Theorem scafeqg 14041

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 14039	. . 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 6053	. . 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 2240	1 ⊢ ((𝑊 ∈ 𝑉 ∧ · Fn (𝐾 × 𝐵)) → ∙ = · )

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1372   ∈ wcel 2175   × cxp 4672   Fn wfn 5265  ‘cfv 5270  (class class class)co 5943   ∈ cmpo 5945  Basecbs 12803  Scalarcsca 12883   ·_𝑠 cvsca 12884   ·_sf cscaf 14021

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-cnex 8015  ax-resscn 8016  ax-1re 8018  ax-addrcl 8021

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-5 9097  df-ndx 12806  df-slot 12807  df-base 12809  df-sca 12896  df-scaf 14023

This theorem is referenced by: (None)