Theorem sraassa 19325

Description: The subring algebra over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.)

Hypothesis

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

Assertion

Ref	Expression
sraassa	|- ( ( W e. CRing /\ S e. (SubRing ` W ) ) -> A e. AssAlg )

Proof of Theorem sraassa

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

Step	Hyp	Ref	Expression
1		sraassa.a	. . . 4 |- A = ( (subringAlg ` W ) ` S )
2	1	a1i 11	. . 3 |- ( ( W e. CRing /\ S e. (SubRing ` W ) ) -> A = ( (subringAlg ` W ) ` S ) )
3		eqid 2622	. . . . 5 |- ( Base ` W ) = ( Base ` W )
4	3	subrgss 18781	. . . 4 |- ( S e. (SubRing ` W ) -> S C_ ( Base ` W ) )
5	4	adantl 482	. . 3 |- ( ( W e. CRing /\ S e. (SubRing ` W ) ) -> S C_ ( Base ` W ) )
6	2, 5	srabase 19178	. 2 |- ( ( W e. CRing /\ S e. (SubRing ` W ) ) -> ( Base ` W ) = ( Base ` A ) )
7	2, 5	srasca 19181	. 2 |- ( ( W e. CRing /\ S e. (SubRing ` W ) ) -> ( W ↾s S ) = (Scalar ` A ) )
8		eqid 2622	. . . 4 |- ( W ↾s S ) = ( W ↾s S )
9	8	subrgbas 18789	. . 3 |- ( S e. (SubRing ` W ) -> S = ( Base ` ( W ↾s S ) ) )
10	9	adantl 482	. 2 |- ( ( W e. CRing /\ S e. (SubRing ` W ) ) -> S = ( Base ` ( W ↾s S ) ) )
11	2, 5	sravsca 19182	. 2 |- ( ( W e. CRing /\ S e. (SubRing ` W ) ) -> ( .r ` W ) = ( .s ` A ) )
12	2, 5	sramulr 19180	. 2 |- ( ( W e. CRing /\ S e. (SubRing ` W ) ) -> ( .r ` W ) = ( .r ` A ) )
13	1	sralmod 19187	. . 3 |- ( S e. (SubRing ` W ) -> A e. LMod )
14	13	adantl 482	. 2 |- ( ( W e. CRing /\ S e. (SubRing ` W ) ) -> A e. LMod )
15		crngring 18558	. . . 4 |- ( W e. CRing -> W e. Ring )
16	15	adantr 481	. . 3 |- ( ( W e. CRing /\ S e. (SubRing ` W ) ) -> W e. Ring )
17		eqidd 2623	. . . 4 |- ( ( W e. CRing /\ S e. (SubRing ` W ) ) -> ( Base ` W ) = ( Base ` W ) )
18	2, 5	sraaddg 19179	. . . . 5 |- ( ( W e. CRing /\ S e. (SubRing ` W ) ) -> ( +g ` W ) = ( +g ` A ) )
19	18	oveqdr 6674	. . . 4 |- ( ( ( W e. CRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` W ) /\ y e. ( Base ` W ) ) ) -> ( x ( +g ` W ) y ) = ( x ( +g ` A ) y ) )
20	12	oveqdr 6674	. . . 4 |- ( ( ( W e. CRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` W ) /\ y e. ( Base ` W ) ) ) -> ( x ( .r ` W ) y ) = ( x ( .r ` A ) y ) )
21	17, 6, 19, 20	ringpropd 18582	. . 3 |- ( ( W e. CRing /\ S e. (SubRing ` W ) ) -> ( W e. Ring <-> A e. Ring ) )
22	16, 21	mpbid 222	. 2 |- ( ( W e. CRing /\ S e. (SubRing ` W ) ) -> A e. Ring )
23	8	subrgcrng 18784	. 2 |- ( ( W e. CRing /\ S e. (SubRing ` W ) ) -> ( W ↾s S ) e. CRing )
24	16	adantr 481	. . 3 |- ( ( ( W e. CRing /\ S e. (SubRing ` W ) ) /\ ( x e. S /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> W e. Ring )
25	5	adantr 481	. . . 4 |- ( ( ( W e. CRing /\ S e. (SubRing ` W ) ) /\ ( x e. S /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> S C_ ( Base ` W ) )
26		simpr1 1067	. . . 4 |- ( ( ( W e. CRing /\ S e. (SubRing ` W ) ) /\ ( x e. S /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> x e. S )
27	25, 26	sseldd 3604	. . 3 |- ( ( ( W e. CRing /\ S e. (SubRing ` W ) ) /\ ( x e. S /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> x e. ( Base ` W ) )
28		simpr2 1068	. . 3 |- ( ( ( W e. CRing /\ S e. (SubRing ` W ) ) /\ ( x e. S /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> y e. ( Base ` W ) )
29		simpr3 1069	. . 3 |- ( ( ( W e. CRing /\ S e. (SubRing ` W ) ) /\ ( x e. S /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> z e. ( Base ` W ) )
30		eqid 2622	. . . 4 |- ( .r ` W ) = ( .r ` W )
31	3, 30	ringass 18564	. . 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 ) ) )
32	24, 27, 28, 29, 31	syl13anc 1328	. 2 |- ( ( ( W e. CRing /\ S e. (SubRing ` W ) ) /\ ( x e. S /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> ( ( x ( .r ` W ) y ) ( .r ` W ) z ) = ( x ( .r ` W ) ( y ( .r ` W ) z ) ) )
33		eqid 2622	. . . . 5 |- (mulGrp ` W ) = (mulGrp ` W )
34	33	crngmgp 18555	. . . 4 |- ( W e. CRing -> (mulGrp ` W ) e. CMnd )
35	34	ad2antrr 762	. . 3 |- ( ( ( W e. CRing /\ S e. (SubRing ` W ) ) /\ ( x e. S /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> (mulGrp ` W ) e. CMnd )
36	33, 3	mgpbas 18495	. . . 4 |- ( Base ` W ) = ( Base ` (mulGrp ` W ) )
37	33, 30	mgpplusg 18493	. . . 4 |- ( .r ` W ) = ( +g ` (mulGrp ` W ) )
38	36, 37	cmn12 18213	. . 3 |- ( ( (mulGrp ` W ) e. CMnd /\ ( y e. ( Base ` W ) /\ x e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> ( y ( .r ` W ) ( x ( .r ` W ) z ) ) = ( x ( .r ` W ) ( y ( .r ` W ) z ) ) )
39	35, 28, 27, 29, 38	syl13anc 1328	. 2 |- ( ( ( W e. CRing /\ S e. (SubRing ` W ) ) /\ ( x e. S /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) ) -> ( y ( .r ` W ) ( x ( .r ` W ) z ) ) = ( x ( .r ` W ) ( y ( .r ` W ) z ) ) )
40	6, 7, 10, 11, 12, 14, 22, 23, 32, 39	isassad 19323	1 |- ( ( W e. CRing /\ S e. (SubRing ` W ) ) -> A e. AssAlg )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   .rcmulr 15942  CMndccmn 18193  mulGrpcmgp 18489   Ringcrg 18547   CRingccrg 18548  SubRingcsubrg 18776   LModclmod 18863  subringAlg csra 19168  AssAlgcasa 19309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-subg 17591  df-cmn 18195  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-subrg 18778  df-lmod 18865  df-sra 19172  df-assa 19312

This theorem is referenced by:  rlmassa  19326