Theorem lcoel0 49484

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 6890	. . . 4 ⊢ (0g‘𝑀) ∈ V
2	1	snid 4623	. . 3 ⊢ (0g‘𝑀) ∈ {(0g‘𝑀)}
3		oveq2 7420	. . . 4 ⊢ (𝑉 = ∅ → (𝑀 LinCo 𝑉) = (𝑀 LinCo ∅))
4		lmodgrp 21122	. . . . . 6 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
5		grpmnd 19131	. . . . . 6 ⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
6		lco0 49483	. . . . . 6 ⊢ (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {(0g‘𝑀)})
7	4, 5, 6	3syl 19	. . . . 5 ⊢ (𝑀 ∈ LMod → (𝑀 LinCo ∅) = {(0g‘𝑀)})
8	7	adantr 486	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo ∅) = {(0g‘𝑀)})
9	3, 8	sylan9eq 2816	. . 3 ⊢ ((𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (𝑀 LinCo 𝑉) = {(0g‘𝑀)})
10	2, 9	eleqtrrid 2868	. 2 ⊢ ((𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (0g‘𝑀) ∈ (𝑀 LinCo 𝑉))
11		eqid 2761	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
12		eqid 2761	. . . . . 6 ⊢ (0g‘𝑀) = (0g‘𝑀)
13	11, 12	lmod0vcl 21146	. . . . 5 ⊢ (𝑀 ∈ LMod → (0g‘𝑀) ∈ (Base‘𝑀))
14	13	adantr 486	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (0g‘𝑀) ∈ (Base‘𝑀))
15	14	adantl 487	. . 3 ⊢ ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (0g‘𝑀) ∈ (Base‘𝑀))
16		eqid 2761	. . . . . 6 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
17		eqid 2761	. . . . . 6 ⊢ (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
18		eqidd 2762	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀)))
19	18	cbvmptv 5209	. . . . . 6 ⊢ (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) = (𝑤 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀)))
20		eqid 2761	. . . . . 6 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
21	11, 16, 17, 12, 19, 20	lcoc0 49478	. . . . 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 487	. . . 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 488	. . . . . . . 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 5106	. . . . . . . . . 10 ⊢ (𝑠 = (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) → (𝑠 finSupp (0g‘(Scalar‘𝑀)) ↔ (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀))))
25		oveq1 7419	. . . . . . . . . . . 12 ⊢ (𝑠 = (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) → (𝑠( linC ‘𝑀)𝑉) = ((𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉))
26	25	eqeq2d 2772	. . . . . . . . . . 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 2768	. . . . . . . . . . 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 290	. . . . . . . . . 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 644	. . . . . . . . 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 487	. . . . . . . 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 418	. . . . . 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 87	. . . . 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 1134	. . . 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 39	. . 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 49468	. . . 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 487	. . 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 726	. 2 ⊢ ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (0g‘𝑀) ∈ (𝑀 LinCo 𝑉))
39	10, 38	pm2.61ian 824	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (0g‘𝑀) ∈ (𝑀 LinCo 𝑉))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3087  ∅c0 4279  𝒫 cpw 4557  {csn 4584   class class class wbr 5103   ↦ cmpt 5186  ‘cfv 6531  (class class class)co 7412   ↑_m cmap 8831   finSupp cfsupp 9337  Basecbs 17367  Scalarcsca 17411  0_gc0g 17590  Mndcmnd 18903  Grpcgrp 19124  LModclmod 21115   linC clinc 49460   LinCo clinco 49461

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-supp 8162  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-map 8833  df-en 8958  df-fin 8961  df-fsupp 9338  df-seq 14125  df-0g 17592  df-gsum 17593  df-mgm 18796  df-sgrp 18888  df-mnd 18904  df-grp 19127  df-ring 20441  df-lmod 21117  df-linc 49462  df-lco 49463

This theorem is used by:  lincolss  49490