Theorem scaffval 19368

Description: The scalar multiplication operation as a function. (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
scaffval	⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦))

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

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

Proof of Theorem scaffval

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		scaffval.a	. 2 ⊢ ∙ = ( ·sf ‘𝑊)
2		fveq2 6493	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
3		scaffval.f	. . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊)
4	2, 3	syl6eqr 2826	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
5	4	fveq2d 6497	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
6		scaffval.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝐹)
7	5, 6	syl6eqr 2826	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
8		fveq2 6493	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
9		scaffval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
10	8, 9	syl6eqr 2826	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
11		fveq2 6493	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
12		scaffval.s	. . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑊)
13	11, 12	syl6eqr 2826	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · )
14	13	oveqd 6987	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥( ·𝑠 ‘𝑤)𝑦) = (𝑥 · 𝑦))
15	7, 10, 14	mpoeq123dv 7041	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠 ‘𝑤)𝑦)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
16		df-scaf 19353	. . . 4 ⊢ ·sf = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠 ‘𝑤)𝑦)))
17		df-ov 6973	. . . . . . . 8 ⊢ (𝑥 · 𝑦) = ( · ‘⟨𝑥, 𝑦⟩)
18		fvrn0 6521	. . . . . . . 8 ⊢ ( · ‘⟨𝑥, 𝑦⟩) ∈ (ran · ∪ {∅})
19	17, 18	eqeltri 2856	. . . . . . 7 ⊢ (𝑥 · 𝑦) ∈ (ran · ∪ {∅})
20	19	rgen2w 3095	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐵 (𝑥 · 𝑦) ∈ (ran · ∪ {∅})
21		eqid 2772	. . . . . . 7 ⊢ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦))
22	21	fmpo 7568	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐵 (𝑥 · 𝑦) ∈ (ran · ∪ {∅}) ↔ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)):(𝐾 × 𝐵)⟶(ran · ∪ {∅}))
23	20, 22	mpbi 222	. . . . 5 ⊢ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)):(𝐾 × 𝐵)⟶(ran · ∪ {∅})
24	6	fvexi 6507	. . . . . 6 ⊢ 𝐾 ∈ V
25	9	fvexi 6507	. . . . . 6 ⊢ 𝐵 ∈ V
26	24, 25	xpex 7287	. . . . 5 ⊢ (𝐾 × 𝐵) ∈ V
27	12	fvexi 6507	. . . . . . 7 ⊢ · ∈ V
28	27	rnex 7426	. . . . . 6 ⊢ ran · ∈ V
29		p0ex 5131	. . . . . 6 ⊢ {∅} ∈ V
30	28, 29	unex 7280	. . . . 5 ⊢ (ran · ∪ {∅}) ∈ V
31		fex2 7447	. . . . 5 ⊢ (((𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)):(𝐾 × 𝐵)⟶(ran · ∪ {∅}) ∧ (𝐾 × 𝐵) ∈ V ∧ (ran · ∪ {∅}) ∈ V) → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) ∈ V)
32	23, 26, 30, 31	mp3an 1440	. . . 4 ⊢ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) ∈ V
33	15, 16, 32	fvmpt 6589	. . 3 ⊢ (𝑊 ∈ V → ( ·sf ‘𝑊) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
34		fvprc 6486	. . . . 5 ⊢ (¬ 𝑊 ∈ V → ( ·sf ‘𝑊) = ∅)
35		mpo0 7049	. . . . 5 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) = ∅
36	34, 35	syl6eqr 2826	. . . 4 ⊢ (¬ 𝑊 ∈ V → ( ·sf ‘𝑊) = (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
37		fvprc 6486	. . . . . . . . 9 ⊢ (¬ 𝑊 ∈ V → (Scalar‘𝑊) = ∅)
38	3, 37	syl5eq 2820	. . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → 𝐹 = ∅)
39	38	fveq2d 6497	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (Base‘𝐹) = (Base‘∅))
40	6, 39	syl5eq 2820	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → 𝐾 = (Base‘∅))
41		base0 16386	. . . . . 6 ⊢ ∅ = (Base‘∅)
42	40, 41	syl6eqr 2826	. . . . 5 ⊢ (¬ 𝑊 ∈ V → 𝐾 = ∅)
43		eqid 2772	. . . . 5 ⊢ 𝐵 = 𝐵
44		mpoeq12 7039	. . . . 5 ⊢ ((𝐾 = ∅ ∧ 𝐵 = 𝐵) → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
45	42, 43, 44	sylancl 577	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
46	36, 45	eqtr4d 2811	. . 3 ⊢ (¬ 𝑊 ∈ V → ( ·sf ‘𝑊) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
47	33, 46	pm2.61i 177	. 2 ⊢ ( ·sf ‘𝑊) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦))
48	1, 47	eqtri 2796	1 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦))

Syntax hints:  ¬ wn 3   = wceq 1507   ∈ wcel 2050  ∀wral 3082  Vcvv 3409   ∪ cun 3821  ∅c0 4172  {csn 4435  ⟨cop 4441   × cxp 5399  ran crn 5402  ⟶wf 6178  ‘cfv 6182  (class class class)co 6970   ∈ cmpo 6972  Basecbs 16333  Scalarcsca 16418   ·_𝑠 cvsca 16419   ·_sf cscaf 19351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5306  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-1st 7495  df-2nd 7496  df-slot 16337  df-base 16339  df-scaf 19353

This theorem is referenced by:  scafval  19369  scafeq  19370  scaffn  19371  lmodscaf  19372  rlmscaf  19696