Theorem sralmod 14457

Description: The subring algebra is a left module. (Contributed by Stefan O'Rear, 27-Nov-2014.)

Hypothesis

Ref	Expression
sralmod.a	|- A = ( (subringAlg ` W ) ` S )

Assertion

Ref	Expression
sralmod	|- ( S e. (SubRing ` W ) -> A e. LMod )

Proof of Theorem sralmod

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

Step	Hyp	Ref	Expression
1		sralmod.a	. . . 4 |- A = ( (subringAlg ` W ) ` S )
2	1	a1i 9	. . 3 |- ( S e. (SubRing ` W ) -> A = ( (subringAlg ` W ) ` S ) )
3		eqid 2229	. . . 4 |- ( Base ` W ) = ( Base ` W )
4	3	subrgss 14229	. . 3 |- ( S e. (SubRing ` W ) -> S C_ ( Base ` W ) )
5		subrgrcl 14233	. . 3 |- ( S e. (SubRing ` W ) -> W e. Ring )
6	2, 4, 5	srabaseg 14446	. 2 |- ( S e. (SubRing ` W ) -> ( Base ` W ) = ( Base ` A ) )
7	2, 4, 5	sraaddgg 14447	. 2 |- ( S e. (SubRing ` W ) -> ( +g ` W ) = ( +g ` A ) )
8	2, 4, 5	srascag 14449	. 2 |- ( S e. (SubRing ` W ) -> ( W ↾s S ) = (Scalar ` A ) )
9	2, 4, 5	sravscag 14450	. 2 |- ( S e. (SubRing ` W ) -> ( .r ` W ) = ( .s ` A ) )
10		eqidd 2230	. . 3 |- ( S e. (SubRing ` W ) -> ( W ↾s S ) = ( W ↾s S ) )
11		eqidd 2230	. . 3 |- ( S e. (SubRing ` W ) -> ( Base ` W ) = ( Base ` W ) )
12		id 19	. . 3 |- ( S e. (SubRing ` W ) -> S e. (SubRing ` W ) )
13	10, 11, 5, 12	ressbasd 13143	. 2 |- ( S e. (SubRing ` W ) -> ( S i^i ( Base ` W ) ) = ( Base ` ( W ↾s S ) ) )
14		eqidd 2230	. . 3 |- ( S e. (SubRing ` W ) -> ( +g ` W ) = ( +g ` W ) )
15	10, 14, 12, 5	ressplusgd 13205	. 2 |- ( S e. (SubRing ` W ) -> ( +g ` W ) = ( +g ` ( W ↾s S ) ) )
16		eqid 2229	. . . 4 |- ( W ↾s S ) = ( W ↾s S )
17		eqid 2229	. . . 4 |- ( .r ` W ) = ( .r ` W )
18	16, 17	ressmulrg 13221	. . 3 |- ( ( S e. (SubRing ` W ) /\ W e. Ring ) -> ( .r ` W ) = ( .r ` ( W ↾s S ) ) )
19	5, 18	mpdan 421	. 2 |- ( S e. (SubRing ` W ) -> ( .r ` W ) = ( .r ` ( W ↾s S ) ) )
20		eqid 2229	. . 3 |- ( 1r ` W ) = ( 1r ` W )
21	16, 20	subrg1 14238	. 2 |- ( S e. (SubRing ` W ) -> ( 1r ` W ) = ( 1r ` ( W ↾s S ) ) )
22	16	subrgring 14231	. 2 |- ( S e. (SubRing ` W ) -> ( W ↾s S ) e. Ring )
23	5	ringgrpd 14011	. . 3 |- ( S e. (SubRing ` W ) -> W e. Grp )
24	7	oveqdr 6041	. . . 4 |- ( ( S e. (SubRing ` W ) /\ ( x e. ( Base ` W ) /\ y e. ( Base ` W ) ) ) -> ( x ( +g ` W ) y ) = ( x ( +g ` A ) y ) )
25	11, 6, 24	grppropd 13593	. . 3 |- ( S e. (SubRing ` W ) -> ( W e. Grp <-> A e. Grp ) )
26	23, 25	mpbid 147	. 2 |- ( S e. (SubRing ` W ) -> A e. Grp )
27	5	3ad2ant1 1042	. . 3 |- ( ( S e. (SubRing ` W ) /\ x e. ( S i^i ( Base ` W ) ) /\ y e. ( Base ` W ) ) -> W e. Ring )
28		elinel2 3392	. . . 4 |- ( x e. ( S i^i ( Base ` W ) ) -> x e. ( Base ` W ) )
