Theorem issubassa 19324

Description: The subalgebras of an associative algebra are exactly the subrings (under the ring multiplication) that are simultaneously subspaces (under the scalar multiplication from the vector space). (Contributed by Mario Carneiro, 7-Jan-2015.)

Hypotheses

Ref	Expression
issubassa.s	|- S = ( W ↾s A )
issubassa.l	|- L = ( LSubSp ` W )
issubassa.v	|- V = ( Base ` W )
issubassa.o	|- .1. = ( 1r ` W )

Assertion

Ref	Expression
issubassa	|- ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) -> ( S e. AssAlg <-> ( A e. (SubRing ` W ) /\ A e. L ) ) )

Proof of Theorem issubassa

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

Step	Hyp	Ref	Expression
1		simpl1 1064	. . . . . 6 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> W e. AssAlg )
2		assaring 19320	. . . . . 6 |- ( W e. AssAlg -> W e. Ring )
3	1, 2	syl 17	. . . . 5 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> W e. Ring )
4		issubassa.s	. . . . . 6 |- S = ( W ↾s A )
5		assaring 19320	. . . . . . 7 |- ( S e. AssAlg -> S e. Ring )
6	5	adantl 482	. . . . . 6 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> S e. Ring )
7	4, 6	syl5eqelr 2706	. . . . 5 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> ( W ↾s A ) e. Ring )
8	3, 7	jca 554	. . . 4 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> ( W e. Ring /\ ( W ↾s A ) e. Ring ) )
9		simpl3 1066	. . . . 5 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> A C_ V )
10		simpl2 1065	. . . . 5 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> .1. e. A )
11	9, 10	jca 554	. . . 4 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> ( A C_ V /\ .1. e. A ) )
12		issubassa.v	. . . . 5 |- V = ( Base ` W )
13		issubassa.o	. . . . 5 |- .1. = ( 1r ` W )
14	12, 13	issubrg 18780	. . . 4 |- ( A e. (SubRing ` W ) <-> ( ( W e. Ring /\ ( W ↾s A ) e. Ring ) /\ ( A C_ V /\ .1. e. A ) ) )
15	8, 11, 14	sylanbrc 698	. . 3 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> A e. (SubRing ` W ) )
16		assalmod 19319	. . . . 5 |- ( S e. AssAlg -> S e. LMod )
17	16	adantl 482	. . . 4 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> S e. LMod )
18		assalmod 19319	. . . . 5 |- ( W e. AssAlg -> W e. LMod )
19		issubassa.l	. . . . . 6 |- L = ( LSubSp ` W )
20	4, 12, 19	islss3 18959	. . . . 5 |- ( W e. LMod -> ( A e. L <-> ( A C_ V /\ S e. LMod ) ) )
21	1, 18, 20	3syl 18	. . . 4 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> ( A e. L <-> ( A C_ V /\ S e. LMod ) ) )
22	9, 17, 21	mpbir2and 957	. . 3 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> A e. L )
23	15, 22	jca 554	. 2 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> ( A e. (SubRing ` W ) /\ A e. L ) )
24	12	subrgss 18781	. . . . . 6 |- ( A e. (SubRing ` W ) -> A C_ V )
25	24	ad2antrl 764	. . . . 5 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> A C_ V )
26	4, 12	ressbas2 15931	. . . . 5 |- ( A C_ V -> A = ( Base ` S ) )
27	25, 26	syl 17	. . . 4 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> A = ( Base ` S ) )
28		eqid 2622	. . . . . 6 |- (Scalar ` W ) = (Scalar ` W )
29	4, 28	resssca 16031	. . . . 5 |- ( A e. (SubRing ` W ) -> (Scalar ` W ) = (Scalar ` S ) )
30	29	ad2antrl 764	. . . 4 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> (Scalar ` W ) = (Scalar ` S ) )
31		eqidd 2623	. . . 4 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) ) )
32		eqid 2622	. . . . . 6 |- ( .s ` W ) = ( .s ` W )
33	4, 32	ressvsca 16032	. . . . 5 |- ( A e. (SubRing ` W ) -> ( .s ` W ) = ( .s ` S ) )
34	33	ad2antrl 764	. . . 4 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> ( .s ` W ) = ( .s ` S ) )
