Theorem ellcoellss 42224

Description: Every linear combination of a subset of a linear subspace is also contained in the linear subspace. (Contributed by AV, 20-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)

Assertion

Ref	Expression
ellcoellss	|- ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> A. x e. ( M LinCo V ) x e. S )

Distinct variable groups:   x, M   x, S   x, V

Proof of Theorem ellcoellss

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1061	. . . 4 |- ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> M e. LMod )
2		eqid 2622	. . . . . . 7 |- ( Base ` M ) = ( Base ` M )
3		eqid 2622	. . . . . . 7 |- ( LSubSp ` M ) = ( LSubSp ` M )
4	2, 3	lssss 18937	. . . . . 6 |- ( S e. ( LSubSp ` M ) -> S C_ ( Base ` M ) )
5	4	3ad2ant2 1083	. . . . 5 |- ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> S C_ ( Base ` M ) )
6		sstr 3611	. . . . . . . 8 |- ( ( V C_ S /\ S C_ ( Base ` M ) ) -> V C_ ( Base ` M ) )
7		fvex 6201	. . . . . . . . . 10 |- ( Base ` M ) e. _V
8	7	ssex 4802	. . . . . . . . 9 |- ( V C_ ( Base ` M ) -> V e. _V )
9		elpwg 4166	. . . . . . . . . 10 |- ( V e. _V -> ( V e. ~P ( Base ` M ) <-> V C_ ( Base ` M ) ) )
10	9	biimprd 238	. . . . . . . . 9 |- ( V e. _V -> ( V C_ ( Base ` M ) -> V e. ~P ( Base ` M ) ) )
11	8, 10	mpcom 38	. . . . . . . 8 |- ( V C_ ( Base ` M ) -> V e. ~P ( Base ` M ) )
12	6, 11	syl 17	. . . . . . 7 |- ( ( V C_ S /\ S C_ ( Base ` M ) ) -> V e. ~P ( Base ` M ) )
13	12	ex 450	. . . . . 6 |- ( V C_ S -> ( S C_ ( Base ` M ) -> V e. ~P ( Base ` M ) ) )
14	13	3ad2ant3 1084	. . . . 5 |- ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> ( S C_ ( Base ` M ) -> V e. ~P ( Base ` M ) ) )
15	5, 14	mpd 15	. . . 4 |- ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> V e. ~P ( Base ` M ) )
16		eqid 2622	. . . . 5 |- (Scalar ` M ) = (Scalar ` M )
17		eqid 2622	. . . . 5 |- ( Base ` (Scalar ` M ) ) = ( Base ` (Scalar ` M ) )
18	2, 16, 17	lcoval 42201	. . . 4 |- ( ( M e. LMod /\ V e. ~P ( Base ` M ) ) -> ( x e. ( M LinCo V ) <-> ( x e. ( Base ` M ) /\ E. f e. ( ( Base ` (Scalar ` M ) ) ^m V ) ( f finSupp ( 0g ` (Scalar ` M ) ) /\ x = ( f ( linC ` M ) V ) ) ) ) )
19	1, 15, 18	syl2anc 693	. . 3 |- ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> ( x e. ( M LinCo V ) <-> ( x e. ( Base ` M ) /\ E. f e. ( ( Base ` (Scalar ` M ) ) ^m V ) ( f finSupp ( 0g ` (Scalar ` M ) ) /\ x = ( f ( linC ` M ) V ) ) ) ) )
20		lincellss 42215	. . . . . . . . . . . . 13 |- ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> ( ( f e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ f finSupp ( 0g ` (Scalar ` M ) ) ) -> ( f ( linC ` M ) V ) e. S ) )
21	20	imp 445	. . . . . . . . . . . 12 |- ( ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) /\ ( f e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ f finSupp ( 0g ` (Scalar ` M ) ) ) ) -> ( f ( linC ` M ) V ) e. S )
22		eleq1 2689	. . . . . . . . . . . 12 |- ( x = ( f ( linC ` M ) V ) -> ( x e. S <-> ( f ( linC ` M ) V ) e. S ) )
23	21, 22	syl5ibr 236	. . . . . . . . . . 11 |- ( x = ( f ( linC ` M ) V ) -> ( ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) /\ ( f e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ f finSupp ( 0g ` (Scalar ` M ) ) ) ) -> x e. S ) )
24	23	expd 452	. . . . . . . . . 10 |- ( x = ( f ( linC ` M ) V ) -> ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> ( ( f e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ f finSupp ( 0g ` (Scalar ` M ) ) ) -> x e. S ) ) )
25	24	com12 32	. . . . . . . . 9 |- ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> ( x = ( f ( linC ` M ) V ) -> ( ( f e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ f finSupp ( 0g ` (Scalar ` M ) ) ) -> x e. S ) ) )
26	25	adantr 481	. . . . . . . 8 |- ( ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) /\ x e. ( Base ` M ) ) -> ( x = ( f ( linC ` M ) V ) -> ( ( f e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ f finSupp ( 0g ` (Scalar ` M ) ) ) -> x e. S ) ) )
27	26	com13 88	. . . . . . 7 |- ( ( f e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ f finSupp ( 0g ` (Scalar ` M ) ) ) -> ( x = ( f ( linC ` M ) V ) -> ( ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) /\ x e. ( Base ` M ) ) -> x e. S ) ) )
28	27	impr 649	. . . . . 6 |- ( ( f e. ( ( Base ` (Scalar ` M ) ) ^m V ) /\ ( f finSupp ( 0g ` (Scalar ` M ) ) /\ x = ( f ( linC ` M ) V ) ) ) -> ( ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) /\ x e. ( Base ` M ) ) -> x e. S ) )
29	28	rexlimiva 3028	. . . . 5 |- ( E. f e. ( ( Base ` (Scalar ` M ) ) ^m V ) ( f finSupp ( 0g ` (Scalar ` M ) ) /\ x = ( f ( linC ` M ) V ) ) -> ( ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) /\ x e. ( Base ` M ) ) -> x e. S ) )
30	29	com12 32	. . . 4 |- ( ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) /\ x e. ( Base ` M ) ) -> ( E. f e. ( ( Base ` (Scalar ` M ) ) ^m V ) ( f finSupp ( 0g ` (Scalar ` M ) ) /\ x = ( f ( linC ` M ) V ) ) -> x e. S ) )
31	30	expimpd 629	. . 3 |- ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> ( ( x e. ( Base ` M ) /\ E. f e. ( ( Base ` (Scalar ` M ) ) ^m V ) ( f finSupp ( 0g ` (Scalar ` M ) ) /\ x = ( f ( linC ` M ) V ) ) ) -> x e. S ) )
32	19, 31	sylbid 230	. 2 |- ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> ( x e. ( M LinCo V ) -> x e. S ) )
33	32	ralrimiv 2965	1 |- ( ( M e. LMod /\ S e. ( LSubSp ` M ) /\ V C_ S ) -> A. x e. ( M LinCo V ) x e. S )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   finSupp cfsupp 8275   Basecbs 15857  Scalarcsca 15944   0gc0g 16100   LModclmod 18863   LSubSpclss 18932   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-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-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-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-lss 18933  df-linc 42195  df-lco 42196

This theorem is referenced by:  lcosslsp  42227