Theorem scaffng 14326

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 2805	. . . . . 6 ⊢ 𝑥 ∈ V
2		vscaslid 13248	. . . . . . 7 ⊢ ( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ)
3	2	slotex 13111	. . . . . 6 ⊢ (𝑊 ∈ 𝑉 → ( ·𝑠 ‘𝑊) ∈ V)
4		vex 2805	. . . . . . 7 ⊢ 𝑦 ∈ V
5	4	a1i 9	. . . . . 6 ⊢ (𝑊 ∈ 𝑉 → 𝑦 ∈ V)
6		ovexg 6052	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ ( ·𝑠 ‘𝑊) ∈ V ∧ 𝑦 ∈ V) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ V)
7	1, 3, 5, 6	mp3an2i 1378	. . . . 5 ⊢ (𝑊 ∈ 𝑉 → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ V)
8	7	ralrimivw 2606	. . . 4 ⊢ (𝑊 ∈ 𝑉 → ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ V)
9	8	ralrimivw 2606	. . 3 ⊢ (𝑊 ∈ 𝑉 → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ V)
10		eqid 2231	. . . 4 ⊢ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦))
11	10	fnmpo 6367	. . 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 2231	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
18	13, 14, 15, 16, 17	scaffvalg 14323	. . 3 ⊢ (𝑊 ∈ 𝑉 → ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦)))
19	18	fneq1d 5420	. 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 1397   ∈ wcel 2202  ∀wral 2510  Vcvv 2802   × cxp 4723   Fn wfn 5321  ‘cfv 5326  (class class class)co 6018   ∈ cmpo 6020  Basecbs 13084  Scalarcsca 13165   ·_𝑠 cvsca 13166   ·_sf cscaf 14305

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8123  ax-resscn 8124  ax-1re 8126  ax-addrcl 8129

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-ndx 13087  df-slot 13088  df-base 13090  df-sca 13178  df-vsca 13179  df-scaf 14307

This theorem is referenced by:  lmodfopnelem1  14341