Theorem lidlrng 41927

Description: A (left) ideal of a ring is a non-unital ring. (Contributed by AV, 17-Feb-2020.)

Hypotheses

Ref	Expression
lidlabl.l	|- L = (LIdeal ` R )
lidlabl.i	|- I = ( R ↾s U )

Assertion

Ref	Expression
lidlrng	|- ( ( R e. Ring /\ U e. L ) -> I e. Rng )

Proof of Theorem lidlrng

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lidlabl.l	. . 3 |- L = (LIdeal ` R )
2		lidlabl.i	. . 3 |- I = ( R ↾s U )
3	1, 2	lidlabl 41924	. 2 |- ( ( R e. Ring /\ U e. L ) -> I e. Abel )
4	1, 2	lidlmsgrp 41926	. 2 |- ( ( R e. Ring /\ U e. L ) -> (mulGrp ` I ) e. SGrp )
5		simpll 790	. . . . . 6 |- ( ( ( R e. Ring /\ U e. L ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> R e. Ring )
6	1, 2	lidlssbas 41922	. . . . . . . . . 10 |- ( U e. L -> ( Base ` I ) C_ ( Base ` R ) )
7	6	sseld 3602	. . . . . . . . 9 |- ( U e. L -> ( a e. ( Base ` I ) -> a e. ( Base ` R ) ) )
8	6	sseld 3602	. . . . . . . . 9 |- ( U e. L -> ( b e. ( Base ` I ) -> b e. ( Base ` R ) ) )
9	6	sseld 3602	. . . . . . . . 9 |- ( U e. L -> ( c e. ( Base ` I ) -> c e. ( Base ` R ) ) )
10	7, 8, 9	3anim123d 1406	. . . . . . . 8 |- ( U e. L -> ( ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) -> ( a e. ( Base ` R ) /\ b e. ( Base ` R ) /\ c e. ( Base ` R ) ) ) )
11	10	adantl 482	. . . . . . 7 |- ( ( R e. Ring /\ U e. L ) -> ( ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) -> ( a e. ( Base ` R ) /\ b e. ( Base ` R ) /\ c e. ( Base ` R ) ) ) )
12	11	imp 445	. . . . . 6 |- ( ( ( R e. Ring /\ U e. L ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( a e. ( Base ` R ) /\ b e. ( Base ` R ) /\ c e. ( Base ` R ) ) )
13		eqid 2622	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
14		eqid 2622	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
15		eqid 2622	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
16	13, 14, 15	ringdi 18566	. . . . . 6 |- ( ( R e. Ring /\ ( a e. ( Base ` R ) /\ b e. ( Base ` R ) /\ c e. ( Base ` R ) ) ) -> ( a ( .r ` R ) ( b ( +g ` R ) c ) ) = ( ( a ( .r ` R ) b ) ( +g ` R ) ( a ( .r ` R ) c ) ) )
17	5, 12, 16	syl2anc 693	. . . . 5 |- ( ( ( R e. Ring /\ U e. L ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( a ( .r ` R ) ( b ( +g ` R ) c ) ) = ( ( a ( .r ` R ) b ) ( +g ` R ) ( a ( .r ` R ) c ) ) )
18	13, 14, 15	ringdir 18567	. . . . . 6 |- ( ( R e. Ring /\ ( a e. ( Base ` R ) /\ b e. ( Base ` R ) /\ c e. ( Base ` R ) ) ) -> ( ( a ( +g ` R ) b ) ( .r ` R ) c ) = ( ( a ( .r ` R ) c ) ( +g ` R ) ( b ( .r ` R ) c ) ) )
19	5, 12, 18	syl2anc 693	. . . . 5 |- ( ( ( R e. Ring /\ U e. L ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( ( a ( +g ` R ) b ) ( .r ` R ) c ) = ( ( a ( .r ` R ) c ) ( +g ` R ) ( b ( .r ` R ) c ) ) )
20	17, 19	jca 554	. . . 4 |- ( ( ( R e. Ring /\ U e. L ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( ( a ( .r ` R ) ( b ( +g ` R ) c ) ) = ( ( a ( .r ` R ) b ) ( +g ` R ) ( a ( .r ` R ) c ) ) /\ ( ( a ( +g ` R ) b ) ( .r ` R ) c ) = ( ( a ( .r ` R ) c ) ( +g ` R ) ( b ( .r ` R ) c ) ) ) )
