Theorem scaffvalg 13582

Description: The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.)

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
scaffvalg	⊢ (𝑊 ∈ 𝑉 → ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥, · ,𝑦   𝑥,𝑊,𝑦   𝑥,𝑉,𝑦

Allowed substitution hints:   ∙ (𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem scaffvalg

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		scaffval.a	. 2 ⊢ ∙ = ( ·sf ‘𝑊)
2		elex 2762	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
3		df-scaf 13566	. . . 4 ⊢ ·sf = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠 ‘𝑤)𝑦)))
4		fveq2 5529	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
5		scaffval.f	. . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊)
6	4, 5	eqtr4di 2239	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
7	6	fveq2d 5533	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
8		scaffval.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝐹)
9	7, 8	eqtr4di 2239	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
10		fveq2 5529	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
11		scaffval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
12	10, 11	eqtr4di 2239	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
13		fveq2 5529	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
14		scaffval.s	. . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑊)
15	13, 14	eqtr4di 2239	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · )
16	15	oveqd 5907	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥( ·𝑠 ‘𝑤)𝑦) = (𝑥 · 𝑦))
17	9, 12, 16	mpoeq123dv 5952	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠 ‘𝑤)𝑦)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
18		elex 2762	. . . 4 ⊢ (𝑊 ∈ V → 𝑊 ∈ V)
19		basfn 12537	. . . . . . 7 ⊢ Base Fn V
20		scaslid 12629	. . . . . . . . 9 ⊢ (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
21	20	slotex 12506	. . . . . . . 8 ⊢ (𝑊 ∈ V → (Scalar‘𝑊) ∈ V)
22	5, 21	eqeltrid 2275	. . . . . . 7 ⊢ (𝑊 ∈ V → 𝐹 ∈ V)
23		funfvex 5546	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝐹 ∈ dom Base) → (Base‘𝐹) ∈ V)
24	23	funfni 5330	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝐹 ∈ V) → (Base‘𝐹) ∈ V)
25	19, 22, 24	sylancr 414	. . . . . 6 ⊢ (𝑊 ∈ V → (Base‘𝐹) ∈ V)
26	8, 25	eqeltrid 2275	. . . . 5 ⊢ (𝑊 ∈ V → 𝐾 ∈ V)
27		funfvex 5546	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑊 ∈ dom Base) → (Base‘𝑊) ∈ V)
28	27	funfni 5330	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑊 ∈ V) → (Base‘𝑊) ∈ V)
29	19, 28	mpan 424	. . . . . 6 ⊢ (𝑊 ∈ V → (Base‘𝑊) ∈ V)
30	11, 29	eqeltrid 2275	. . . . 5 ⊢ (𝑊 ∈ V → 𝐵 ∈ V)
31		mpoexga 6230	. . . . 5 ⊢ ((𝐾 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) ∈ V)
32	26, 30, 31	syl2anc 411	. . . 4 ⊢ (𝑊 ∈ V → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) ∈ V)
33	3, 17, 18, 32	fvmptd3 5624	. . 3 ⊢ (𝑊 ∈ V → ( ·sf ‘𝑊) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
34	2, 33	syl 14	. 2 ⊢ (𝑊 ∈ 𝑉 → ( ·sf ‘𝑊) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
35	1, 34	eqtrid 2233	1 ⊢ (𝑊 ∈ 𝑉 → ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))

Syntax hints:   → wi 4   = wceq 1363   ∈ wcel 2159  Vcvv 2751   Fn wfn 5225  ‘cfv 5230  (class class class)co 5890   ∈ cmpo 5892  Basecbs 12479  Scalarcsca 12557   ·_𝑠 cvsca 12558   ·_sf cscaf 13564

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-cnex 7919  ax-resscn 7920  ax-1re 7922  ax-addrcl 7925

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-reu 2474  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-id 4307  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-inn 8937  df-2 8995  df-3 8996  df-4 8997  df-5 8998  df-ndx 12482  df-slot 12483  df-base 12485  df-sca 12570  df-scaf 13566

This theorem is referenced by:  scafvalg  13583  scafeqg  13584  scaffng  13585  lmodscaf  13586