Theorem lcoel0 49014

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 6876	. . . 4 ⊢ (0g‘𝑀) ∈ V
2	1	snid 4620	. . 3 ⊢ (0g‘𝑀) ∈ {(0g‘𝑀)}
3		oveq2 7400	. . . 4 ⊢ (𝑉 = ∅ → (𝑀 LinCo 𝑉) = (𝑀 LinCo ∅))
4		lmodgrp 20914	. . . . . 6 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
5		grpmnd 18965	. . . . . 6 ⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
6		lco0 49013	. . . . . 6 ⊢ (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {(0g‘𝑀)})
7	4, 5, 6	3syl 18	. . . . 5 ⊢ (𝑀 ∈ LMod → (𝑀 LinCo ∅) = {(0g‘𝑀)})
8	7	adantr 484	. . . 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 20938	. . . . 5 ⊢ (𝑀 ∈ LMod → (0g‘𝑀) ∈ (Base‘𝑀))
14	13	adantr 484	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (0g‘𝑀) ∈ (Base‘𝑀))
15	14	adantl 485	. . 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 5203	. . . . . 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 49008	. . . . 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 485	. . . 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 486	. . . . . . . 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 5102	. . . . . . . . . 10 ⊢ (𝑠 = (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) → (𝑠 finSupp (0g‘(Scalar‘𝑀)) ↔ (𝑣 ∈ 𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀))))
25		oveq1 7399	. . . . . . . . . . . 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 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 641	. . . . . . . . 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 485	. . . . . . . 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 3574	. . . . . . 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 416	. . . . . 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 1128	. . . 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 48998	. . . 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 485	. . 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 723	. 2 ⊢ ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (0g‘𝑀) ∈ (𝑀 LinCo 𝑉))
39	10, 38	pm2.61ian 821	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (0g‘𝑀) ∈ (𝑀 LinCo 𝑉))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∃wrex 3085  ∅c0 4285  𝒫 cpw 4554  {csn 4581   class class class wbr 5099   ↦ cmpt 5180  ‘cfv 6517  (class class class)co 7392   ↑_m cmap 8803   finSupp cfsupp 9304  Basecbs 17228  Scalarcsca 17272  0_gc0g 17451  Mndcmnd 18751  Grpcgrp 18958  LModclmod 20907   linC clinc 48990   LinCo clinco 48991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-supp 8136  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-map 8805  df-en 8924  df-fin 8927  df-fsupp 9305  df-seq 14012  df-0g 17453  df-gsum 17454  df-mgm 18657  df-sgrp 18736  df-mnd 18752  df-grp 18961  df-ring 20264  df-lmod 20909  df-linc 48992  df-lco 48993

This theorem is referenced by:  lincolss  49020