Theorem lcoss 42225

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 e. LMod /\ V e. ~P ( Base ` M ) ) -> V C_ ( M LinCo V ) )

Proof of Theorem lcoss

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

Step	Hyp	Ref	Expression
1		elelpwi 4171	. . . . . . 7 |- ( ( v e. V /\ V e. ~P ( Base ` M ) ) -> v e. ( Base ` M ) )
2	1	expcom 451	. . . . . 6 |- ( V e. ~P ( Base ` M ) -> ( v e. V -> v e. ( Base ` M ) ) )
3	2	adantl 482	. . . . 5 |- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( v e. V -> v e. ( Base ` M ) ) )
4	3	imp 445	. . . 4 |- ( ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) /\ v e. V ) -> v e. ( Base ` M ) )
5		eqid 2622	. . . . . . 7 |- ( Base ` M ) = ( Base ` M )
6		eqid 2622	. . . . . . 7 |- (Scalar ` M ) = (Scalar ` M )
7		eqid 2622	. . . . . . 7 |- ( 0g ` (Scalar ` M ) ) = ( 0g ` (Scalar ` M ) )
8		eqid 2622	. . . . . . 7 |- ( 1r ` (Scalar ` M ) ) = ( 1r ` (Scalar ` M ) )
9		equequ1 1952	. . . . . . . . 9 |- ( x = y -> ( x = v <-> y = v ) )
10	9	ifbid 4108	. . . . . . . 8 |- ( x = y -> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) = if ( y = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) )
11	10	cbvmptv 4750	. . . . . . 7 |- ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) = ( y e. V |-> if ( y = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) )
12	5, 6, 7, 8, 11	mptcfsupp 42161	. . . . . 6 |- ( ( M e. LMod /\ V e. ~P ( Base ` M ) /\ v e. V ) -> ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) finSupp ( 0g ` (Scalar ` M ) ) )
13	12	3expa 1265	. . . . 5 |- ( ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) /\ v e. V ) -> ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) finSupp ( 0g ` (Scalar ` M ) ) )
14		eqid 2622	. . . . . . . 8 |- ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) = ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) )
15	5, 6, 7, 8, 14	linc1 42214	. . . . . . 7 |- ( ( M e. LMod /\ V e. ~P ( Base ` M ) /\ v e. V ) -> ( ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ( linC ` M ) V ) = v )
16	15	3expa 1265	. . . . . 6 |- ( ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) /\ v e. V ) -> ( ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ( linC ` M ) V ) = v )
17	16	eqcomd 2628	. . . . 5 |- ( ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) /\ v e. V ) -> v = ( ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ( linC ` M ) V ) )
18		eqid 2622	. . . . . . . . . . 11 |- ( Base ` (Scalar ` M ) ) = ( Base ` (Scalar ` M ) )
19	6, 18, 8	lmod1cl 18890	. . . . . . . . . 10 |- ( M e. LMod -> ( 1r ` (Scalar ` M ) ) e. ( Base ` (Scalar ` M ) ) )
20	6, 18, 7	lmod0cl 18889	. . . . . . . . . 10 |- ( M e. LMod -> ( 0g ` (Scalar ` M ) ) e. ( Base ` (Scalar ` M ) ) )
21	19, 20	ifcld 4131	. . . . . . . . 9 |- ( M e. LMod -> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) e. ( Base ` (Scalar ` M ) ) )
22	21	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) /\ v e. V ) /\ x e. V ) -> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) e. ( Base ` (Scalar ` M ) ) )
23	22, 14	fmptd 6385	. . . . . . 7 |- ( ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) /\ v e. V ) -> ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) : V --> ( Base ` (Scalar ` M ) ) )
24		fvex 6201	. . . . . . . 8 |- ( Base ` (Scalar ` M ) ) e. _V
25		simplr 792	. . . . . . . 8 |- ( ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) /\ v e. V ) -> V e. ~P ( Base ` M ) )
26		elmapg 7870	. . . . . . . 8 |- ( ( ( Base ` (Scalar ` M ) ) e. _V /\ V e. ~P ( Base ` M ) ) -> ( ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) e. ( ( Base ` (Scalar ` M ) ) ^m V ) <-> ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) : V --> ( Base ` (Scalar ` M ) ) ) )
