Theorem sralmod 14006

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 2196	. . . 4 |- ( Base ` W ) = ( Base ` W )
4	3	subrgss 13778	. . 3 |- ( S e. (SubRing ` W ) -> S C_ ( Base ` W ) )
5		subrgrcl 13782	. . 3 |- ( S e. (SubRing ` W ) -> W e. Ring )
6	2, 4, 5	srabaseg 13995	. 2 |- ( S e. (SubRing ` W ) -> ( Base ` W ) = ( Base ` A ) )
7	2, 4, 5	sraaddgg 13996	. 2 |- ( S e. (SubRing ` W ) -> ( +g ` W ) = ( +g ` A ) )
8	2, 4, 5	srascag 13998	. 2 |- ( S e. (SubRing ` W ) -> ( W ↾s S ) = (Scalar ` A ) )
9	2, 4, 5	sravscag 13999	. 2 |- ( S e. (SubRing ` W ) -> ( .r ` W ) = ( .s ` A ) )
10		eqidd 2197	. . 3 |- ( S e. (SubRing ` W ) -> ( W ↾s S ) = ( W ↾s S ) )
11		eqidd 2197	. . 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 12745	. 2 |- ( S e. (SubRing ` W ) -> ( S i^i ( Base ` W ) ) = ( Base ` ( W ↾s S ) ) )
14		eqidd 2197	. . 3 |- ( S e. (SubRing ` W ) -> ( +g ` W ) = ( +g ` W ) )
15	10, 14, 12, 5	ressplusgd 12806	. 2 |- ( S e. (SubRing ` W ) -> ( +g ` W ) = ( +g ` ( W ↾s S ) ) )
16		eqid 2196	. . . 4 |- ( W ↾s S ) = ( W ↾s S )
17		eqid 2196	. . . 4 |- ( .r ` W ) = ( .r ` W )
18	16, 17	ressmulrg 12822	. . 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 2196	. . 3 |- ( 1r ` W ) = ( 1r ` W )
21	16, 20	subrg1 13787	. 2 |- ( S e. (SubRing ` W ) -> ( 1r ` W ) = ( 1r ` ( W ↾s S ) ) )
22	16	subrgring 13780	. 2 |- ( S e. (SubRing ` W ) -> ( W ↾s S ) e. Ring )
23	5	ringgrpd 13561	. . 3 |- ( S e. (SubRing ` W ) -> W e. Grp )
24	7	oveqdr 5950	. . . 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 13149	. . 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 3350	. . . 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 13569	. . 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 3353	. . 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 2196	. . . 4 |- ( +g ` W ) = ( +g ` W )
39	3, 38, 17	ringdi 13574	. . 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 3353	. . 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 3353	. . 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 13575	. . 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 13572	. . 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 13579	. . 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 13849	1 |- ( S e. (SubRing ` W ) -> A e. LMod )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167   i^i cin 3156   ` cfv 5258  (class class class)co 5922   Basecbs 12678   ↾_s cress 12679   +g cplusg 12755   .rcmulr 12756   Grpcgrp 13132   1rcur 13515   Ringcrg 13552  SubRingcsubrg 13773   LModclmod 13843  subringAlg csra 13989

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-5 9052  df-6 9053  df-7 9054  df-8 9055  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-sca 12771  df-vsca 12772  df-ip 12773  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-subg 13300  df-mgp 13477  df-ur 13516  df-ring 13554  df-subrg 13775  df-lmod 13845  df-sra 13991

This theorem is referenced by:  rlmlmod  14020