Theorem issubassa2 15143

Description: A subring of a unital algebra is a subspace and thus a subalgebra iff it contains all scalar multiples of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.)

Hypotheses

Ref	Expression
issubassa2.a	|- A = (algSc ` W )
issubassa2.l	|- L = ( LSubSp ` W )

Assertion

Ref	Expression
issubassa2	|- ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) -> ( S e. L <-> ran A C_ S ) )

Proof of Theorem issubassa2

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

Step	Hyp	Ref	Expression
1		issubassa2.a	. . . . 5 |- A = (algSc ` W )
2		eqid 2238	. . . . 5 |- ( 1r ` W ) = ( 1r ` W )
3		eqid 2238	. . . . 5 |- ( LSpan ` W ) = ( LSpan ` W )
4	1, 2, 3	rnascl 15142	. . . 4 |- ( W e. AssAlg -> ran A = ( ( LSpan ` W ) ` { ( 1r ` W ) } ) )
5	4	ad2antrr 492	. . 3 |- ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ S e. L ) -> ran A = ( ( LSpan ` W ) ` { ( 1r ` W ) } ) )
6		issubassa2.l	. . . 4 |- L = ( LSubSp ` W )
7		assalmod 15114	. . . . 5 |- ( W e. AssAlg -> W e. LMod )
8	7	ad2antrr 492	. . . 4 |- ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ S e. L ) -> W e. LMod )
9		simpr 110	. . . 4 |- ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ S e. L ) -> S e. L )
10	2	subrg1cl 14645	. . . . 5 |- ( S e. (SubRing ` W ) -> ( 1r ` W ) e. S )
11	10	ad2antlr 493	. . . 4 |- ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ S e. L ) -> ( 1r ` W ) e. S )
12	6, 3, 8, 9, 11	ellspsn5 14855	. . 3 |- ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ S e. L ) -> ( ( LSpan ` W ) ` { ( 1r ` W ) } ) C_ S )
13	5, 12	eqsstrd 3284	. 2 |- ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ S e. L ) -> ran A C_ S )
14		subrgsubg 14643	. . . 4 |- ( S e. (SubRing ` W ) -> S e. (SubGrp ` W ) )
15	14	ad2antlr 493	. . 3 |- ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ ran A C_ S ) -> S e. (SubGrp ` W ) )
16		simplll 539	. . . . . 6 |- ( ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ ran A C_ S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. S ) ) -> W e. AssAlg )
17		simprl 535	. . . . . 6 |- ( ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ ran A C_ S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. S ) ) -> x e. ( Base ` (Scalar ` W ) ) )
18		eqid 2238	. . . . . . . . . 10 |- ( Base ` W ) = ( Base ` W )
19	18	subrgss 14638	. . . . . . . . 9 |- ( S e. (SubRing ` W ) -> S C_ ( Base ` W ) )
20	19	ad2antlr 493	. . . . . . . 8 |- ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ ran A C_ S ) -> S C_ ( Base ` W ) )
21	20	sselda 3248	. . . . . . 7 |- ( ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ ran A C_ S ) /\ y e. S ) -> y e. ( Base ` W ) )
22	21	adantrl 482	. . . . . 6 |- ( ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ ran A C_ S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. S ) ) -> y e. ( Base ` W ) )
23		eqid 2238	. . . . . . 7 |- (Scalar ` W ) = (Scalar ` W )
24		eqid 2238	. . . . . . 7 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
25		eqid 2238	. . . . . . 7 |- ( .r ` W ) = ( .r ` W )
26		eqid 2238	. . . . . . 7 |- ( .s ` W ) = ( .s ` W )
27	1, 23, 24, 18, 25, 26	asclmul1 15137	. . . . . 6 |- ( ( W e. AssAlg /\ x e. ( Base ` (Scalar ` W ) ) /\ y e. ( Base ` W ) ) -> ( ( A ` x ) ( .r ` W ) y ) = ( x ( .s ` W ) y ) )
28	16, 17, 22, 27	syl3anc 1278	. . . . 5 |- ( ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ ran A C_ S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. S ) ) -> ( ( A ` x ) ( .r ` W ) y ) = ( x ( .s ` W ) y ) )
29		simpllr 540	. . . . . 6 |- ( ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ ran A C_ S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. S ) ) -> S e. (SubRing ` W ) )
30		simplr 533	. . . . . . . 8 |- ( ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ ran A C_ S ) /\ x e. ( Base ` (Scalar ` W ) ) ) -> ran A C_ S )