21	20	ralrimivvva 2972	. . 3 |- ( ( R e. Ring /\ U e. L ) -> A. a e. ( Base ` I ) A. b e. ( Base ` I ) A. c e. ( Base ` I ) ( ( a ( .r ` R ) ( b ( +g ` R ) c ) ) = ( ( a ( .r ` R ) b ) ( +g ` R ) ( a ( .r ` R ) c ) ) /\ ( ( a ( +g ` R ) b ) ( .r ` R ) c ) = ( ( a ( .r ` R ) c ) ( +g ` R ) ( b ( .r ` R ) c ) ) ) )
22	2, 15	ressmulr 16006	. . . . . . . . . 10 |- ( U e. L -> ( .r ` R ) = ( .r ` I ) )
23	22	eqcomd 2628	. . . . . . . . 9 |- ( U e. L -> ( .r ` I ) = ( .r ` R ) )
24		eqidd 2623	. . . . . . . . 9 |- ( U e. L -> a = a )
25	2, 14	ressplusg 15993	. . . . . . . . . . 11 |- ( U e. L -> ( +g ` R ) = ( +g ` I ) )
26	25	eqcomd 2628	. . . . . . . . . 10 |- ( U e. L -> ( +g ` I ) = ( +g ` R ) )
27	26	oveqd 6667	. . . . . . . . 9 |- ( U e. L -> ( b ( +g ` I ) c ) = ( b ( +g ` R ) c ) )
28	23, 24, 27	oveq123d 6671	. . . . . . . 8 |- ( U e. L -> ( a ( .r ` I ) ( b ( +g ` I ) c ) ) = ( a ( .r ` R ) ( b ( +g ` R ) c ) ) )
29	23	oveqd 6667	. . . . . . . . 9 |- ( U e. L -> ( a ( .r ` I ) b ) = ( a ( .r ` R ) b ) )
30	23	oveqd 6667	. . . . . . . . 9 |- ( U e. L -> ( a ( .r ` I ) c ) = ( a ( .r ` R ) c ) )
31	26, 29, 30	oveq123d 6671	. . . . . . . 8 |- ( U e. L -> ( ( a ( .r ` I ) b ) ( +g ` I ) ( a ( .r ` I ) c ) ) = ( ( a ( .r ` R ) b ) ( +g ` R ) ( a ( .r ` R ) c ) ) )
32	28, 31	eqeq12d 2637	. . . . . . 7 |- ( U e. L -> ( ( a ( .r ` I ) ( b ( +g ` I ) c ) ) = ( ( a ( .r ` I ) b ) ( +g ` I ) ( a ( .r ` I ) c ) ) <-> ( a ( .r ` R ) ( b ( +g ` R ) c ) ) = ( ( a ( .r ` R ) b ) ( +g ` R ) ( a ( .r ` R ) c ) ) ) )
33	26	oveqd 6667	. . . . . . . . 9 |- ( U e. L -> ( a ( +g ` I ) b ) = ( a ( +g ` R ) b ) )
34		eqidd 2623	. . . . . . . . 9 |- ( U e. L -> c = c )
35	23, 33, 34	oveq123d 6671	. . . . . . . 8 |- ( U e. L -> ( ( a ( +g ` I ) b ) ( .r ` I ) c ) = ( ( a ( +g ` R ) b ) ( .r ` R ) c ) )
36	23	oveqd 6667	. . . . . . . . 9 |- ( U e. L -> ( b ( .r ` I ) c ) = ( b ( .r ` R ) c ) )
37	26, 30, 36	oveq123d 6671	. . . . . . . 8 |- ( U e. L -> ( ( a ( .r ` I ) c ) ( +g ` I ) ( b ( .r ` I ) c ) ) = ( ( a ( .r ` R ) c ) ( +g ` R ) ( b ( .r ` R ) c ) ) )
38	35, 37	eqeq12d 2637	. . . . . . 7 |- ( U e. L -> ( ( ( a ( +g ` I ) b ) ( .r ` I ) c ) = ( ( a ( .r ` I ) c ) ( +g ` I ) ( b ( .r ` I ) c ) ) <-> ( ( a ( +g ` R ) b ) ( .r ` R ) c ) = ( ( a ( .r ` R ) c ) ( +g ` R ) ( b ( .r ` R ) c ) ) ) )
39	32, 38	anbi12d 747	. . . . . 6 |- ( U e. L -> ( ( ( a ( .r ` I ) ( b ( +g ` I ) c ) ) = ( ( a ( .r ` I ) b ) ( +g ` I ) ( a ( .r ` I ) c ) ) /\ ( ( a ( +g ` I ) b ) ( .r ` I ) c ) = ( ( a ( .r ` I ) c ) ( +g ` I ) ( b ( .r ` I ) c ) ) ) <-> ( ( a ( .r ` R ) ( b ( +g ` R ) c ) ) = ( ( a ( .r ` R ) b ) ( +g ` R ) ( a ( .r ` R ) c ) ) /\ ( ( a ( +g ` R ) b ) ( .r ` R ) c ) = ( ( a ( .r ` R ) c ) ( +g ` R ) ( b ( .r ` R ) c ) ) ) ) )
