Theorem sralmod 13696

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 2187	. . . 4 |- ( Base ` W ) = ( Base ` W )
4	3	subrgss 13499	. . 3 |- ( S e. (SubRing ` W ) -> S C_ ( Base ` W ) )
5		subrgrcl 13503	. . 3 |- ( S e. (SubRing ` W ) -> W e. Ring )
6	2, 4, 5	srabaseg 13685	. 2 |- ( S e. (SubRing ` W ) -> ( Base ` W ) = ( Base ` A ) )
7	2, 4, 5	sraaddgg 13686	. 2 |- ( S e. (SubRing ` W ) -> ( +g ` W ) = ( +g ` A ) )
8	2, 4, 5	srascag 13688	. 2 |- ( S e. (SubRing ` W ) -> ( W ↾s S ) = (Scalar ` A ) )
9	2, 4, 5	sravscag 13689	. 2 |- ( S e. (SubRing ` W ) -> ( .r ` W ) = ( .s ` A ) )
10		eqidd 2188	. . 3 |- ( S e. (SubRing ` W ) -> ( W ↾s S ) = ( W ↾s S ) )
11		eqidd 2188	. . 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 12541	. 2 |- ( S e. (SubRing ` W ) -> ( S i^i ( Base ` W ) ) = ( Base ` ( W ↾s S ) ) )
14		eqidd 2188	. . 3 |- ( S e. (SubRing ` W ) -> ( +g ` W ) = ( +g ` W ) )
15	10, 14, 12, 5	ressplusgd 12602	. 2 |- ( S e. (SubRing ` W ) -> ( +g ` W ) = ( +g ` ( W ↾s S ) ) )
16		eqid 2187	. . . 4 |- ( W ↾s S ) = ( W ↾s S )
17		eqid 2187	. . . 4 |- ( .r ` W ) = ( .r ` W )
18	16, 17	ressmulrg 12618	. . 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 2187	. . 3 |- ( 1r ` W ) = ( 1r ` W )
21	16, 20	subrg1 13508	. 2 |- ( S e. (SubRing ` W ) -> ( 1r ` W ) = ( 1r ` ( W ↾s S ) ) )
22	16	subrgring 13501	. 2 |- ( S e. (SubRing ` W ) -> ( W ↾s S ) e. Ring )
23	5	ringgrpd 13314	. . 3 |- ( S e. (SubRing ` W ) -> W e. Grp )
24	7	oveqdr 5916	. . . 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 12923	. . 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 1019	. . 3 |- ( ( S e. (SubRing ` W ) /\ x e. ( S i^i ( Base ` W ) ) /\ y e. ( Base ` W ) ) -> W e. Ring )
28		elinel2 3334	. . . 4 |- ( x e. ( S i^i ( Base ` W ) ) -> x e. ( Base ` W ) )
29	28	3ad2ant2 1020	. . 3 |- ( ( S e. (SubRing ` W ) /\ x e. ( S i^i ( Base ` W ) ) /\ y e. ( Base ` W ) ) -> x e. ( Base ` W ) )
30		simp3 1000	. . 3 |- ( ( S e. (SubRing ` W ) /\ x e. ( S i^i ( Base ` W ) ) /\ y e. ( Base ` W ) ) -> y e. ( Base ` W ) )
31	3, 17	ringcl 13322	. . 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 1248	. 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 1004	. . . 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 3337	. . 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 1005	. . 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 1006	. . 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 2187	. . . 4 |- ( +g ` W ) = ( +g ` W )
39	3, 38, 17	ringdi 13327	. . 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 1250	. 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 1004	. . . 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 3337	. . 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 1005	. . . 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 3337	. . 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 1006	. . 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 13328	. . 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 1250	. 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 13325	. . 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 1250	. 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 13332	. . 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 13539	1 |- ( S e. (SubRing ` W ) -> A e. LMod )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 979   = wceq 1363   e. wcel 2158   i^i cin 3140   ` cfv 5228  (class class class)co 5888   Basecbs 12476   ↾_s cress 12477   +g cplusg 12551   .rcmulr 12552   Grpcgrp 12906   1rcur 13268   Ringcrg 13305  SubRingcsubrg 13494   LModclmod 13533  subringAlg csra 13679

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-addcom 7925  ax-addass 7927  ax-i2m1 7930  ax-0lt1 7931  ax-0id 7933  ax-rnegex 7934  ax-pre-ltirr 7937  ax-pre-lttrn 7939  ax-pre-ltadd 7941

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8008  df-mnf 8009  df-ltxr 8011  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-5 8995  df-6 8996  df-7 8997  df-8 8998  df-ndx 12479  df-slot 12480  df-base 12482  df-sets 12483  df-iress 12484  df-plusg 12564  df-mulr 12565  df-sca 12567  df-vsca 12568  df-ip 12569  df-0g 12725  df-mgm 12794  df-sgrp 12827  df-mnd 12840  df-grp 12909  df-subg 13070  df-mgp 13230  df-ur 13269  df-ring 13307  df-subrg 13496  df-lmod 13535  df-sra 13681

This theorem is referenced by:  rlmlmod  13710