Theorem sralmod 13946

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 2193	. . . 4 |- ( Base ` W ) = ( Base ` W )
4	3	subrgss 13718	. . 3 |- ( S e. (SubRing ` W ) -> S C_ ( Base ` W ) )
5		subrgrcl 13722	. . 3 |- ( S e. (SubRing ` W ) -> W e. Ring )
6	2, 4, 5	srabaseg 13935	. 2 |- ( S e. (SubRing ` W ) -> ( Base ` W ) = ( Base ` A ) )
7	2, 4, 5	sraaddgg 13936	. 2 |- ( S e. (SubRing ` W ) -> ( +g ` W ) = ( +g ` A ) )
8	2, 4, 5	srascag 13938	. 2 |- ( S e. (SubRing ` W ) -> ( W ↾s S ) = (Scalar ` A ) )
9	2, 4, 5	sravscag 13939	. 2 |- ( S e. (SubRing ` W ) -> ( .r ` W ) = ( .s ` A ) )
10		eqidd 2194	. . 3 |- ( S e. (SubRing ` W ) -> ( W ↾s S ) = ( W ↾s S ) )
11		eqidd 2194	. . 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 12685	. 2 |- ( S e. (SubRing ` W ) -> ( S i^i ( Base ` W ) ) = ( Base ` ( W ↾s S ) ) )
14		eqidd 2194	. . 3 |- ( S e. (SubRing ` W ) -> ( +g ` W ) = ( +g ` W ) )
15	10, 14, 12, 5	ressplusgd 12746	. 2 |- ( S e. (SubRing ` W ) -> ( +g ` W ) = ( +g ` ( W ↾s S ) ) )
16		eqid 2193	. . . 4 |- ( W ↾s S ) = ( W ↾s S )
17		eqid 2193	. . . 4 |- ( .r ` W ) = ( .r ` W )
18	16, 17	ressmulrg 12762	. . 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 2193	. . 3 |- ( 1r ` W ) = ( 1r ` W )
21	16, 20	subrg1 13727	. 2 |- ( S e. (SubRing ` W ) -> ( 1r ` W ) = ( 1r ` ( W ↾s S ) ) )
22	16	subrgring 13720	. 2 |- ( S e. (SubRing ` W ) -> ( W ↾s S ) e. Ring )
23	5	ringgrpd 13501	. . 3 |- ( S e. (SubRing ` W ) -> W e. Grp )
24	7	oveqdr 5946	. . . 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 13089	. . 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 1020	. . 3 |- ( ( S e. (SubRing ` W ) /\ x e. ( S i^i ( Base ` W ) ) /\ y e. ( Base ` W ) ) -> W e. Ring )
28		elinel2 3346	. . . 4 |- ( x e. ( S i^i ( Base ` W ) ) -> x e. ( Base ` W ) )
29	28	3ad2ant2 1021	. . 3 |- ( ( S e. (SubRing ` W ) /\ x e. ( S i^i ( Base ` W ) ) /\ y e. ( Base ` W ) ) -> x e. ( Base ` W ) )
30		simp3 1001	. . 3 |- ( ( S e. (SubRing ` W ) /\ x e. ( S i^i ( Base ` W ) ) /\ y e. ( Base ` W ) ) -> y e. ( Base ` W ) )
31	3, 17	ringcl 13509	. . 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 1249	. 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 1005	. . . 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 3349	. . 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 1006	. . 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 1007	. . 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 2193	. . . 4 |- ( +g ` W ) = ( +g ` W )
39	3, 38, 17	ringdi 13514	. . 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 1251	. 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 1005	. . . 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 3349	. . 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 1006	. . . 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 3349	. . 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 1007	. . 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 13515	. . 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 1251	. 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 13512	. . 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 1251	. 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 13519	. . 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 13789	1 |- ( S e. (SubRing ` W ) -> A e. LMod )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2164   i^i cin 3152   ` cfv 5254  (class class class)co 5918   Basecbs 12618   ↾_s cress 12619   +g cplusg 12695   .rcmulr 12696   Grpcgrp 13072   1rcur 13455   Ringcrg 13492  SubRingcsubrg 13713   LModclmod 13783  subringAlg csra 13929

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045  df-7 9046  df-8 9047  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-mulr 12709  df-sca 12711  df-vsca 12712  df-ip 12713  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-subg 13240  df-mgp 13417  df-ur 13456  df-ring 13494  df-subrg 13715  df-lmod 13785  df-sra 13931

This theorem is referenced by:  rlmlmod  13960