Theorem scaffval 18881

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 6191	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
3		scaffval.f	. . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊)
4	2, 3	syl6eqr 2674	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
5	4	fveq2d 6195	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
6		scaffval.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝐹)
7	5, 6	syl6eqr 2674	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
8		fveq2 6191	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
9		scaffval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
10	8, 9	syl6eqr 2674	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
11		fveq2 6191	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
12		scaffval.s	. . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑊)
13	11, 12	syl6eqr 2674	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · )
14	13	oveqd 6667	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥( ·𝑠 ‘𝑤)𝑦) = (𝑥 · 𝑦))
15	7, 10, 14	mpt2eq123dv 6717	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠 ‘𝑤)𝑦)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
16		df-scaf 18866	. . . 4 ⊢ ·sf = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)), 𝑦 ∈ (Base‘𝑤) ↦ (𝑥( ·𝑠 ‘𝑤)𝑦)))
17		df-ov 6653	. . . . . . . 8 ⊢ (𝑥 · 𝑦) = ( · ‘⟨𝑥, 𝑦⟩)
18		fvrn0 6216	. . . . . . . 8 ⊢ ( · ‘⟨𝑥, 𝑦⟩) ∈ (ran · ∪ {∅})
19	17, 18	eqeltri 2697	. . . . . . 7 ⊢ (𝑥 · 𝑦) ∈ (ran · ∪ {∅})
20	19	rgen2w 2925	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐵 (𝑥 · 𝑦) ∈ (ran · ∪ {∅})
21		eqid 2622	. . . . . . 7 ⊢ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦))
22	21	fmpt2 7237	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐵 (𝑥 · 𝑦) ∈ (ran · ∪ {∅}) ↔ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)):(𝐾 × 𝐵)⟶(ran · ∪ {∅}))
23	20, 22	mpbi 220	. . . . 5 ⊢ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)):(𝐾 × 𝐵)⟶(ran · ∪ {∅})
24		fvex 6201	. . . . . . 7 ⊢ (Base‘𝐹) ∈ V
25	6, 24	eqeltri 2697	. . . . . 6 ⊢ 𝐾 ∈ V
26		fvex 6201	. . . . . . 7 ⊢ (Base‘𝑊) ∈ V
27	9, 26	eqeltri 2697	. . . . . 6 ⊢ 𝐵 ∈ V
28	25, 27	xpex 6962	. . . . 5 ⊢ (𝐾 × 𝐵) ∈ V
29		fvex 6201	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝑊) ∈ V
30	12, 29	eqeltri 2697	. . . . . . 7 ⊢ · ∈ V
31	30	rnex 7100	. . . . . 6 ⊢ ran · ∈ V
32		p0ex 4853	. . . . . 6 ⊢ {∅} ∈ V
33	31, 32	unex 6956	. . . . 5 ⊢ (ran · ∪ {∅}) ∈ V
34		fex2 7121	. . . . 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 1424	. . . 4 ⊢ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) ∈ V
36	15, 16, 35	fvmpt 6282	. . 3 ⊢ (𝑊 ∈ V → ( ·sf ‘𝑊) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
37		fvprc 6185	. . . . 5 ⊢ (¬ 𝑊 ∈ V → ( ·sf ‘𝑊) = ∅)
38		mpt20 6725	. . . . 5 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) = ∅
39	37, 38	syl6eqr 2674	. . . 4 ⊢ (¬ 𝑊 ∈ V → ( ·sf ‘𝑊) = (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
40		fvprc 6185	. . . . . . . . 9 ⊢ (¬ 𝑊 ∈ V → (Scalar‘𝑊) = ∅)
41	3, 40	syl5eq 2668	. . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → 𝐹 = ∅)
42	41	fveq2d 6195	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (Base‘𝐹) = (Base‘∅))
43	6, 42	syl5eq 2668	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → 𝐾 = (Base‘∅))
44		base0 15912	. . . . . 6 ⊢ ∅ = (Base‘∅)
45	43, 44	syl6eqr 2674	. . . . 5 ⊢ (¬ 𝑊 ∈ V → 𝐾 = ∅)
46		eqid 2622	. . . . 5 ⊢ 𝐵 = 𝐵
47		mpt2eq12 6715	. . . . 5 ⊢ ((𝐾 = ∅ ∧ 𝐵 = 𝐵) → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
48	45, 46, 47	sylancl 694	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
49	39, 48	eqtr4d 2659	. . 3 ⊢ (¬ 𝑊 ∈ V → ( ·sf ‘𝑊) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
50	36, 49	pm2.61i 176	. 2 ⊢ ( ·sf ‘𝑊) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦))
51	1, 50	eqtri 2644	1 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦))

Syntax hints:  ¬ wn 3   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∪ cun 3572  ∅c0 3915  {csn 4177  ⟨cop 4183   × cxp 5112  ran crn 5115  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  Basecbs 15857  Scalarcsca 15944   ·_𝑠 cvsca 15945   ·_sf cscaf 18864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-slot 15861  df-base 15863  df-scaf 18866

This theorem is referenced by:  scafval  18882  scafeq  18883  scaffn  18884  lmodscaf  18885  rlmscaf  19208