40	39	adantl 482	. . . . 5 |- ( ( R e. Ring /\ U e. L ) -> ( ( ( a ( .r ` I ) ( b ( +g ` I ) c ) ) = ( ( a ( .r ` I ) b ) ( +g ` I ) ( a ( .r ` I ) c ) ) /\ ( ( a ( +g ` I ) b ) ( .r ` I ) c ) = ( ( a ( .r ` I ) c ) ( +g ` I ) ( b ( .r ` I ) c ) ) ) <-> ( ( a ( .r ` R ) ( b ( +g ` R ) c ) ) = ( ( a ( .r ` R ) b ) ( +g ` R ) ( a ( .r ` R ) c ) ) /\ ( ( a ( +g ` R ) b ) ( .r ` R ) c ) = ( ( a ( .r ` R ) c ) ( +g ` R ) ( b ( .r ` R ) c ) ) ) ) )
41	40	ralbidv 2986	. . . 4 |- ( ( R e. Ring /\ U e. L ) -> ( A. c e. ( Base ` I ) ( ( a ( .r ` I ) ( b ( +g ` I ) c ) ) = ( ( a ( .r ` I ) b ) ( +g ` I ) ( a ( .r ` I ) c ) ) /\ ( ( a ( +g ` I ) b ) ( .r ` I ) c ) = ( ( a ( .r ` I ) c ) ( +g ` I ) ( b ( .r ` I ) c ) ) ) <-> A. c e. ( Base ` I ) ( ( a ( .r ` R ) ( b ( +g ` R ) c ) ) = ( ( a ( .r ` R ) b ) ( +g ` R ) ( a ( .r ` R ) c ) ) /\ ( ( a ( +g ` R ) b ) ( .r ` R ) c ) = ( ( a ( .r ` R ) c ) ( +g ` R ) ( b ( .r ` R ) c ) ) ) ) )
42	41	2ralbidv 2989	. . 3 |- ( ( R e. Ring /\ U e. L ) -> ( A. a e. ( Base ` I ) A. b e. ( Base ` I ) A. c e. ( Base ` I ) ( ( a ( .r ` I ) ( b ( +g ` I ) c ) ) = ( ( a ( .r ` I ) b ) ( +g ` I ) ( a ( .r ` I ) c ) ) /\ ( ( a ( +g ` I ) b ) ( .r ` I ) c ) = ( ( a ( .r ` I ) c ) ( +g ` I ) ( b ( .r ` I ) c ) ) ) <-> A. a e. ( Base ` I ) A. b e. ( Base ` I ) A. c e. ( Base ` I ) ( ( a ( .r ` R ) ( b ( +g ` R ) c ) ) = ( ( a ( .r ` R ) b ) ( +g ` R ) ( a ( .r ` R ) c ) ) /\ ( ( a ( +g ` R ) b ) ( .r ` R ) c ) = ( ( a ( .r ` R ) c ) ( +g ` R ) ( b ( .r ` R ) c ) ) ) ) )
43	21, 42	mpbird 247	. 2 |- ( ( R e. Ring /\ U e. L ) -> A. a e. ( Base ` I ) A. b e. ( Base ` I ) A. c e. ( Base ` I ) ( ( a ( .r ` I ) ( b ( +g ` I ) c ) ) = ( ( a ( .r ` I ) b ) ( +g ` I ) ( a ( .r ` I ) c ) ) /\ ( ( a ( +g ` I ) b ) ( .r ` I ) c ) = ( ( a ( .r ` I ) c ) ( +g ` I ) ( b ( .r ` I ) c ) ) ) )
44		eqid 2622	. . 3 |- ( Base ` I ) = ( Base ` I )
45		eqid 2622	. . 3 |- (mulGrp ` I ) = (mulGrp ` I )
46		eqid 2622	. . 3 |- ( +g ` I ) = ( +g ` I )
47		eqid 2622	. . 3 |- ( .r ` I ) = ( .r ` I )
48	44, 45, 46, 47	isrng 41876	. 2 |- ( I e. Rng <-> ( I e. Abel /\ (mulGrp ` I ) e. SGrp /\ A. a e. ( Base ` I ) A. b e. ( Base ` I ) A. c e. ( Base ` I ) ( ( a ( .r ` I ) ( b ( +g ` I ) c ) ) = ( ( a ( .r ` I ) b ) ( +g ` I ) ( a ( .r ` I ) c ) ) /\ ( ( a ( +g ` I ) b ) ( .r ` I ) c ) = ( ( a ( .r ` I ) c ) ( +g ` I ) ( b ( .r ` I ) c ) ) ) ) )
49	3, 4, 43, 48	syl3anbrc 1246	1 |- ( ( R e. Ring /\ U e. L ) -> I e. Rng )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   .rcmulr 15942  SGrpcsgrp 17283   Abelcabl 18194  mulGrpcmgp 18489   Ringcrg 18547  LIdealclidl 19170  Rngcrng 41874

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-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-minusg 17426  df-sbg 17427  df-subg 17591  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-subrg 18778  df-lmod 18865  df-lss 18933  df-sra 19172  df-rgmod 19173  df-lidl 19174  df-rng0 41875

This theorem is referenced by:  zlidlring  41928  uzlidlring  41929  2zrng  41935