Theorem linds0 48505

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 4460	. . . . . 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 5243	. . . . . 6 ⊢ ∅ ∈ V
4		breq1 5092	. . . . . . . . 9 ⊢ (𝑓 = ∅ → (𝑓 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ finSupp (0g‘(Scalar‘𝑀))))
5		oveq1 7353	. . . . . . . . . 10 ⊢ (𝑓 = ∅ → (𝑓( linC ‘𝑀)∅) = (∅( linC ‘𝑀)∅))
6	5	eqeq1d 2733	. . . . . . . . 9 ⊢ (𝑓 = ∅ → ((𝑓( linC ‘𝑀)∅) = (0g‘𝑀) ↔ (∅( linC ‘𝑀)∅) = (0g‘𝑀)))
7	4, 6	anbi12d 632	. . . . . . . 8 ⊢ (𝑓 = ∅ → ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝑓( linC ‘𝑀)∅) = (0g‘𝑀)) ↔ (∅ finSupp (0g‘(Scalar‘𝑀)) ∧ (∅( linC ‘𝑀)∅) = (0g‘𝑀))))
8		fveq1 6821	. . . . . . . . . 10 ⊢ (𝑓 = ∅ → (𝑓‘𝑥) = (∅‘𝑥))
9	8	eqeq1d 2733	. . . . . . . . 9 ⊢ (𝑓 = ∅ → ((𝑓‘𝑥) = (0g‘(Scalar‘𝑀)) ↔ (∅‘𝑥) = (0g‘(Scalar‘𝑀))))
10	9	ralbidv 3155	. . . . . . . 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 4625	. . . . . 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 6835	. . . . . 6 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
16		map0e 8806	. . . . . 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 8392	. . . . 5 ⊢ 1o = {∅}
19	17, 18	eqtrdi 2782	. . . 4 ⊢ (𝑀 ∈ 𝑉 → ((Base‘(Scalar‘𝑀)) ↑m ∅) = {∅})
20	14, 19	raleqtrrdv 3296	. . 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 5292	. . 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 2731	. . . 4 ⊢ (Base‘𝑀) = (Base‘𝑀)
24		eqid 2731	. . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀)
25		eqid 2731	. . . 4 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
26		eqid 2731	. . . 4 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
27		eqid 2731	. . . 4 ⊢ (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
28	23, 24, 25, 26, 27	islininds 48486	. . 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 690	. 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 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436  ∅c0 4280  𝒫 cpw 4547  {csn 4573   class class class wbr 5089  ‘cfv 6481  (class class class)co 7346  1_oc1o 8378   ↑_m cmap 8750   finSupp cfsupp 9245  Basecbs 17120  Scalarcsca 17164  0_gc0g 17343   linC clinc 48444   linIndS clininds 48480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1o 8385  df-map 8752  df-lininds 48482

This theorem is referenced by: (None)