Theorem linds0 45806

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 4443	. . . . . 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 5231	. . . . . 6 ⊢ ∅ ∈ V
4		breq1 5077	. . . . . . . . 9 ⊢ (𝑓 = ∅ → (𝑓 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ finSupp (0g‘(Scalar‘𝑀))))
5		oveq1 7282	. . . . . . . . . 10 ⊢ (𝑓 = ∅ → (𝑓( linC ‘𝑀)∅) = (∅( linC ‘𝑀)∅))
6	5	eqeq1d 2740	. . . . . . . . 9 ⊢ (𝑓 = ∅ → ((𝑓( linC ‘𝑀)∅) = (0g‘𝑀) ↔ (∅( linC ‘𝑀)∅) = (0g‘𝑀)))
7	4, 6	anbi12d 631	. . . . . . . 8 ⊢ (𝑓 = ∅ → ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g‘𝑀)) ↔ (∅ finSupp (0g‘(Scalar‘𝑀)) ∧ (∅( linC ‘𝑀)∅) = (0g‘𝑀))))
8		fveq1 6773	. . . . . . . . . 10 ⊢ (𝑓 = ∅ → (𝑓‘𝑥) = (∅‘𝑥))
9	8	eqeq1d 2740	. . . . . . . . 9 ⊢ (𝑓 = ∅ → ((𝑓‘𝑥) = (0g‘(Scalar‘𝑀)) ↔ (∅‘𝑥) = (0g‘(Scalar‘𝑀))))
10	9	ralbidv 3112	. . . . . . . 8 ⊢ (𝑓 = ∅ → (∀𝑥 ∈ ∅ (𝑓‘𝑥) = (0g‘(Scalar‘𝑀)) ↔ ∀𝑥 ∈ ∅ (∅‘𝑥) = (0g‘(Scalar‘𝑀))))
11	7, 10	imbi12d 345	. . . . . . 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 4609	. . . . . 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 13	. . . . 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 256	. . . 4 ⊢ (𝑀 ∈ 𝑉 → ∀𝑓 ∈ {∅} ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g‘𝑀)) → ∀𝑥 ∈ ∅ (𝑓‘𝑥) = (0g‘(Scalar‘𝑀))))
15		fvex 6787	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
16		map0e 8670	. . . . . . 7 ⊢ ((Base‘(Scalar‘𝑀)) ∈ V → ((Base‘(Scalar‘𝑀)) ↑m ∅) = 1o)
17	15, 16	mp1i 13	. . . . . 6 ⊢ (𝑀 ∈ 𝑉 → ((Base‘(Scalar‘𝑀)) ↑m ∅) = 1o)
18		df1o2 8304	. . . . . 6 ⊢ 1o = {∅}
19	17, 18	eqtrdi 2794	. . . . 5 ⊢ (𝑀 ∈ 𝑉 → ((Base‘(Scalar‘𝑀)) ↑m ∅) = {∅})
20	19	raleqdv 3348	. . . 4 ⊢ (𝑀 ∈ 𝑉 → (∀𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g‘𝑀)) → ∀𝑥 ∈ ∅ (𝑓‘𝑥) = (0g‘(Scalar‘𝑀))) ↔ ∀𝑓 ∈ {∅} ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g‘𝑀)) → ∀𝑥 ∈ ∅ (𝑓‘𝑥) = (0g‘(Scalar‘𝑀)))))
21	14, 20	mpbird 256	. . 3 ⊢ (𝑀 ∈ 𝑉 → ∀𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g‘𝑀)) → ∀𝑥 ∈ ∅ (𝑓‘𝑥) = (0g‘(Scalar‘𝑀))))
22		0elpw 5278	. . 3 ⊢ ∅ ∈ 𝒫 (Base‘𝑀)
23	21, 22	jctil 520	. 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)))))
24		eqid 2738	. . . 4 ⊢ (Base‘𝑀) = (Base‘𝑀)
25		eqid 2738	. . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀)
26		eqid 2738	. . . 4 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
27		eqid 2738	. . . 4 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
28		eqid 2738	. . . 4 ⊢ (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
29	24, 25, 26, 27, 28	islininds 45787	. . 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))))))
30	3, 29	mpan 687	. 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))))))
31	23, 30	mpbird 256	1 ⊢ (𝑀 ∈ 𝑉 → ∅ linIndS 𝑀)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432  ∅c0 4256  𝒫 cpw 4533  {csn 4561   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275  1_oc1o 8290   ↑_m cmap 8615   finSupp cfsupp 9128  Basecbs 16912  Scalarcsca 16965  0_gc0g 17150   linC clinc 45745   linIndS clininds 45781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1o 8297  df-map 8617  df-lininds 45783

This theorem is referenced by: (None)