Theorem issubassa 14996

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

Step	Hyp	Ref	Expression
1		simpl1 1031	. . . . 5 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> W e. AssAlg )
2		assaring 14990	. . . . 5 |- ( W e. AssAlg -> W e. Ring )
3	1, 2	syl 14	. . . 4 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> W e. Ring )
4		issubassa.s	. . . . 5 |- S = ( W ↾s A )
5		assaring 14990	. . . . . 6 |- ( S e. AssAlg -> S e. Ring )
6	5	adantl 277	. . . . 5 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> S e. Ring )
7	4, 6	eqeltrrid 2326	. . . 4 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> ( W ↾s A ) e. Ring )
8		simpl3 1033	. . . . 5 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> A C_ V )
9		simpl2 1032	. . . . 5 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> .1. e. A )
10	8, 9	jca 306	. . . 4 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> ( A C_ V /\ .1. e. A ) )
11		issubassa.v	. . . . 5 |- V = ( Base ` W )
12		issubassa.o	. . . . 5 |- .1. = ( 1r ` W )
13	11, 12	issubrg 14512	. . . 4 |- ( A e. (SubRing ` W ) <-> ( ( W e. Ring /\ ( W ↾s A ) e. Ring ) /\ ( A C_ V /\ .1. e. A ) ) )
14	3, 7, 10, 13	syl21anbrc 1213	. . 3 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> A e. (SubRing ` W ) )
15		assalmod 14989	. . . . 5 |- ( S e. AssAlg -> S e. LMod )
16	15	adantl 277	. . . 4 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> S e. LMod )
17		assalmod 14989	. . . . 5 |- ( W e. AssAlg -> W e. LMod )
18		issubassa.l	. . . . . 6 |- L = ( LSubSp ` W )
19	4, 11, 18	islss3 14699	. . . . 5 |- ( W e. LMod -> ( A e. L <-> ( A C_ V /\ S e. LMod ) ) )
20	1, 17, 19	3syl 17	. . . 4 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> ( A e. L <-> ( A C_ V /\ S e. LMod ) ) )
21	8, 16, 20	mpbir2and 957	. . 3 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> A e. L )
22	14, 21	jca 306	. 2 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ S e. AssAlg ) -> ( A e. (SubRing ` W ) /\ A e. L ) )
23	4, 18	issubassa3 14995	. . 3 |- ( ( W e. AssAlg /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> S e. AssAlg )
24	23	3ad2antl1 1190	. 2 |- ( ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) /\ ( A e. (SubRing ` W ) /\ A e. L ) ) -> S e. AssAlg )
25	22, 24	impbida 604	1 |- ( ( W e. AssAlg /\ .1. e. A /\ A C_ V ) -> ( S e. AssAlg <-> ( A e. (SubRing ` W ) /\ A e. L ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   C_ wss 3220   ` cfv 5375  (class class class)co 6079   Basecbs 13335   ↾_s cress 13336   1rcur 14245   Ringcrg 14283  SubRingcsubrg 14508   LModclmod 14606   LSubSpclss 14672  AssAlgcasa 14979

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-1st 6368  df-2nd 6369  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-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-0g 13595  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-grp 13791  df-minusg 13792  df-sbg 13793  df-subg 13956  df-mgp 14201  df-ur 14246  df-ring 14285  df-subrg 14510  df-lmod 14608  df-lssm 14673  df-assa 14982

This theorem is referenced by: (None)