29	28	3ad2ant2 1043	. . 3 |- ( ( S e. (SubRing ` W ) /\ x e. ( S i^i ( Base ` W ) ) /\ y e. ( Base ` W ) ) -> x e. ( Base ` W ) )
30		simp3 1023	. . 3 |- ( ( S e. (SubRing ` W ) /\ x e. ( S i^i ( Base ` W ) ) /\ y e. ( Base ` W ) ) -> y e. ( Base ` W ) )
31	3, 17	ringcl 14019	. . 3 |- ( ( W e. Ring /\ x e. ( Base ` W ) /\ y e. ( Base ` W ) ) -> ( x ( .r ` W ) y ) e. ( Base ` W ) )
32	27, 29, 30, 31	syl3anc 1271	. 2 |- ( ( S e. (SubRing ` W ) /\ x e. ( S i^i ( Base ` W ) ) /\ y e. ( Base ` W ) ) -> ( x ( .r ` W ) y ) e. ( Base ` W ) )
33	5	adantr 276	. . 3 |- ( ( S e. (SubRing ` W ) /\ ( x e. ( S i^i ( Base ` W ) ) /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> W e. Ring )
34		simpr1 1027	. . . 4 |- ( ( S e. (SubRing ` W ) /\ ( x e. ( S i^i ( Base ` W ) ) /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> x e. ( S i^i ( Base ` W ) ) )
35	34	elin2d 3395	. . 3 |- ( ( S e. (SubRing ` W ) /\ ( x e. ( S i^i ( Base ` W ) ) /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> x e. ( Base ` W ) )
36		simpr2 1028	. . 3 |- ( ( S e. (SubRing ` W ) /\ ( x e. ( S i^i ( Base ` W ) ) /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> y e. ( Base ` W ) )
37		simpr3 1029	. . 3 |- ( ( S e. (SubRing ` W ) /\ ( x e. ( S i^i ( Base ` W ) ) /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> z e. ( Base ` W ) )
38		eqid 2229	. . . 4 |- ( +g ` W ) = ( +g ` W )
39	3, 38, 17	ringdi 14024	. . 3 |- ( ( W e. Ring /\ ( x e. ( Base ` W ) /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> ( x ( .r ` W ) ( y ( +g ` W ) z ) ) = ( ( x ( .r ` W ) y ) ( +g ` W ) ( x ( .r ` W ) z ) ) )
40	33, 35, 36, 37, 39	syl13anc 1273	. 2 |- ( ( S e. (SubRing ` W ) /\ ( x e. ( S i^i ( Base ` W ) ) /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> ( x ( .r ` W ) ( y ( +g ` W ) z ) ) = ( ( x ( .r ` W ) y ) ( +g ` W ) ( x ( .r ` W ) z ) ) )
41	5	adantr 276	. . 3 |- ( ( S e. (SubRing ` W ) /\ ( x e. ( S i^i ( Base ` W ) ) /\ y e. ( S i^i ( Base ` W ) ) /\ z e. ( Base ` W ) ) ) -> W e. Ring )
42		simpr1 1027	. . . 4 |- ( ( S e. (SubRing ` W ) /\ ( x e. ( S i^i ( Base ` W ) ) /\ y e. ( S i^i ( Base ` W ) ) /\ z e. ( Base ` W ) ) ) -> x e. ( S i^i ( Base ` W ) ) )
43	42	elin2d 3395	. . 3 |- ( ( S e. (SubRing ` W ) /\ ( x e. ( S i^i ( Base ` W ) ) /\ y e. ( S i^i ( Base ` W ) ) /\ z e. ( Base ` W ) ) ) -> x e. ( Base ` W ) )
44		simpr2 1028	. . . 4 |- ( ( S e. (SubRing ` W ) /\ ( x e. ( S i^i ( Base ` W ) ) /\ y e. ( S i^i ( Base ` W ) ) /\ z e. ( Base ` W ) ) ) -> y e. ( S i^i ( Base ` W ) ) )