31	7	ad2antrr 492	. . . . . . . . . 10 |- ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ ran A C_ S ) -> W e. LMod )
32		assaring 15115	. . . . . . . . . . 11 |- ( W e. AssAlg -> W e. Ring )
33	32	ad2antrr 492	. . . . . . . . . 10 |- ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ ran A C_ S ) -> W e. Ring )
34	1, 23, 24, 31, 33	asclfnd 15131	. . . . . . . . 9 |- ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ ran A C_ S ) -> A Fn ( Base ` (Scalar ` W ) ) )
35		fnfvelrn 5843	. . . . . . . . 9 |- ( ( A Fn ( Base ` (Scalar ` W ) ) /\ x e. ( Base ` (Scalar ` W ) ) ) -> ( A ` x ) e. ran A )
36	34, 35	sylan 283	. . . . . . . 8 |- ( ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ ran A C_ S ) /\ x e. ( Base ` (Scalar ` W ) ) ) -> ( A ` x ) e. ran A )
37	30, 36	sseldd 3249	. . . . . . 7 |- ( ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ ran A C_ S ) /\ x e. ( Base ` (Scalar ` W ) ) ) -> ( A ` x ) e. S )
38	37	adantrr 483	. . . . . 6 |- ( ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ ran A C_ S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. S ) ) -> ( A ` x ) e. S )
39		simprr 537	. . . . . 6 |- ( ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ ran A C_ S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. S ) ) -> y e. S )
40	25	subrgmcl 14649	. . . . . 6 |- ( ( S e. (SubRing ` W ) /\ ( A ` x ) e. S /\ y e. S ) -> ( ( A ` x ) ( .r ` W ) y ) e. S )
41	29, 38, 39, 40	syl3anc 1278	. . . . 5 |- ( ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ ran A C_ S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. S ) ) -> ( ( A ` x ) ( .r ` W ) y ) e. S )
42	28, 41	eqeltrrd 2316	. . . 4 |- ( ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ ran A C_ S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. S ) ) -> ( x ( .s ` W ) y ) e. S )
43	42	ralrimivva 2632	. . 3 |- ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ ran A C_ S ) -> A. x e. ( Base ` (Scalar ` W ) ) A. y e. S ( x ( .s ` W ) y ) e. S )
44	23, 24, 18, 26, 6	islss4 14827	. . . . 5 |- ( W e. LMod -> ( S e. L <-> ( S e. (SubGrp ` W ) /\ A. x e. ( Base ` (Scalar ` W ) ) A. y e. S ( x ( .s ` W ) y ) e. S ) ) )
45	7, 44	syl 14	. . . 4 |- ( W e. AssAlg -> ( S e. L <-> ( S e. (SubGrp ` W ) /\ A. x e. ( Base ` (Scalar ` W ) ) A. y e. S ( x ( .s ` W ) y ) e. S ) ) )
46	45	ad2antrr 492	. . 3 |- ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ ran A C_ S ) -> ( S e. L <-> ( S e. (SubGrp ` W ) /\ A. x e. ( Base ` (Scalar ` W ) ) A. y e. S ( x ( .s ` W ) y ) e. S ) ) )
47	15, 43, 46	mpbir2and 957	. 2 |- ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ ran A C_ S ) -> S e. L )
48	13, 47	impbida 604	1 |- ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) -> ( S e. L <-> ran A C_ S ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   {csn 3709   ran crn 4777   Fn wfn 5375   ` cfv 5380  (class class class)co 6088   Basecbs 13409   .rcmulr 13490  Scalarcsca 13492   .scvsca 13493  SubGrpcsubg 14028   1rcur 14368   Ringcrg 14406  SubRingcsubrg 14633   LModclmod 14731   LSubSpclss 14797   LSpanclspn 14831  AssAlgcasa 15104  algSccascl 15106

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 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-pre-ltirr 8297  ax-pre-lttrn 8299  ax-pre-ltadd 8301

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 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-pnf 8368  df-mnf 8369  df-ltxr 8371  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-ndx 13412  df-slot 13413  df-base 13415  df-sets 13416  df-iress 13417  df-plusg 13502  df-mulr 13503  df-sca 13505  df-vsca 13506  df-0g 13670  df-mgm 13734  df-sgrp 13775  df-mnd 13788  df-grp 13866  df-minusg 13867  df-sbg 13868  df-subg 14031  df-mgp 14324  df-ur 14369  df-ring 14408  df-subrg 14635  df-lmod 14733  df-lssm 14798  df-lsp 14832  df-assa 15107  df-ascl 15109

This theorem is used by:  rnasclassa  15146