Theorem linds0 48941

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 4438	. . . . . 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 5242	. . . . . 6 ⊢ ∅ ∈ V
4		breq1 5088	. . . . . . . . 9 ⊢ (𝑓 = ∅ → (𝑓 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ finSupp (0g‘(Scalar‘𝑀))))
5		oveq1 7374	. . . . . . . . . 10 ⊢ (𝑓 = ∅ → (𝑓( linC ‘𝑀)∅) = (∅( linC ‘𝑀)∅))
6	5	eqeq1d 2738	. . . . . . . . 9 ⊢ (𝑓 = ∅ → ((𝑓( linC ‘𝑀)∅) = (0g‘𝑀) ↔ (∅( linC ‘𝑀)∅) = (0g‘𝑀)))
7	4, 6	anbi12d 633	. . . . . . . 8 ⊢ (𝑓 = ∅ → ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g‘𝑀)) ↔ (∅ finSupp (0g‘(Scalar‘𝑀)) ∧ (∅( linC ‘𝑀)∅) = (0g‘𝑀))))
8		fveq1 6839	. . . . . . . . . 10 ⊢ (𝑓 = ∅ → (𝑓‘𝑥) = (∅‘𝑥))
9	8	eqeq1d 2738	. . . . . . . . 9 ⊢ (𝑓 = ∅ → ((𝑓‘𝑥) = (0g‘(Scalar‘𝑀)) ↔ (∅‘𝑥) = (0g‘(Scalar‘𝑀))))
10	9	ralbidv 3160	. . . . . . . 8 ⊢ (𝑓 = ∅ → (∀𝑥 ∈ ∅ (𝑓‘𝑥) = (0g‘(Scalar‘𝑀)) ↔ ∀𝑥 ∈ ∅ (∅‘𝑥) = (0g‘(Scalar‘𝑀))))
11	7, 10	imbi12d 344	. . . . . . 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 4619	. . . . . 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 257	. . . 4 ⊢ (𝑀 ∈ 𝑉 → ∀𝑓 ∈ {∅} ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g‘𝑀)) → ∀𝑥 ∈ ∅ (𝑓‘𝑥) = (0g‘(Scalar‘𝑀))))
15		fvex 6853	. . . . . 6 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
16		map0e 8830	. . . . . 6 ⊢ ((Base‘(Scalar‘𝑀)) ∈ V → ((Base‘(Scalar‘𝑀)) ↑m ∅) = 1o)
17	15, 16	mp1i 13	. . . . 5 ⊢ (𝑀 ∈ 𝑉 → ((Base‘(Scalar‘𝑀)) ↑m ∅) = 1o)
18		df1o2 8412	. . . . 5 ⊢ 1o = {∅}
19	17, 18	eqtrdi 2787	. . . 4 ⊢ (𝑀 ∈ 𝑉 → ((Base‘(Scalar‘𝑀)) ↑m ∅) = {∅})
20	14, 19	raleqtrrdv 3299	. . 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 5297	. . 3 ⊢ ∅ ∈ 𝒫 (Base‘𝑀)
22	20, 21	jctil 519	. 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 2736	. . . 4 ⊢ (Base‘𝑀) = (Base‘𝑀)
24		eqid 2736	. . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀)
25		eqid 2736	. . . 4 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
26		eqid 2736	. . . 4 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
27		eqid 2736	. . . 4 ⊢ (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
28	23, 24, 25, 26, 27	islininds 48922	. . 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 691	. 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 257	1 ⊢ (𝑀 ∈ 𝑉 → ∅ linIndS 𝑀)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  Vcvv 3429  ∅c0 4273  𝒫 cpw 4541  {csn 4567   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  1_oc1o 8398   ↑_m cmap 8773   finSupp cfsupp 9274  Basecbs 17179  Scalarcsca 17223  0_gc0g 17402   linC clinc 48880   linIndS clininds 48916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1o 8405  df-map 8775  df-lininds 48918

This theorem is referenced by: (None)