Theorem aspsubrg 15018

Description: The algebraic span of a set of vectors is a subring of the algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)

Hypotheses

Ref	Expression
aspval.a	|- A = (AlgSpan ` W )
aspval.v	|- V = ( Base ` W )

Assertion

Ref	Expression
aspsubrg	|- ( ( W e. AssAlg /\ S C_ V ) -> ( A ` S ) e. (SubRing ` W ) )

Proof of Theorem aspsubrg

Dummy variables t w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		aspval.a	. . 3 |- A = (AlgSpan ` W )
2		aspval.v	. . 3 |- V = ( Base ` W )
3		eqid 2238	. . 3 |- ( LSubSp ` W ) = ( LSubSp ` W )
4	1, 2, 3	aspval 15015	. 2 |- ( ( W e. AssAlg /\ S C_ V ) -> ( A ` S ) = |^| { t e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) | S C_ t } )
5		ssrab2 3333	. . . 4 |- { t e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) | S C_ t } C_ ( (SubRing ` W ) i^i ( LSubSp ` W ) )
6		inss1 3451	. . . 4 |- ( (SubRing ` W ) i^i ( LSubSp ` W ) ) C_ (SubRing ` W )
7	5, 6	sstri 3257	. . 3 |- { t e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) | S C_ t } C_ (SubRing ` W )
8		sseq2 3272	. . . . 5 |- ( t = V -> ( S C_ t <-> S C_ V ) )
9		assaring 15007	. . . . . . . 8 |- ( W e. AssAlg -> W e. Ring )
10	2	subrgid 14531	. . . . . . . 8 |- ( W e. Ring -> V e. (SubRing ` W ) )
11	9, 10	syl 14	. . . . . . 7 |- ( W e. AssAlg -> V e. (SubRing ` W ) )
12	11	adantr 276	. . . . . 6 |- ( ( W e. AssAlg /\ S C_ V ) -> V e. (SubRing ` W ) )
13		assalmod 15006	. . . . . . . 8 |- ( W e. AssAlg -> W e. LMod )
14	2, 3	lss1 14699	. . . . . . . 8 |- ( W e. LMod -> V e. ( LSubSp ` W ) )
15	13, 14	syl 14	. . . . . . 7 |- ( W e. AssAlg -> V e. ( LSubSp ` W ) )
16	15	adantr 276	. . . . . 6 |- ( ( W e. AssAlg /\ S C_ V ) -> V e. ( LSubSp ` W ) )
17	12, 16	elind 3414	. . . . 5 |- ( ( W e. AssAlg /\ S C_ V ) -> V e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) )
18		simpr 110	. . . . 5 |- ( ( W e. AssAlg /\ S C_ V ) -> S C_ V )
19	8, 17, 18	elrabd 2984	. . . 4 |- ( ( W e. AssAlg /\ S C_ V ) -> V e. { t e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) | S C_ t } )
20		elex2 2838	. . . 4 |- ( V e. { t e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) | S C_ t } -> E. w w e. { t e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) | S C_ t } )
21	19, 20	syl 14	. . 3 |- ( ( W e. AssAlg /\ S C_ V ) -> E. w w e. { t e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) | S C_ t } )
22		subrgintm 14551	. . 3 |- ( ( { t e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) | S C_ t } C_ (SubRing ` W ) /\ E. w w e. { t e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) | S C_ t } ) -> |^| { t e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) | S C_ t } e. (SubRing ` W ) )
23	7, 21, 22	sylancr 418	. 2 |- ( ( W e. AssAlg /\ S C_ V ) -> |^| { t e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) | S C_ t } e. (SubRing ` W ) )
24	4, 23	eqeltrd 2315	1 |- ( ( W e. AssAlg /\ S C_ V ) -> ( A ` S ) e. (SubRing ` W ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   {crab 2532   i^i cin 3219   C_ wss 3220   |^|cint 3970   ` cfv 5377   Basecbs 13352   Ringcrg 14300  SubRingcsubrg 14525   LModclmod 14623   LSubSpclss 14689  AssAlgcasa 14996  AlgSpancasp 14997

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-plusg 13444  df-mulr 13445  df-sca 13447  df-vsca 13448  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-minusg 13809  df-subg 13973  df-cmn 14089  df-abl 14090  df-mgp 14218  df-ur 14263  df-srg 14268  df-ring 14302  df-subrg 14527  df-lmod 14625  df-lssm 14690  df-assa 14999  df-asp 15000

This theorem is used by: (None)