45	44	elin2d 3395	. . 3 |- ( ( S e. (SubRing ` W ) /\ ( x e. ( S i^i ( Base ` W ) ) /\ y e. ( S i^i ( Base ` W ) ) /\ z e. ( Base ` W ) ) ) -> y e. ( Base ` W ) )
46		simpr3 1029	. . 3 |- ( ( S e. (SubRing ` W ) /\ ( x e. ( S i^i ( Base ` W ) ) /\ y e. ( S i^i ( Base ` W ) ) /\ z e. ( Base ` W ) ) ) -> z e. ( Base ` W ) )
47	3, 38, 17	ringdir 14025	. . 3 |- ( ( W e. Ring /\ ( x e. ( Base ` W ) /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> ( ( x ( +g ` W ) y ) ( .r ` W ) z ) = ( ( x ( .r ` W ) z ) ( +g ` W ) ( y ( .r ` W ) z ) ) )
48	41, 43, 45, 46, 47	syl13anc 1273	. 2 |- ( ( S e. (SubRing ` W ) /\ ( x e. ( S i^i ( Base ` W ) ) /\ y e. ( S i^i ( Base ` W ) ) /\ z e. ( Base ` W ) ) ) -> ( ( x ( +g ` W ) y ) ( .r ` W ) z ) = ( ( x ( .r ` W ) z ) ( +g ` W ) ( y ( .r ` W ) z ) ) )
49	3, 17	ringass 14022	. . 3 |- ( ( W e. Ring /\ ( x e. ( Base ` W ) /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> ( ( x ( .r ` W ) y ) ( .r ` W ) z ) = ( x ( .r ` W ) ( y ( .r ` W ) z ) ) )
50	41, 43, 45, 46, 49	syl13anc 1273	. 2 |- ( ( S e. (SubRing ` W ) /\ ( x e. ( S i^i ( Base ` W ) ) /\ y e. ( S i^i ( Base ` W ) ) /\ z e. ( Base ` W ) ) ) -> ( ( x ( .r ` W ) y ) ( .r ` W ) z ) = ( x ( .r ` W ) ( y ( .r ` W ) z ) ) )
51	3, 17, 20	ringlidm 14029	. . 3 |- ( ( W e. Ring /\ x e. ( Base ` W ) ) -> ( ( 1r ` W ) ( .r ` W ) x ) = x )
52	5, 51	sylan 283	. 2 |- ( ( S e. (SubRing ` W ) /\ x e. ( Base ` W ) ) -> ( ( 1r ` W ) ( .r ` W ) x ) = x )
53	6, 7, 8, 9, 13, 15, 19, 21, 22, 26, 32, 40, 48, 50, 52	islmodd 14300	1 |- ( S e. (SubRing ` W ) -> A e. LMod )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   i^i cin 3197   ` cfv 5324  (class class class)co 6013   Basecbs 13075   ↾_s cress 13076   +g cplusg 13153   .rcmulr 13154   Grpcgrp 13576   1rcur 13965   Ringcrg 14002  SubRingcsubrg 14224   LModclmod 14294  subringAlg csra 14440

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-addass 8127  ax-i2m1 8130  ax-0lt1 8131  ax-0id 8133  ax-rnegex 8134  ax-pre-ltirr 8137  ax-pre-lttrn 8139  ax-pre-ltadd 8141

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8209  df-mnf 8210  df-ltxr 8212  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-5 9198  df-6 9199  df-7 9200  df-8 9201  df-ndx 13078  df-slot 13079  df-base 13081  df-sets 13082  df-iress 13083  df-plusg 13166  df-mulr 13167  df-sca 13169  df-vsca 13170  df-ip 13171  df-0g 13334  df-mgm 13432  df-sgrp 13478  df-mnd 13493  df-grp 13579  df-subg 13750  df-mgp 13927  df-ur 13966  df-ring 14004  df-subrg 14226  df-lmod 14296  df-sra 14442

This theorem is referenced by:  rlmlmod  14471