35		eqid 2622	. . . . . 6 |- ( .r ` W ) = ( .r ` W )
36	4, 35	ressmulr 16006	. . . . 5 |- ( A e. (SubRing ` W ) -> ( .r ` W ) = ( .r ` S ) )
37	36	ad2antrl 764	. . . 4 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> ( .r ` W ) = ( .r ` S ) )
38		simpr 477	. . . . 5 |- ( ( A e. (SubRing ` W ) /\ A e. L ) -> A e. L )
39	4, 19	lsslmod 18960	. . . . 5 |- ( ( W e. LMod /\ A e. L ) -> S e. LMod )
40	18, 38, 39	syl2an 494	. . . 4 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> S e. LMod )
41	4	subrgring 18783	. . . . 5 |- ( A e. (SubRing ` W ) -> S e. Ring )
42	41	ad2antrl 764	. . . 4 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> S e. Ring )
43	28	assasca 19321	. . . . 5 |- ( W e. AssAlg -> (Scalar ` W ) e. CRing )
44	43	adantr 481	. . . 4 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> (Scalar ` W ) e. CRing )
45		simpll 790	. . . . 5 |- ( ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. A /\ z e. A ) ) -> W e. AssAlg )
46		simpr1 1067	. . . . 5 |- ( ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. A /\ z e. A ) ) -> x e. ( Base ` (Scalar ` W ) ) )
47	25	adantr 481	. . . . . 6 |- ( ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. A /\ z e. A ) ) -> A C_ V )
48		simpr2 1068	. . . . . 6 |- ( ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. A /\ z e. A ) ) -> y e. A )
49	47, 48	sseldd 3604	. . . . 5 |- ( ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. A /\ z e. A ) ) -> y e. V )
50		simpr3 1069	. . . . . 6 |- ( ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. A /\ z e. A ) ) -> z e. A )
51	47, 50	sseldd 3604	. . . . 5 |- ( ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. A /\ z e. A ) ) -> z e. V )
52		eqid 2622	. . . . . 6 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
53	12, 28, 52, 32, 35	assaass 19317	. . . . 5 |- ( ( W e. AssAlg /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. V /\ z e. V ) ) -> ( ( x ( .s ` W ) y ) ( .r ` W ) z ) = ( x ( .s ` W ) ( y ( .r ` W ) z ) ) )
54	45, 46, 49, 51, 53	syl13anc 1328	. . . 4 |- ( ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. A /\ z e. A ) ) -> ( ( x ( .s ` W ) y ) ( .r ` W ) z ) = ( x ( .s ` W ) ( y ( .r ` W ) z ) ) )
55	12, 28, 52, 32, 35	assaassr 19318	. . . . 5 |- ( ( W e. AssAlg /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. V /\ z e. V ) ) -> ( y ( .r ` W ) ( x ( .s ` W ) z ) ) = ( x ( .s ` W ) ( y ( .r ` W ) z ) ) )
56	45, 46, 49, 51, 55	syl13anc 1328	. . . 4 |- ( ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. A /\ z e. A ) ) -> ( y ( .r ` W ) ( x ( .s ` W ) z ) ) = ( x ( .s ` W ) ( y ( .r ` W ) z ) ) )
57	27, 30, 31, 34, 37, 40, 42, 44, 54, 56	isassad 19323	. . 3 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> S e. AssAlg )
58	57	3ad2antl1 1223	. 2 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> S e. AssAlg )
59	23, 58	impbida 877	1 |- ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) -> ( S e. AssAlg <-> ( A e. (SubRing ` W ) /\ A e. L ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   1rcur 18501   Ringcrg 18547   CRingccrg 18548  SubRingcsubrg 18776   LModclmod 18863   LSubSpclss 18932  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-1st 7168  df-2nd 7169  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-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-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-mgp 18490  df-ur 18502  df-ring 18549  df-subrg 18778  df-lmod 18865  df-lss 18933  df-assa 19312

This theorem is referenced by:  mplassa  19454  ply1assa  19569