Theorem linds0 49302

Description: The empty set is always a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)

Assertion

Ref	Expression
linds0	⊢ (𝑀 ∈ 𝑉 → ∅ linIndS 𝑀)

Proof of Theorem linds0

Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ral0 4461	. . . . . 6 ⊢ ∀𝑥 ∈ ∅ (∅‘𝑥) = (0g‘(Scalar‘𝑀))
2	1	2a1i 12	. . . . 5 ⊢ (𝑀 ∈ 𝑉 → ((∅ finSupp (0g‘(Scalar‘𝑀)) ∧ (∅( linC ‘𝑀)∅) = (0g‘𝑀)) → ∀𝑥 ∈ ∅ (∅‘𝑥) = (0g‘(Scalar‘𝑀))))
3		0ex 5272	. . . . . 6 ⊢ ∅ ∈ V
4		breq1 5114	. . . . . . . . 9 ⊢ (𝑓 = ∅ → (𝑓 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ finSupp (0g‘(Scalar‘𝑀))))
5		oveq1 7426	. . . . . . . . . 10 ⊢ (𝑓 = ∅ → (𝑓( linC ‘𝑀)∅) = (∅( linC ‘𝑀)∅))
6	5	eqeq1d 2767	. . . . . . . . 9 ⊢ (𝑓 = ∅ → ((𝑓( linC ‘𝑀)∅) = (0g‘𝑀) ↔ (∅( linC ‘𝑀)∅) = (0g‘𝑀)))
7	4, 6	anbi12d 644	. . . . . . . 8 ⊢ (𝑓 = ∅ → ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g‘𝑀)) ↔ (∅ finSupp (0g‘(Scalar‘𝑀)) ∧ (∅( linC ‘𝑀)∅) = (0g‘𝑀))))
8		fveq1 6884	. . . . . . . . . 10 ⊢ (𝑓 = ∅ → (𝑓‘𝑥) = (∅‘𝑥))
9	8	eqeq1d 2767	. . . . . . . . 9 ⊢ (𝑓 = ∅ → ((𝑓‘𝑥) = (0g‘(Scalar‘𝑀)) ↔ (∅‘𝑥) = (0g‘(Scalar‘𝑀))))
10	9	ralbidv 3190	. . . . . . . 8 ⊢ (𝑓 = ∅ → (∀𝑥 ∈ ∅ (𝑓‘𝑥) = (0g‘(Scalar‘𝑀)) ↔ ∀𝑥 ∈ ∅ (∅‘𝑥) = (0g‘(Scalar‘𝑀))))
11	7, 10	imbi12d 347	. . . . . . 7 ⊢ (𝑓 = ∅ → (((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g‘𝑀)) → ∀𝑥 ∈ ∅ (𝑓‘𝑥) = (0g‘(Scalar‘𝑀))) ↔ ((∅ finSupp (0g‘(Scalar‘𝑀)) ∧ (∅( linC ‘𝑀)∅) = (0g‘𝑀)) → ∀𝑥 ∈ ∅ (∅‘𝑥) = (0g‘(Scalar‘𝑀)))))
12	11	ralsng 4643	. . . . . 6 ⊢ (∅ ∈ V → (∀𝑓 ∈ {∅} ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g‘𝑀)) → ∀𝑥 ∈ ∅ (𝑓‘𝑥) = (0g‘(Scalar‘𝑀))) ↔ ((∅ finSupp (0g‘(Scalar‘𝑀)) ∧ (∅( linC ‘𝑀)∅) = (0g‘𝑀)) → ∀𝑥 ∈ ∅ (∅‘𝑥) = (0g‘(Scalar‘𝑀)))))
13	3, 12	mp1i 14	. . . . 5 ⊢ (𝑀 ∈ 𝑉 → (∀𝑓 ∈ {∅} ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g‘𝑀)) → ∀𝑥 ∈ ∅ (𝑓‘𝑥) = (0g‘(Scalar‘𝑀))) ↔ ((∅ finSupp (0g‘(Scalar‘𝑀)) ∧ (∅( linC ‘𝑀)∅) = (0g‘𝑀)) → ∀𝑥 ∈ ∅ (∅‘𝑥) = (0g‘(Scalar‘𝑀)))))
14	2, 13	mpbird 260	. . . 4 ⊢ (𝑀 ∈ 𝑉 → ∀𝑓 ∈ {∅} ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g‘𝑀)) → ∀𝑥 ∈ ∅ (𝑓‘𝑥) = (0g‘(Scalar‘𝑀))))
15		fvex 6898	. . . . . 6 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
16		map0e 8886	. . . . . 6 ⊢ ((Base‘(Scalar‘𝑀)) ∈ V → ((Base‘(Scalar‘𝑀)) ↑m ∅) = 1o)
17	15, 16	mp1i 14	. . . . 5 ⊢ (𝑀 ∈ 𝑉 → ((Base‘(Scalar‘𝑀)) ↑m ∅) = 1o)
18		df1o2 8466	. . . . 5 ⊢ 1o = {∅}
19	17, 18	eqtrdi 2816	. . . 4 ⊢ (𝑀 ∈ 𝑉 → ((Base‘(Scalar‘𝑀)) ↑m ∅) = {∅})
20	14, 19	raleqtrrdv 3329	. . 3 ⊢ (𝑀 ∈ 𝑉 → ∀𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g‘𝑀)) → ∀𝑥 ∈ ∅ (𝑓‘𝑥) = (0g‘(Scalar‘𝑀))))
21		0elpw 5328	. . 3 ⊢ ∅ ∈ 𝒫 (Base‘𝑀)
22	20, 21	jctil 529	. 2 ⊢ (𝑀 ∈ 𝑉 → (∅ ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g‘𝑀)) → ∀𝑥 ∈ ∅ (𝑓‘𝑥) = (0g‘(Scalar‘𝑀)))))
23		eqid 2765	. . . 4 ⊢ (Base‘𝑀) = (Base‘𝑀)
24		eqid 2765	. . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀)
25		eqid 2765	. . . 4 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
26		eqid 2765	. . . 4 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
27		eqid 2765	. . . 4 ⊢ (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
28	23, 24, 25, 26, 27	islininds 49283	. . 3 ⊢ ((∅ ∈ V ∧ 𝑀 ∈ 𝑉) → (∅ linIndS 𝑀 ↔ (∅ ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g‘𝑀)) → ∀𝑥 ∈ ∅ (𝑓‘𝑥) = (0g‘(Scalar‘𝑀))))))
29	3, 28	mpan 703	. 2 ⊢ (𝑀 ∈ 𝑉 → (∅ linIndS 𝑀 ↔ (∅ ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g‘𝑀)) → ∀𝑥 ∈ ∅ (𝑓‘𝑥) = (0g‘(Scalar‘𝑀))))))
30	22, 29	mpbird 260	1 ⊢ (𝑀 ∈ 𝑉 → ∅ linIndS 𝑀)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  Vcvv 3457  ∅c0 4286  𝒫 cpw 4564  {csn 4591   class class class wbr 5111  ‘cfv 6540  (class class class)co 7419  1_oc1o 8452   ↑_m cmap 8830   finSupp cfsupp 9328  Basecbs 17291  Scalarcsca 17335  0_gc0g 17514   linC clinc 49241   linIndS clininds 49277

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1o 8459  df-map 8832  df-lininds 49279

This theorem is used by: (None)