Theorem scaffn 20869

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
scaffn	⊢ ∙ Fn (𝐾 × 𝐵)

Proof of Theorem scaffn

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

Step	Hyp	Ref	Expression
1		scaffval.b	. . 3 ⊢ 𝐵 = (Base‘𝑊)
2		scaffval.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
3		scaffval.k	. . 3 ⊢ 𝐾 = (Base‘𝐹)
4		scaffval.a	. . 3 ⊢ ∙ = ( ·sf ‘𝑊)
5		eqid 2737	. . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
6	1, 2, 3, 4, 5	scaffval 20866	. 2 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦))
7		ovex 7393	. 2 ⊢ (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ V
8	6, 7	fnmpoi 8016	1 ⊢ ∙ Fn (𝐾 × 𝐵)

Syntax hints:   = wceq 1542   × cxp 5622   Fn wfn 6487  ‘cfv 6492  (class class class)co 7360  Basecbs 17170  Scalarcsca 17214   ·_𝑠 cvsca 17215   ·_sf cscaf 20847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-scaf 20849

This theorem is referenced by: (None)