Theorem scafeqg 14070

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 14068	. . 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 6054	. . 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 2241	1 ⊢ ((𝑊 ∈ 𝑉 ∧ · Fn (𝐾 × 𝐵)) → ∙ = · )

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2176   × cxp 4673   Fn wfn 5266  ‘cfv 5271  (class class class)co 5944   ∈ cmpo 5946  Basecbs 12832  Scalarcsca 12912   ·_𝑠 cvsca 12913   ·_sf cscaf 14050

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-5 9098  df-ndx 12835  df-slot 12836  df-base 12838  df-sca 12925  df-scaf 14052

This theorem is referenced by: (None)