Theorem scaffvalg 14344

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 2813	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
3		df-scaf 14328	. . . 4 ⊢ ·sf = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠 ‘𝑤)𝑦)))
4		fveq2 5642	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
5		scaffval.f	. . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊)
6	4, 5	eqtr4di 2281	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
7	6	fveq2d 5646	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
8		scaffval.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝐹)
9	7, 8	eqtr4di 2281	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
10		fveq2 5642	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
11		scaffval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
12	10, 11	eqtr4di 2281	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
13		fveq2 5642	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
14		scaffval.s	. . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑊)
15	13, 14	eqtr4di 2281	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · )
16	15	oveqd 6040	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥( ·𝑠 ‘𝑤)𝑦) = (𝑥 · 𝑦))
17	9, 12, 16	mpoeq123dv 6088	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠 ‘𝑤)𝑦)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
18		elex 2813	. . . 4 ⊢ (𝑊 ∈ V → 𝑊 ∈ V)
19		basfn 13164	. . . . . . 7 ⊢ Base Fn V
20		scaslid 13259	. . . . . . . . 9 ⊢ (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
21	20	slotex 13132	. . . . . . . 8 ⊢ (𝑊 ∈ V → (Scalar‘𝑊) ∈ V)
22	5, 21	eqeltrid 2317	. . . . . . 7 ⊢ (𝑊 ∈ V → 𝐹 ∈ V)
23		funfvex 5659	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝐹 ∈ dom Base) → (Base‘𝐹) ∈ V)
24	23	funfni 5434	. . . . . . 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 2317	. . . . 5 ⊢ (𝑊 ∈ V → 𝐾 ∈ V)
27		funfvex 5659	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑊 ∈ dom Base) → (Base‘𝑊) ∈ V)
28	27	funfni 5434	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑊 ∈ V) → (Base‘𝑊) ∈ V)
29	19, 28	mpan 424	. . . . . 6 ⊢ (𝑊 ∈ V → (Base‘𝑊) ∈ V)
30	11, 29	eqeltrid 2317	. . . . 5 ⊢ (𝑊 ∈ V → 𝐵 ∈ V)
31		mpoexga 6382	. . . . 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 5743	. . 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 2275	1 ⊢ (𝑊 ∈ 𝑉 → ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))

Syntax hints:   → wi 4   = wceq 1397   ∈ wcel 2201  Vcvv 2801   Fn wfn 5323  ‘cfv 5328  (class class class)co 6023   ∈ cmpo 6025  Basecbs 13105  Scalarcsca 13186   ·_𝑠 cvsca 13187   ·_sf cscaf 14326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-cnex 8128  ax-resscn 8129  ax-1re 8131  ax-addrcl 8134

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-5 9210  df-ndx 13108  df-slot 13109  df-base 13111  df-sca 13199  df-scaf 14328

This theorem is referenced by:  scafvalg  14345  scafeqg  14346  scaffng  14347  lmodscaf  14348