Theorem scafeqg 13807

Description: If the scalar multiplication operation is already a function, the functionalization of it is equal to the original operation. (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)
scaffval.s	⊢ · = ( ·𝑠 ‘𝑊)

Assertion

Ref	Expression
scafeqg	⊢ ((𝑊 ∈ 𝑉 ∧ · Fn (𝐾 × 𝐵)) → ∙ = · )

Proof of Theorem scafeqg

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

Step	Hyp	Ref	Expression
1		scaffval.b	. . . 4 ⊢ 𝐵 = (Base‘𝑊)
2		scaffval.f	. . . 4 ⊢ 𝐹 = (Scalar‘𝑊)
3		scaffval.k	. . . 4 ⊢ 𝐾 = (Base‘𝐹)
4		scaffval.a	. . . 4 ⊢ ∙ = ( ·sf ‘𝑊)
5		scaffval.s	. . . 4 ⊢ · = ( ·𝑠 ‘𝑊)
6	1, 2, 3, 4, 5	scaffvalg 13805	. . 3 ⊢ (𝑊 ∈ 𝑉 → ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
7	6	adantr 276	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ · Fn (𝐾 × 𝐵)) → ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
8		fnovim 6028	. . 3 ⊢ ( · Fn (𝐾 × 𝐵) → · = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
9	8	adantl 277	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ · Fn (𝐾 × 𝐵)) → · = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
10	7, 9	eqtr4d 2229	1 ⊢ ((𝑊 ∈ 𝑉 ∧ · Fn (𝐾 × 𝐵)) → ∙ = · )

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164   × cxp 4658   Fn wfn 5250  ‘cfv 5255  (class class class)co 5919   ∈ cmpo 5921  Basecbs 12621  Scalarcsca 12701   ·_𝑠 cvsca 12702   ·_sf cscaf 13787

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-5 9046  df-ndx 12624  df-slot 12625  df-base 12627  df-sca 12714  df-scaf 13789

This theorem is referenced by: (None)