Theorem lcoel0 48273

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 6919	. . . 4 ⊢ (0g‘𝑀) ∈ V
2	1	snid 4666	. . 3 ⊢ (0g‘𝑀) ∈ {(0g‘𝑀)}
3		oveq2 7438	. . . 4 ⊢ (𝑉 = ∅ → (𝑀 LinCo 𝑉) = (𝑀 LinCo ∅))
4		lmodgrp 20881	. . . . . 6 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
5		grpmnd 18970	. . . . . 6 ⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
6		lco0 48272	. . . . . 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 2794	. . 3 ⊢ ((𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (𝑀 LinCo 𝑉) = {(0g‘𝑀)})
10	2, 9	eleqtrrid 2845	. 2 ⊢ ((𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (0g‘𝑀) ∈ (𝑀 LinCo 𝑉))
11		eqid 2734	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
12		eqid 2734	. . . . . 6 ⊢ (0g‘𝑀) = (0g‘𝑀)
13	11, 12	lmod0vcl 20905	. . . . 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 2734	. . . . . 6 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
17		eqid 2734	. . . . . 6 ⊢ (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
18		eqidd 2735	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀)))
19	18	cbvmptv 5260	. . . . . 6 ⊢ (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) = (𝑤 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀)))
20		eqid 2734	. . . . . 6 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
21	11, 16, 17, 12, 19, 20	lcoc0 48267	. . . . 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 5150	. . . . . . . . . 10 ⊢ (𝑠 = (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) → (𝑠 finSupp (0g‘(Scalar‘𝑀)) ↔ (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀))))
25		oveq1 7437	. . . . . . . . . . . 12 ⊢ (𝑠 = (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) → (𝑠( linC ‘𝑀)𝑉) = ((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉))
26	25	eqeq2d 2745	. . . . . . . . . . 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 2741	. . . . . . . . . . 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 3614	. . . . . . 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 1115	. . . 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 48257	. . . 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 812	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 1536   ∈ wcel 2105  ∃wrex 3067  ∅c0 4338  𝒫 cpw 4604  {csn 4630   class class class wbr 5147   ↦ cmpt 5230  ‘cfv 6562  (class class class)co 7430   ↑_m cmap 8864   finSupp cfsupp 9398  Basecbs 17244  Scalarcsca 17300  0_gc0g 17485  Mndcmnd 18759  Grpcgrp 18963  LModclmod 20874   linC clinc 48249   LinCo clinco 48250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-map 8866  df-en 8984  df-fin 8987  df-fsupp 9399  df-seq 14039  df-0g 17487  df-gsum 17488  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-grp 18966  df-ring 20252  df-lmod 20876  df-linc 48251  df-lco 48252

This theorem is referenced by:  lincolss  48279