Theorem lcoel0 48459

Description: The zero vector is always a linear combination. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)

Assertion

Ref	Expression
lcoel0	⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (0g‘𝑀) ∈ (𝑀 LinCo 𝑉))

Proof of Theorem lcoel0

Dummy variables 𝑠 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6835	. . . 4 ⊢ (0g‘𝑀) ∈ V
2	1	snid 4615	. . 3 ⊢ (0g‘𝑀) ∈ {(0g‘𝑀)}
3		oveq2 7354	. . . 4 ⊢ (𝑉 = ∅ → (𝑀 LinCo 𝑉) = (𝑀 LinCo ∅))
4		lmodgrp 20798	. . . . . 6 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
5		grpmnd 18850	. . . . . 6 ⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
6		lco0 48458	. . . . . 6 ⊢ (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {(0g‘𝑀)})
7	4, 5, 6	3syl 18	. . . . 5 ⊢ (𝑀 ∈ LMod → (𝑀 LinCo ∅) = {(0g‘𝑀)})
8	7	adantr 480	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo ∅) = {(0g‘𝑀)})
9	3, 8	sylan9eq 2786	. . 3 ⊢ ((𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (𝑀 LinCo 𝑉) = {(0g‘𝑀)})
10	2, 9	eleqtrrid 2838	. 2 ⊢ ((𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (0g‘𝑀) ∈ (𝑀 LinCo 𝑉))
11		eqid 2731	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
12		eqid 2731	. . . . . 6 ⊢ (0g‘𝑀) = (0g‘𝑀)
13	11, 12	lmod0vcl 20822	. . . . 5 ⊢ (𝑀 ∈ LMod → (0g‘𝑀) ∈ (Base‘𝑀))
14	13	adantr 480	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (0g‘𝑀) ∈ (Base‘𝑀))
15	14	adantl 481	. . 3 ⊢ ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (0g‘𝑀) ∈ (Base‘𝑀))
16		eqid 2731	. . . . . 6 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
17		eqid 2731	. . . . . 6 ⊢ (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
18		eqidd 2732	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀)))
19	18	cbvmptv 5195	. . . . . 6 ⊢ (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) = (𝑤 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀)))
20		eqid 2731	. . . . . 6 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
21	11, 16, 17, 12, 19, 20	lcoc0 48453	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀)) ∧ ((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g‘𝑀)))
22	21	adantl 481	. . . 4 ⊢ ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → ((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀)) ∧ ((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g‘𝑀)))
23		simpl 482	. . . . . . . 8 ⊢ (((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ (¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))) → (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
24		breq1 5094	. . . . . . . . . 10 ⊢ (𝑠 = (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) → (𝑠 finSupp (0g‘(Scalar‘𝑀)) ↔ (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀))))
25		oveq1 7353	. . . . . . . . . . . 12 ⊢ (𝑠 = (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) → (𝑠( linC ‘𝑀)𝑉) = ((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉))
26	25	eqeq2d 2742	. . . . . . . . . . 11 ⊢ (𝑠 = (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) → ((0g‘𝑀) = (𝑠( linC ‘𝑀)𝑉) ↔ (0g‘𝑀) = ((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉)))
27		eqcom 2738	. . . . . . . . . . 11 ⊢ ((0g‘𝑀) = ((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) ↔ ((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g‘𝑀))
28	26, 27	bitrdi 287	. . . . . . . . . 10 ⊢ (𝑠 = (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) → ((0g‘𝑀) = (𝑠( linC ‘𝑀)𝑉) ↔ ((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g‘𝑀)))
29	24, 28	anbi12d 632	. . . . . . . . 9 ⊢ (𝑠 = (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) → ((𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (0g‘𝑀) = (𝑠( linC ‘𝑀)𝑉)) ↔ ((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀)) ∧ ((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g‘𝑀))))
30	29	adantl 481	. . . . . . . 8 ⊢ ((((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ (¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))) ∧ 𝑠 = (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀)))) → ((𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (0g‘𝑀) = (𝑠( linC ‘𝑀)𝑉)) ↔ ((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀)) ∧ ((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g‘𝑀))))
31	23, 30	rspcedv 3570	. . . . . . 7 ⊢ (((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ (¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))) → (((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀)) ∧ ((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g‘𝑀)) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (0g‘𝑀) = (𝑠( linC ‘𝑀)𝑉))))
32	31	ex 412	. . . . . 6 ⊢ ((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) → ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀)) ∧ ((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g‘𝑀)) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (0g‘𝑀) = (𝑠( linC ‘𝑀)𝑉)))))
33	32	com23 86	. . . . 5 ⊢ ((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) → (((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀)) ∧ ((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g‘𝑀)) → ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (0g‘𝑀) = (𝑠( linC ‘𝑀)𝑉)))))
34	33	3impib 1116	. . . 4 ⊢ (((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀)) ∧ ((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g‘𝑀)) → ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (0g‘𝑀) = (𝑠( linC ‘𝑀)𝑉))))
35	22, 34	mpcom 38	. . 3 ⊢ ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (0g‘𝑀) = (𝑠( linC ‘𝑀)𝑉)))
36	11, 16, 20	lcoval 48443	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((0g‘𝑀) ∈ (𝑀 LinCo 𝑉) ↔ ((0g‘𝑀) ∈ (Base‘𝑀) ∧ ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (0g‘𝑀) = (𝑠( linC ‘𝑀)𝑉)))))
37	36	adantl 481	. . 3 ⊢ ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → ((0g‘𝑀) ∈ (𝑀 LinCo 𝑉) ↔ ((0g‘𝑀) ∈ (Base‘𝑀) ∧ ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (0g‘𝑀) = (𝑠( linC ‘𝑀)𝑉)))))
38	15, 35, 37	mpbir2and 713	. 2 ⊢ ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (0g‘𝑀) ∈ (𝑀 LinCo 𝑉))
39	10, 38	pm2.61ian 811	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (0g‘𝑀) ∈ (𝑀 LinCo 𝑉))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  ∅c0 4283  𝒫 cpw 4550  {csn 4576   class class class wbr 5091   ↦ cmpt 5172  ‘cfv 6481  (class class class)co 7346   ↑_m cmap 8750   finSupp cfsupp 9245  Basecbs 17117  Scalarcsca 17161  0_gc0g 17340  Mndcmnd 18639  Grpcgrp 18843  LModclmod 20791   linC clinc 48435   LinCo clinco 48436

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-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-supp 8091  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-map 8752  df-en 8870  df-fin 8873  df-fsupp 9246  df-seq 13906  df-0g 17342  df-gsum 17343  df-mgm 18545  df-sgrp 18624  df-mnd 18640  df-grp 18846  df-ring 20151  df-lmod 20793  df-linc 48437  df-lco 48438

This theorem is referenced by:  lincolss  48465