Theorem linds0 47533

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 4513	. . . . . 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 5307	. . . . . 6 ⊢ ∅ ∈ V
4		breq1 5151	. . . . . . . . 9 ⊢ (𝑓 = ∅ → (𝑓 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ finSupp (0g‘(Scalar‘𝑀))))
5		oveq1 7427	. . . . . . . . . 10 ⊢ (𝑓 = ∅ → (𝑓( linC ‘𝑀)∅) = (∅( linC ‘𝑀)∅))
6	5	eqeq1d 2730	. . . . . . . . 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 6896	. . . . . . . . . 10 ⊢ (𝑓 = ∅ → (𝑓‘𝑥) = (∅‘𝑥))
9	8	eqeq1d 2730	. . . . . . . . 9 ⊢ (𝑓 = ∅ → ((𝑓‘𝑥) = (0g‘(Scalar‘𝑀)) ↔ (∅‘𝑥) = (0g‘(Scalar‘𝑀))))
10	9	ralbidv 3174	. . . . . . . 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 4678	. . . . . 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 6910	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
16		map0e 8901	. . . . . . 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 8494	. . . . . 6 ⊢ 1o = {∅}
19	17, 18	eqtrdi 2784	. . . . 5 ⊢ (𝑀 ∈ 𝑉 → ((Base‘(Scalar‘𝑀)) ↑m ∅) = {∅})
20	19	raleqdv 3322	. . . 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 257	. . 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 5356	. . 3 ⊢ ∅ ∈ 𝒫 (Base‘𝑀)
23	21, 22	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)))))
24		eqid 2728	. . . 4 ⊢ (Base‘𝑀) = (Base‘𝑀)
25		eqid 2728	. . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀)
26		eqid 2728	. . . 4 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
27		eqid 2728	. . . 4 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
28		eqid 2728	. . . 4 ⊢ (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
29	24, 25, 26, 27, 28	islininds 47514	. . 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 689	. 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 257	1 ⊢ (𝑀 ∈ 𝑉 → ∅ linIndS 𝑀)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3058  Vcvv 3471  ∅c0 4323  𝒫 cpw 4603  {csn 4629   class class class wbr 5148  ‘cfv 6548  (class class class)co 7420  1_oc1o 8480   ↑_m cmap 8845   finSupp cfsupp 9386  Basecbs 17180  Scalarcsca 17236  0_gc0g 17421   linC clinc 47472   linIndS clininds 47508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1o 8487  df-map 8847  df-lininds 47510

This theorem is referenced by: (None)