Theorem scaffng 14629

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 2824	. . . . . 6 ⊢ 𝑥 ∈ V
2		vscaslid 13500	. . . . . . 7 ⊢ ( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ)
3	2	slotex 13362	. . . . . 6 ⊢ (𝑊 ∈ 𝑉 → ( ·𝑠 ‘𝑊) ∈ V)
4		vex 2824	. . . . . . 7 ⊢ 𝑦 ∈ V
5	4	a1i 9	. . . . . 6 ⊢ (𝑊 ∈ 𝑉 → 𝑦 ∈ V)
6		ovexg 6113	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ ( ·𝑠 ‘𝑊) ∈ V ∧ 𝑦 ∈ V) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ V)
7	1, 3, 5, 6	mp3an2i 1383	. . . . 5 ⊢ (𝑊 ∈ 𝑉 → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ V)
8	7	ralrimivw 2624	. . . 4 ⊢ (𝑊 ∈ 𝑉 → ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ V)
9	8	ralrimivw 2624	. . 3 ⊢ (𝑊 ∈ 𝑉 → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ V)
10		eqid 2238	. . . 4 ⊢ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦))
11	10	fnmpo 6432	. . 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 2238	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
18	13, 14, 15, 16, 17	scaffvalg 14626	. . 3 ⊢ (𝑊 ∈ 𝑉 → ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦)))
19	18	fneq1d 5469	. 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 1402   ∈ wcel 2209  ∀wral 2528  Vcvv 2821   × cxp 4770   Fn wfn 5370  ‘cfv 5375  (class class class)co 6079   ∈ cmpo 6081  Basecbs 13335  Scalarcsca 13417   ·_𝑠 cvsca 13418   ·_sf cscaf 14607

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-ndx 13338  df-slot 13339  df-base 13341  df-sca 13430  df-vsca 13431  df-scaf 14609

This theorem is referenced by:  lmodfopnelem1  14644