Theorem lcoss 42019

Description: A set of vectors of a module is a subset of the set of all linear combinations of the set. (Contributed by AV, 18-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)

Assertion

Ref	Expression
lcoss	⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ⊆ (𝑀 LinCo 𝑉))

Proof of Theorem lcoss

Dummy variables 𝑓 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elelpwi 4119	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑣 ∈ (Base‘𝑀))
2	1	expcom 450	. . . . . 6 ⊢ (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑣 ∈ 𝑉 → 𝑣 ∈ (Base‘𝑀)))
3	2	adantl 481	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑣 ∈ 𝑉 → 𝑣 ∈ (Base‘𝑀)))
4	3	imp 444	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ (Base‘𝑀))
5		eqid 2610	. . . . . . 7 ⊢ (Base‘𝑀) = (Base‘𝑀)
6		eqid 2610	. . . . . . 7 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
7		eqid 2610	. . . . . . 7 ⊢ (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
8		eqid 2610	. . . . . . 7 ⊢ (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))
9		equequ1 1939	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑣 ↔ 𝑦 = 𝑣))
10	9	ifbid 4058	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) = if(𝑦 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))
11	10	cbvmptv 4678	. . . . . . 7 ⊢ (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) = (𝑦 ∈ 𝑉 ↦ if(𝑦 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))
12	5, 6, 7, 8, 11	mptcfsupp 41955	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑣 ∈ 𝑉) → (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) finSupp (0g‘(Scalar‘𝑀)))
13	12	3expa 1257	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝑣 ∈ 𝑉) → (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) finSupp (0g‘(Scalar‘𝑀)))
14		eqid 2610	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) = (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))
15	5, 6, 7, 8, 14	linc1 42008	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑣 ∈ 𝑉) → ((𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑉) = 𝑣)
16	15	3expa 1257	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝑣 ∈ 𝑉) → ((𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑉) = 𝑣)
17	16	eqcomd 2616	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝑣 ∈ 𝑉) → 𝑣 = ((𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑉))
18		eqid 2610	. . . . . . . . . . 11 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
19	6, 18, 8	lmod1cl 18713	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → (1r‘(Scalar‘𝑀)) ∈ (Base‘(Scalar‘𝑀)))
20	6, 18, 7	lmod0cl 18712	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → (0g‘(Scalar‘𝑀)) ∈ (Base‘(Scalar‘𝑀)))
21	19, 20	ifcld 4081	. . . . . . . . 9 ⊢ (𝑀 ∈ LMod → if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) ∈ (Base‘(Scalar‘𝑀)))
22	21	ad3antrrr 762	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝑣 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))) ∈ (Base‘(Scalar‘𝑀)))
23	22, 14	fmptd 6292	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝑣 ∈ 𝑉) → (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))):𝑉⟶(Base‘(Scalar‘𝑀)))
24		fvex 6113	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
25		simplr 788	. . . . . . . 8 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝑣 ∈ 𝑉) → 𝑉 ∈ 𝒫 (Base‘𝑀))
26		elmapg 7757	. . . . . . . 8 ⊢ (((Base‘(Scalar‘𝑀)) ∈ V ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ↔ (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))):𝑉⟶(Base‘(Scalar‘𝑀))))
27	24, 25, 26	sylancr 694	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝑣 ∈ 𝑉) → ((𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ↔ (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))):𝑉⟶(Base‘(Scalar‘𝑀))))
28	23, 27	mpbird 246	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝑣 ∈ 𝑉) → (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
29		breq1 4586	. . . . . . . 8 ⊢ (𝑓 = (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) → (𝑓 finSupp (0g‘(Scalar‘𝑀)) ↔ (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) finSupp (0g‘(Scalar‘𝑀))))
30		oveq1 6556	. . . . . . . . 9 ⊢ (𝑓 = (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) → (𝑓( linC ‘𝑀)𝑉) = ((𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑉))
31	30	eqeq2d 2620	. . . . . . . 8 ⊢ (𝑓 = (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) → (𝑣 = (𝑓( linC ‘𝑀)𝑉) ↔ 𝑣 = ((𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑉)))
32	29, 31	anbi12d 743	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) → ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑓( linC ‘𝑀)𝑉)) ↔ ((𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = ((𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑉))))
33	32	adantl 481	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝑣 ∈ 𝑉) ∧ 𝑓 = (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))) → ((𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑓( linC ‘𝑀)𝑉)) ↔ ((𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = ((𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑉))))
34	28, 33	rspcedv 3286	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝑣 ∈ 𝑉) → (((𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀)))) finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = ((𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑣, (1r‘(Scalar‘𝑀)), (0g‘(Scalar‘𝑀))))( linC ‘𝑀)𝑉)) → ∃𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑓( linC ‘𝑀)𝑉))))
35	13, 17, 34	mp2and 711	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝑣 ∈ 𝑉) → ∃𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑓( linC ‘𝑀)𝑉)))
36	5, 6, 18	lcoval 41995	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑣 ∈ (𝑀 LinCo 𝑉) ↔ (𝑣 ∈ (Base‘𝑀) ∧ ∃𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑓( linC ‘𝑀)𝑉)))))
37	36	adantr 480	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝑣 ∈ 𝑉) → (𝑣 ∈ (𝑀 LinCo 𝑉) ↔ (𝑣 ∈ (Base‘𝑀) ∧ ∃𝑓 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑓 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑓( linC ‘𝑀)𝑉)))))
38	4, 35, 37	mpbir2and 959	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ (𝑀 LinCo 𝑉))
39	38	ex 449	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑣 ∈ 𝑉 → 𝑣 ∈ (𝑀 LinCo 𝑉)))
40	39	ssrdv 3574	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ⊆ (𝑀 LinCo 𝑉))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ifcif 4036  𝒫 cpw 4108   class class class wbr 4583   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744   finSupp cfsupp 8158  Basecbs 15695  Scalarcsca 15771  0_gc0g 15923  1_rcur 18324  LModclmod 18686   linC clinc 41987   LinCo clinco 41988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-gsum 15926  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-mgp 18313  df-ur 18325  df-ring 18372  df-lmod 18688  df-linc 41989  df-lco 41990

This theorem is referenced by:  lspsslco  42020