⊢ ( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ)
3	2	slotex 12903	. . . . . 6 ⊢ (𝑊 ∈ 𝑉 → ( ·𝑠 ‘𝑊) ∈ V)
4		vex 2776	. . . . . . 7 ⊢ 𝑦 ∈ V
5	4	a1i 9	. . . . . 6 ⊢ (𝑊 ∈ 𝑉 → 𝑦 ∈ V)
6		ovexg 5985	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ ( ·𝑠 ‘𝑊) ∈ V ∧ 𝑦 ∈ V) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ V)
7	1, 3, 5, 6	mp3an2i 1355	. . . . 5 ⊢ (𝑊 ∈ 𝑉 → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ V)
8	7	ralrimivw 2581	. . . 4 ⊢ (𝑊 ∈ 𝑉 → ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ V)
9	8	ralrimivw 2581	. . 3 ⊢ (𝑊 ∈ 𝑉 → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ V)
10		eqid 2206	. . . 4 ⊢ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦))
11	10	fnmpo 6295	. . 3 ⊢ (∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ V → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦)) Fn (𝐾 × 𝐵))
12	9, 11	syl 14	. 2 ⊢ (𝑊 ∈ 𝑉 → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦)) Fn (𝐾 × 𝐵))
13		scaffval.b	. . . 4 ⊢ 𝐵 = (Base‘𝑊)
14		scaffval.f	. . . 4 ⊢ 𝐹 = (Scalar‘𝑊)
15		scaffval.k	. . . 4 ⊢ 𝐾 = (Base‘𝐹)
16		scaffval.a	. . . 4 ⊢ ∙ = ( ·sf ‘𝑊)
17		eqid 2206	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
18	13, 14, 15, 16, 17	scaffvalg 14112	. . 3 ⊢ (𝑊 ∈ 𝑉 → ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦)))
19	18	fneq1d 5369	. 2 ⊢ (𝑊 ∈ 𝑉 → ( ∙ Fn (𝐾 × 𝐵) ↔ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦)) Fn (𝐾 × 𝐵)))
20	12, 19	mpbird 167	1 ⊢ (𝑊 ∈ 𝑉 → ∙ Fn (𝐾 × 𝐵))

Syntax hints:   → wi 4   = wceq 1373   ∈ wcel 2177  ∀wral 2485  Vcvv 2773   × cxp 4677   Fn wfn 5271  ‘cfv 5276  (class class class)co 5951   ∈ cmpo 5953  Basecbs 12876  Scalarcsca 12956   ·_𝑠 cvsca 12957   ·_sf cscaf 14094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-cnex 8023  ax-resscn 8024  ax-1re 8026  ax-addrcl 8029

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-5 9105  df-6 9106  df-ndx 12879  df-slot 12880  df-base 12882  df-sca 12969  df-vsca 12970  df-scaf 14096

This theorem is referenced by:  lmodfopnelem1  14130