Theorem issubassa2 15018

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 15017	. . . 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 14989	. . . . 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 14520	. . . . 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 14730	. . 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 14518	. . . 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 14513	. . . . . . . . 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 15012	. . . . . 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 14990	. . . . . . . . . . 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 15006	. . . . . . . . 9 |- ( ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) /\ ran A C_ S ) -> A Fn ( Base ` (Scalar ` W ) ) )
35		fnfvelrn 5834	. . . . . . . . 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 14524	. . . . . 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 14702	. . . . 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 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   {csn 3708   ran crn 4773   Fn wfn 5370   ` cfv 5375  (class class class)co 6079   Basecbs 13335   .rcmulr 13415  Scalarcsca 13417   .scvsca 13418  SubGrpcsubg 13953   1rcur 14245   Ringcrg 14283  SubRingcsubrg 14508   LModclmod 14606   LSubSpclss 14672   LSpanclspn 14706  AssAlgcasa 14979  algSccascl 14981

This theorem was proved from 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 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-pre-ltirr 8285  ax-pre-lttrn 8287  ax-pre-ltadd 8289

This theorem 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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-pnf 8356  df-mnf 8357  df-ltxr 8359  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-iress 13343  df-plusg 13427  df-mulr 13428  df-sca 13430  df-vsca 13431  df-0g 13595  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-grp 13791  df-minusg 13792  df-sbg 13793  df-subg 13956  df-mgp 14201  df-ur 14246  df-ring 14285  df-subrg 14510  df-lmod 14608  df-lssm 14673  df-lsp 14707  df-assa 14982  df-ascl 14984

This theorem is referenced by:  rnasclassa  15021