27	24, 25, 26	sylancr 695	. . . . . . 7 |- ( ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) /\ v e. V ) -> ( ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) e. ( ( Base ` (Scalar ` M ) ) ^m V ) <-> ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) : V --> ( Base ` (Scalar ` M ) ) ) )
28	23, 27	mpbird 247	. . . . . 6 |- ( ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) /\ v e. V ) -> ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) e. ( ( Base ` (Scalar ` M ) ) ^m V ) )
29		breq1 4656	. . . . . . . 8 |- ( f = ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) -> ( f finSupp ( 0g ` (Scalar ` M ) ) <-> ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) finSupp ( 0g ` (Scalar ` M ) ) ) )
30		oveq1 6657	. . . . . . . . 9 |- ( f = ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) -> ( f ( linC ` M ) V ) = ( ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ( linC ` M ) V ) )
31	30	eqeq2d 2632	. . . . . . . 8 |- ( f = ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) -> ( v = ( f ( linC ` M ) V ) <-> v = ( ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ( linC ` M ) V ) ) )
32	29, 31	anbi12d 747	. . . . . . 7 |- ( f = ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) -> ( ( f finSupp ( 0g ` (Scalar ` M ) ) /\ v = ( f ( linC ` M ) V ) ) <-> ( ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) finSupp ( 0g ` (Scalar ` M ) ) /\ v = ( ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ( linC ` M ) V ) ) ) )
33	32	adantl 482	. . . . . 6 |- ( ( ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) /\ v e. V ) /\ f = ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ) -> ( ( f finSupp ( 0g ` (Scalar ` M ) ) /\ v = ( f ( linC ` M ) V ) ) <-> ( ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) finSupp ( 0g ` (Scalar ` M ) ) /\ v = ( ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ( linC ` M ) V ) ) ) )
34	28, 33	rspcedv 3313	. . . . 5 |- ( ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) /\ v e. V ) -> ( ( ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) finSupp ( 0g ` (Scalar ` M ) ) /\ v = ( ( x e. V |-> if ( x = v , ( 1r ` (Scalar ` M ) ) , ( 0g ` (Scalar ` M ) ) ) ) ( linC ` M ) V ) ) -> E. f e. ( ( Base ` (Scalar ` M ) ) ^m V ) ( f finSupp ( 0g ` (Scalar ` M ) ) /\ v = ( f ( linC ` M ) V ) ) ) )
35	13, 17, 34	mp2and 715	. . . 4 |- ( ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) /\ v e. V ) -> E. f e. ( ( Base ` (Scalar ` M ) ) ^m V ) ( f finSupp ( 0g ` (Scalar ` M ) ) /\ v = ( f ( linC ` M ) V ) ) )
36	5, 6, 18	lcoval 42201	. . . . 5 |- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( v e. ( M LinCo V ) <-> ( v e. ( Base ` M ) /\ E. f e. ( ( Base ` (Scalar ` M ) ) ^m V ) ( f finSupp ( 0g ` (Scalar ` M ) ) /\ v = ( f ( linC ` M ) V ) ) ) ) )
37	36	adantr 481	. . . 4 |- ( ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) /\ v e. V ) -> ( v e. ( M LinCo V ) <-> ( v e. ( Base ` M ) /\ E. f e. ( ( Base ` (Scalar ` M ) ) ^m V ) ( f finSupp ( 0g ` (Scalar ` M ) ) /\ v = ( f ( linC ` M ) V ) ) ) ) )
38	4, 35, 37	mpbir2and 957	. . 3 |- ( ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) /\ v e. V ) -> v e. ( M LinCo V ) )
39	38	ex 450	. 2 |- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( v e. V -> v e. ( M LinCo V ) ) )
40	39	ssrdv 3609	1 |- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> V C_ ( M LinCo V ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C_ wss 3574   ifcif 4086   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   finSupp cfsupp 8275   Basecbs 15857  Scalarcsca 15944   0gc0g 16100   1rcur 18501   LModclmod 18863   linC clinc 42193   LinCo clinco 42194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-linc 42195  df-lco 42196

This theorem is referenced by:  lspsslco  42226