Theorem scaffval 18704

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 6103	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
3		scaffval.f	. . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊)
4	2, 3	syl6eqr 2662	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
5	4	fveq2d 6107	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
6		scaffval.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝐹)
7	5, 6	syl6eqr 2662	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
8		fveq2 6103	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
9		scaffval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
10	8, 9	syl6eqr 2662	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
11		fveq2 6103	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
12		scaffval.s	. . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑊)
13	11, 12	syl6eqr 2662	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · )
14	13	oveqd 6566	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥( ·𝑠 ‘𝑤)𝑦) = (𝑥 · 𝑦))
15	7, 10, 14	mpt2eq123dv 6615	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠 ‘𝑤)𝑦)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
16		df-scaf 18689	. . . 4 ⊢ ·sf = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠 ‘𝑤)𝑦)))
17		df-ov 6552	. . . . . . . 8 ⊢ (𝑥 · 𝑦) = ( · ‘⟨𝑥, 𝑦⟩)
18		fvrn0 6126	. . . . . . . 8 ⊢ ( · ‘⟨𝑥, 𝑦⟩) ∈ (ran · ∪ {∅})
19	17, 18	eqeltri 2684	. . . . . . 7 ⊢ (𝑥 · 𝑦) ∈ (ran · ∪ {∅})
20	19	rgen2w 2909	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐵 (𝑥 · 𝑦) ∈ (ran · ∪ {∅})
21		eqid 2610	. . . . . . 7 ⊢ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦))
22	21	fmpt2 7126	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐵 (𝑥 · 𝑦) ∈ (ran · ∪ {∅}) ↔ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)):(𝐾 × 𝐵)⟶(ran · ∪ {∅}))
23	20, 22	mpbi 219	. . . . 5 ⊢ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)):(𝐾 × 𝐵)⟶(ran · ∪ {∅})
24		fvex 6113	. . . . . . 7 ⊢ (Base‘𝐹) ∈ V
25	6, 24	eqeltri 2684	. . . . . 6 ⊢ 𝐾 ∈ V
26		fvex 6113	. . . . . . 7 ⊢ (Base‘𝑊) ∈ V
27	9, 26	eqeltri 2684	. . . . . 6 ⊢ 𝐵 ∈ V
28	25, 27	xpex 6860	. . . . 5 ⊢ (𝐾 × 𝐵) ∈ V
29		fvex 6113	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝑊) ∈ V
30	12, 29	eqeltri 2684	. . . . . . 7 ⊢ · ∈ V
31	30	rnex 6992	. . . . . 6 ⊢ ran · ∈ V
32		p0ex 4779	. . . . . 6 ⊢ {∅} ∈ V
33	31, 32	unex 6854	. . . . 5 ⊢ (ran · ∪ {∅}) ∈ V
34		fex2 7014	. . . . 5 ⊢ (((𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)):(𝐾 × 𝐵)⟶(ran · ∪ {∅}) ∧ (𝐾 × 𝐵) ∈ V ∧ (ran · ∪ {∅}) ∈ V) → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) ∈ V)
35	23, 28, 33, 34	mp3an 1416	. . . 4 ⊢ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) ∈ V
36	15, 16, 35	fvmpt 6191	. . 3 ⊢ (𝑊 ∈ V → ( ·sf ‘𝑊) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
37		fvprc 6097	. . . . 5 ⊢ (¬ 𝑊 ∈ V → ( ·sf ‘𝑊) = ∅)
38		mpt20 6623	. . . . 5 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) = ∅
39	37, 38	syl6eqr 2662	. . . 4 ⊢ (¬ 𝑊 ∈ V → ( ·sf ‘𝑊) = (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
40		fvprc 6097	. . . . . . . . 9 ⊢ (¬ 𝑊 ∈ V → (Scalar‘𝑊) = ∅)
41	3, 40	syl5eq 2656	. . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → 𝐹 = ∅)
42	41	fveq2d 6107	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (Base‘𝐹) = (Base‘∅))
43	6, 42	syl5eq 2656	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → 𝐾 = (Base‘∅))
44		base0 15740	. . . . . 6 ⊢ ∅ = (Base‘∅)
45	43, 44	syl6eqr 2662	. . . . 5 ⊢ (¬ 𝑊 ∈ V → 𝐾 = ∅)
46		eqid 2610	. . . . 5 ⊢ 𝐵 = 𝐵
47		mpt2eq12 6613	. . . . 5 ⊢ ((𝐾 = ∅ ∧ 𝐵 = 𝐵) → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
48	45, 46, 47	sylancl 693	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
49	39, 48	eqtr4d 2647	. . 3 ⊢ (¬ 𝑊 ∈ V → ( ·sf ‘𝑊) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
50	36, 49	pm2.61i 175	. 2 ⊢ ( ·sf ‘𝑊) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦))
51	1, 50	eqtri 2632	1 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦))

Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∪ cun 3538  ∅c0 3874  {csn 4125  ⟨cop 4131   × cxp 5036  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Basecbs 15695  Scalarcsca 15771   ·_𝑠 cvsca 15772   ·_sf cscaf 18687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-slot 15699  df-base 15700  df-scaf 18689

This theorem is referenced by:  scafval  18705  scafeq  18706  scaffn  18707  lmodscaf  18708  rlmscaf  19029