Theorem scaffng 13404

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 2742	. . . . . 6 ⊢ 𝑥 ∈ V
2		vscaslid 12623	. . . . . . 7 ⊢ ( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ)
3	2	slotex 12491	. . . . . 6 ⊢ (𝑊 ∈ 𝑉 → ( ·𝑠 ‘𝑊) ∈ V)
4		vex 2742	. . . . . . 7 ⊢ 𝑦 ∈ V
5	4	a1i 9	. . . . . 6 ⊢ (𝑊 ∈ 𝑉 → 𝑦 ∈ V)
6		ovexg 5911	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ ( ·𝑠 ‘𝑊) ∈ V ∧ 𝑦 ∈ V) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ V)
7	1, 3, 5, 6	mp3an2i 1342	. . . . 5 ⊢ (𝑊 ∈ 𝑉 → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ V)
8	7	ralrimivw 2551	. . . 4 ⊢ (𝑊 ∈ 𝑉 → ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ V)
9	8	ralrimivw 2551	. . 3 ⊢ (𝑊 ∈ 𝑉 → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ V)
10		eqid 2177	. . . 4 ⊢ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦))
11	10	fnmpo 6205	. . 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 2177	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
18	13, 14, 15, 16, 17	scaffvalg 13401	. . 3 ⊢ (𝑊 ∈ 𝑉 → ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦)))
19	18	fneq1d 5308	. 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 1353   ∈ wcel 2148  ∀wral 2455  Vcvv 2739   × cxp 4626   Fn wfn 5213  ‘cfv 5218  (class class class)co 5877   ∈ cmpo 5879  Basecbs 12464  Scalarcsca 12541   ·_𝑠 cvsca 12542   ·_sf cscaf 13383

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-5 8983  df-6 8984  df-ndx 12467  df-slot 12468  df-base 12470  df-sca 12554  df-vsca 12555  df-scaf 13385

This theorem is referenced by:  lmodfopnelem1  13419