Theorem scafeqg 14628

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 14626	. . 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 6191	. . 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 2274	1 ⊢ ((𝑊 ∈ 𝑉 ∧ · Fn (𝐾 × 𝐵)) → ∙ = · )

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209   × cxp 4770   Fn wfn 5370  ‘cfv 5375  (class class class)co 6079   ∈ cmpo 6081  Basecbs 13335  Scalarcsca 13417   ·_𝑠 cvsca 13418   ·_sf cscaf 14607

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-ndx 13338  df-slot 13339  df-base 13341  df-sca 13430  df-scaf 14609

This theorem is referenced by: (None)