Theorem issubassa3 14995

Description: A subring that is also a subspace is a subalgebra. The key theorem is islss3 14699. (Contributed by Mario Carneiro, 7-Jan-2015.)

Hypotheses

Ref	Expression
issubassa.s	|- S = ( W ↾s A )
issubassa.l	|- L = ( LSubSp ` W )

Assertion

Ref	Expression
issubassa3	|- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> S e. AssAlg )

Proof of Theorem issubassa3

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

Step	Hyp	Ref	Expression
1		issubassa.s	. . . 4 |- S = ( W ↾s A )
2	1	subrgbas 14521	. . 3 |- ( A e. (SubRing ` W ) -> A = ( Base ` S ) )
3	2	ad2antrl 494	. 2 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> A = ( Base ` S ) )
4		eqid 2238	. . . 4 |- (Scalar ` W ) = (Scalar ` W )
5	1, 4	ressscag 13520	. . 3 |- ( ( W e. AssAlg /\ A e. (SubRing ` W ) ) -> (Scalar ` W ) = (Scalar ` S ) )
6	5	adantrr 483	. 2 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> (Scalar ` W ) = (Scalar ` S ) )
7		eqidd 2239	. 2 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) ) )
8		eqid 2238	. . . 4 |- ( .s ` W ) = ( .s ` W )
9	1, 8	ressvscag 13521	. . 3 |- ( ( W e. AssAlg /\ A e. (SubRing ` W ) ) -> ( .s ` W ) = ( .s ` S ) )
10	9	adantrr 483	. 2 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> ( .s ` W ) = ( .s ` S ) )
11		simprl 535	. . 3 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> A e. (SubRing ` W ) )
12		simpl 109	. . 3 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> W e. AssAlg )
13		eqid 2238	. . . 4 |- ( .r ` W ) = ( .r ` W )
14	1, 13	ressmulrg 13482	. . 3 |- ( ( A e. (SubRing ` W ) /\ W e. AssAlg ) -> ( .r ` W ) = ( .r ` S ) )
15	11, 12, 14	syl2anc 415	. 2 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> ( .r ` W ) = ( .r ` S ) )
16		assalmod 14989	. . 3 |- ( W e. AssAlg -> W e. LMod )
17		simpr 110	. . 3 |- ( ( A e. (SubRing ` W ) /\ A e. L ) -> A e. L )
18		issubassa.l	. . . 4 |- L = ( LSubSp ` W )
19	1, 18	lsslmod 14700	. . 3 |- ( ( W e. LMod /\ A e. L ) -> S e. LMod )
20	16, 17, 19	syl2an 289	. 2 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> S e. LMod )
21	1	subrgring 14515	. . 3 |- ( A e. (SubRing ` W ) -> S e. Ring )
22	21	ad2antrl 494	. 2 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> S e. Ring )
23		idd 21	. . . . 5 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> ( x e. ( Base ` (Scalar ` W ) ) -> x e. ( Base ` (Scalar ` W ) ) ) )
24		eqid 2238	. . . . . . . 8 |- ( Base ` W ) = ( Base ` W )
25	24	subrgss 14513	. . . . . . 7 |- ( A e. (SubRing ` W ) -> A C_ ( Base ` W ) )
26	25	ad2antrl 494	. . . . . 6 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> A C_ ( Base ` W ) )
27	26	sseld 3247	. . . . 5 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> ( y e. A -> y e. ( Base ` W ) ) )
28	26	sseld 3247	. . . . 5 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> ( z e. A -> z e. ( Base ` W ) ) )
29	23, 27, 28	3anim123d 1360	. . . 4 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> ( ( x e. ( Base ` (Scalar ` W ) ) /\ y e. A /\ z e. A ) -> ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) )
30	29	imp 124	. . 3 |- ( ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. A /\ z e. A ) ) -> ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) )
31		eqid 2238	. . . . 5 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
32	24, 4, 31, 8, 13	assaass 14987	. . . 4 |- ( ( W e. AssAlg /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> ( ( x ( .s ` W ) y ) ( .r ` W ) z ) = ( x ( .s ` W ) ( y ( .r ` W ) z ) ) )
33	32	adantlr 481	. . 3 |- ( ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> ( ( x ( .s ` W ) y ) ( .r ` W ) z ) = ( x ( .s ` W ) ( y ( .r ` W ) z ) ) )
34	30, 33	syldan 282	. 2 |- ( ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. A /\ z e. A ) ) -> ( ( x ( .s ` W ) y ) ( .r ` W ) z ) = ( x ( .s ` W ) ( y ( .r ` W ) z ) ) )
35	24, 4, 31, 8, 13	assaassr 14988	. . . 4 |- ( ( W e. AssAlg /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> ( y ( .r ` W ) ( x ( .s ` W ) z ) ) = ( x ( .s ` W ) ( y ( .r ` W ) z ) ) )
36	35	adantlr 481	. . 3 |- ( ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> ( y ( .r ` W ) ( x ( .s ` W ) z ) ) = ( x ( .s ` W ) ( y ( .r ` W ) z ) ) )
37	30, 36	syldan 282	. 2 |- ( ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. A /\ z e. A ) ) -> ( y ( .r ` W ) ( x ( .s ` W ) z ) ) = ( x ( .s ` W ) ( y ( .r ` W ) z ) ) )
38	3, 6, 7, 10, 15, 20, 22, 34, 37	isassad 14994	1 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> S e. AssAlg )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   C_ wss 3220   ` cfv 5375  (class class class)co 6079   Basecbs 13335   ↾_s cress 13336   .rcmulr 13415  Scalarcsca 13417   .scvsca 13418   Ringcrg 14283  SubRingcsubrg 14508   LModclmod 14606   LSubSpclss 14672  AssAlgcasa 14979

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-assa 14982

This theorem is referenced by:  issubassa  14996  rnasclassa  15021