Theorem sralmod 14770

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 2238	. . . 4 |- ( Base ` W ) = ( Base ` W )
4	3	subrgss 14513	. . 3 |- ( S e. (SubRing ` W ) -> S C_ ( Base ` W ) )
5		subrgrcl 14517	. . 3 |- ( S e. (SubRing ` W ) -> W e. Ring )
6	2, 4, 5	srabaseg 14759	. 2 |- ( S e. (SubRing ` W ) -> ( Base ` W ) = ( Base ` A ) )
7	2, 4, 5	sraaddgg 14760	. 2 |- ( S e. (SubRing ` W ) -> ( +g ` W ) = ( +g ` A ) )
8	2, 4, 5	srascag 14762	. 2 |- ( S e. (SubRing ` W ) -> ( W ↾s S ) = (Scalar ` A ) )
9	2, 4, 5	sravscag 14763	. 2 |- ( S e. (SubRing ` W ) -> ( .r ` W ) = ( .s ` A ) )
10		eqidd 2239	. . 3 |- ( S e. (SubRing ` W ) -> ( W ↾s S ) = ( W ↾s S ) )
11		eqidd 2239	. . 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 13404	. 2 |- ( S e. (SubRing ` W ) -> ( S i^i ( Base ` W ) ) = ( Base ` ( W ↾s S ) ) )
14		eqidd 2239	. . 3 |- ( S e. (SubRing ` W ) -> ( +g ` W ) = ( +g ` W ) )
15	10, 14, 12, 5	ressplusgd 13466	. 2 |- ( S e. (SubRing ` W ) -> ( +g ` W ) = ( +g ` ( W ↾s S ) ) )
16		eqid 2238	. . . 4 |- ( W ↾s S ) = ( W ↾s S )
17		eqid 2238	. . . 4 |- ( .r ` W ) = ( .r ` W )
18	16, 17	ressmulrg 13482	. . 3 |- ( ( S e. (SubRing ` W ) /\ W e. Ring ) -> ( .r ` W ) = ( .r ` ( W ↾s S ) ) )
19	5, 18	mpdan 425	. 2 |- ( S e. (SubRing ` W ) -> ( .r ` W ) = ( .r ` ( W ↾s S ) ) )
20		eqid 2238	. . 3 |- ( 1r ` W ) = ( 1r ` W )
21	16, 20	subrg1 14522	. 2 |- ( S e. (SubRing ` W ) -> ( 1r ` W ) = ( 1r ` ( W ↾s S ) ) )
22	16	subrgring 14515	. 2 |- ( S e. (SubRing ` W ) -> ( W ↾s S ) e. Ring )
23	5	ringgrpd 14292	. . 3 |- ( S e. (SubRing ` W ) -> W e. Grp )
24	7	oveqdr 6107	. . . 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 13805	. . 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 1049	. . 3 |- ( ( S e. (SubRing ` W ) /\ x e. ( S i^i ( Base ` W ) ) /\ y e. ( Base ` W ) ) -> W e. Ring )
28		elinel2 3416	. . . 4 |- ( x e. ( S i^i ( Base ` W ) ) -> x e. ( Base ` W ) )
29	28	3ad2ant2 1050	. . 3 |- ( ( S e. (SubRing ` W ) /\ x e. ( S i^i ( Base ` W ) ) /\ y e. ( Base ` W ) ) -> x e. ( Base ` W ) )
30		simp3 1030	. . 3 |- ( ( S e. (SubRing ` W ) /\ x e. ( S i^i ( Base ` W ) ) /\ y e. ( Base ` W ) ) -> y e. ( Base ` W ) )
31	3, 17	ringcl 14300	. . 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 1278	. 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 1034	. . . 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 3419	. . 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 1035	. . 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 1036	. . 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 2238	. . . 4 |- ( +g ` W ) = ( +g ` W )
39	3, 38, 17	ringdi 14306	. . 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 1280	. 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 1034	. . . 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 3419	. . 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 1035	. . . 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 3419	. . 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 1036	. . 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 14307	. . 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 1280	. 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 14303	. . 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 1280	. 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 14311	. . 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 14612	1 |- ( S e. (SubRing ` W ) -> A e. LMod )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   i^i cin 3219   ` cfv 5375  (class class class)co 6079   Basecbs 13335   ↾_s cress 13336   +g cplusg 13414   .rcmulr 13415   Grpcgrp 13788   1rcur 14245   Ringcrg 14283  SubRingcsubrg 14508   LModclmod 14606  subringAlg csra 14753

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-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-7 9351  df-8 9352  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-ip 13432  df-0g 13595  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-grp 13791  df-subg 13956  df-mgp 14201  df-ur 14246  df-ring 14285  df-subrg 14510  df-lmod 14608  df-sra 14755

This theorem is referenced by:  rlmlmod  14784