Theorem lcoel0 44477

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 6677	. . . 4 ⊢ (0g‘𝑀) ∈ V
2	1	snid 4594	. . 3 ⊢ (0g‘𝑀) ∈ {(0g‘𝑀)}
3		oveq2 7158	. . . 4 ⊢ (𝑉 = ∅ → (𝑀 LinCo 𝑉) = (𝑀 LinCo ∅))
4		lmodgrp 19635	. . . . . 6 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
5		grpmnd 18104	. . . . . 6 ⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
6		lco0 44476	. . . . . 6 ⊢ (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {(0g‘𝑀)})
7	4, 5, 6	3syl 18	. . . . 5 ⊢ (𝑀 ∈ LMod → (𝑀 LinCo ∅) = {(0g‘𝑀)})
8	7	adantr 483	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo ∅) = {(0g‘𝑀)})
9	3, 8	sylan9eq 2876	. . 3 ⊢ ((𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (𝑀 LinCo 𝑉) = {(0g‘𝑀)})
10	2, 9	eleqtrrid 2920	. 2 ⊢ ((𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (0g‘𝑀) ∈ (𝑀 LinCo 𝑉))
11		eqid 2821	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
12		eqid 2821	. . . . . 6 ⊢ (0g‘𝑀) = (0g‘𝑀)
13	11, 12	lmod0vcl 19657	. . . . 5 ⊢ (𝑀 ∈ LMod → (0g‘𝑀) ∈ (Base‘𝑀))
14	13	adantr 483	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (0g‘𝑀) ∈ (Base‘𝑀))
15	14	adantl 484	. . 3 ⊢ ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (0g‘𝑀) ∈ (Base‘𝑀))
16		eqid 2821	. . . . . 6 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
17		eqid 2821	. . . . . 6 ⊢ (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
18		eqidd 2822	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀)))
19	18	cbvmptv 5161	. . . . . 6 ⊢ (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) = (𝑤 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀)))
20		eqid 2821	. . . . . 6 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
21	11, 16, 17, 12, 19, 20	lcoc0 44471	. . . . 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 484	. . . 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 485	. . . . . . . 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 5061	. . . . . . . . . 10 ⊢ (𝑠 = (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) → (𝑠 finSupp (0g‘(Scalar‘𝑀)) ↔ (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀))))
25		oveq1 7157	. . . . . . . . . . . 12 ⊢ (𝑠 = (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) → (𝑠( linC ‘𝑀)𝑉) = ((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉))
26	25	eqeq2d 2832	. . . . . . . . . . 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 2828	. . . . . . . . . . 11 ⊢ ((0g‘𝑀) = ((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) ↔ ((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g‘𝑀))
28	26, 27	syl6bb 289	. . . . . . . . . 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 484	. . . . . . . 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 3615	. . . . . . 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 415	. . . . . 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 1112	. . . 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 44461	. . . 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 484	. . 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 711	. 2 ⊢ ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (0g‘𝑀) ∈ (𝑀 LinCo 𝑉))
39	10, 38	pm2.61ian 810	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (0g‘𝑀) ∈ (𝑀 LinCo 𝑉))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∃wrex 3139  ∅c0 4290  𝒫 cpw 4538  {csn 4560   class class class wbr 5058   ↦ cmpt 5138  ‘cfv 6349  (class class class)co 7150   ↑_m cmap 8400   finSupp cfsupp 8827  Basecbs 16477  Scalarcsca 16562  0_gc0g 16707  Mndcmnd 17905  Grpcgrp 18097  LModclmod 19628   linC clinc 44453   LinCo clinco 44454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-supp 7825  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-map 8402  df-en 8504  df-fin 8507  df-fsupp 8828  df-seq 13364  df-0g 16709  df-gsum 16710  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-grp 18100  df-ring 19293  df-lmod 19630  df-linc 44455  df-lco 44456

This theorem is referenced by:  lincolss  44483