Theorem zlidlring 41928

Description: The zero (left) ideal of a non-unital ring is a unital ring (the zero ring). (Contributed by AV, 16-Feb-2020.)

Hypotheses

Ref	Expression
lidlabl.l	|- L = (LIdeal ` R )
lidlabl.i	|- I = ( R ↾s U )
zlidlring.b	|- B = ( Base ` R )
zlidlring.0	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
zlidlring	|- ( ( R e. Ring /\ U = { .0. } ) -> I e. Ring )

Proof of Theorem zlidlring

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

Step	Hyp	Ref	Expression
1		simpl 473	. . . . 5 |- ( ( R e. Ring /\ U = { .0. } ) -> R e. Ring )
2		lidlabl.l	. . . . . . . 8 |- L = (LIdeal ` R )
3		zlidlring.0	. . . . . . . 8 |- .0. = ( 0g ` R )
4	2, 3	lidl0 19219	. . . . . . 7 |- ( R e. Ring -> { .0. } e. L )
5	4	adantr 481	. . . . . 6 |- ( ( R e. Ring /\ U = { .0. } ) -> { .0. } e. L )
6		eleq1 2689	. . . . . . 7 |- ( U = { .0. } -> ( U e. L <-> { .0. } e. L ) )
7	6	adantl 482	. . . . . 6 |- ( ( R e. Ring /\ U = { .0. } ) -> ( U e. L <-> { .0. } e. L ) )
8	5, 7	mpbird 247	. . . . 5 |- ( ( R e. Ring /\ U = { .0. } ) -> U e. L )
9	1, 8	jca 554	. . . 4 |- ( ( R e. Ring /\ U = { .0. } ) -> ( R e. Ring /\ U e. L ) )
10		lidlabl.i	. . . . 5 |- I = ( R ↾s U )
11	2, 10	lidlrng 41927	. . . 4 |- ( ( R e. Ring /\ U e. L ) -> I e. Rng )
12	9, 11	syl 17	. . 3 |- ( ( R e. Ring /\ U = { .0. } ) -> I e. Rng )
13		eleq1 2689	. . . . . . 7 |- ( { .0. } = U -> ( { .0. } e. L <-> U e. L ) )
14	13	eqcoms 2630	. . . . . 6 |- ( U = { .0. } -> ( { .0. } e. L <-> U e. L ) )
15	14	adantl 482	. . . . 5 |- ( ( R e. Ring /\ U = { .0. } ) -> ( { .0. } e. L <-> U e. L ) )
16		id 22	. . . . . . . . . . . . 13 |- ( R e. Ring -> R e. Ring )
17		eqid 2622	. . . . . . . . . . . . . 14 |- ( Base ` R ) = ( Base ` R )
18	17, 3	ring0cl 18569	. . . . . . . . . . . . 13 |- ( R e. Ring -> .0. e. ( Base ` R ) )
19	16, 18	jca 554	. . . . . . . . . . . 12 |- ( R e. Ring -> ( R e. Ring /\ .0. e. ( Base ` R ) ) )
20		eqid 2622	. . . . . . . . . . . . . 14 |- ( .r ` R ) = ( .r ` R )
21	17, 20, 3	ringlz 18587	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ .0. e. ( Base ` R ) ) -> ( .0. ( .r ` R ) .0. ) = .0. )
22	21, 21	jca 554	. . . . . . . . . . . 12 |- ( ( R e. Ring /\ .0. e. ( Base ` R ) ) -> ( ( .0. ( .r ` R ) .0. ) = .0. /\ ( .0. ( .r ` R ) .0. ) = .0. ) )
23	19, 22	syl 17	. . . . . . . . . . 11 |- ( R e. Ring -> ( ( .0. ( .r ` R ) .0. ) = .0. /\ ( .0. ( .r ` R ) .0. ) = .0. ) )
24		fvex 6201	. . . . . . . . . . . . . 14 |- ( 0g ` R ) e. _V
25	3, 24	eqeltri 2697	. . . . . . . . . . . . 13 |- .0. e. _V
26	25	a1i 11	. . . . . . . . . . . 12 |- ( R e. Ring -> .0. e. _V )
27		oveq2 6658	. . . . . . . . . . . . . . 15 |- ( y = .0. -> ( .0. ( .r ` R ) y ) = ( .0. ( .r ` R ) .0. ) )
28		id 22	. . . . . . . . . . . . . . 15 |- ( y = .0. -> y = .0. )
29	27, 28	eqeq12d 2637	. . . . . . . . . . . . . 14 |- ( y = .0. -> ( ( .0. ( .r ` R ) y ) = y <-> ( .0. ( .r ` R ) .0. ) = .0. ) )
30		oveq1 6657	. . . . . . . . . . . . . . 15 |- ( y = .0. -> ( y ( .r ` R ) .0. ) = ( .0. ( .r ` R ) .0. ) )
31	30, 28	eqeq12d 2637	. . . . . . . . . . . . . 14 |- ( y = .0. -> ( ( y ( .r ` R ) .0. ) = y <-> ( .0. ( .r ` R ) .0. ) = .0. ) )
32	29, 31	anbi12d 747	. . . . . . . . . . . . 13 |- ( y = .0. -> ( ( ( .0. ( .r ` R ) y ) = y /\ ( y ( .r ` R ) .0. ) = y ) <-> ( ( .0. ( .r ` R ) .0. ) = .0. /\ ( .0. ( .r ` R ) .0. ) = .0. ) ) )
33	32	ralsng 4218	. . . . . . . . . . . 12 |- ( .0. e. _V -> ( A. y e. { .0. } ( ( .0. ( .r ` R ) y ) = y /\ ( y ( .r ` R ) .0. ) = y ) <-> ( ( .0. ( .r ` R ) .0. ) = .0. /\ ( .0. ( .r ` R ) .0. ) = .0. ) ) )
34	26, 33	syl 17	. . . . . . . . . . 11 |- ( R e. Ring -> ( A. y e. { .0. } ( ( .0. ( .r ` R ) y ) = y /\ ( y ( .r ` R ) .0. ) = y ) <-> ( ( .0. ( .r ` R ) .0. ) = .0. /\ ( .0. ( .r ` R ) .0. ) = .0. ) ) )
35	23, 34	mpbird 247	. . . . . . . . . 10 |- ( R e. Ring -> A. y e. { .0. } ( ( .0. ( .r ` R ) y ) = y /\ ( y ( .r ` R ) .0. ) = y ) )
36		oveq1 6657	. . . . . . . . . . . . . . 15 |- ( x = .0. -> ( x ( .r ` R ) y ) = ( .0. ( .r ` R ) y ) )
37	36	eqeq1d 2624	. . . . . . . . . . . . . 14 |- ( x = .0. -> ( ( x ( .r ` R ) y ) = y <-> ( .0. ( .r ` R ) y ) = y ) )
38		oveq2 6658	. . . . . . . . . . . . . . 15 |- ( x = .0. -> ( y ( .r ` R ) x ) = ( y ( .r ` R ) .0. ) )
39	38	eqeq1d 2624	. . . . . . . . . . . . . 14 |- ( x = .0. -> ( ( y ( .r ` R ) x ) = y <-> ( y ( .r ` R ) .0. ) = y ) )
40	37, 39	anbi12d 747	. . . . . . . . . . . . 13 |- ( x = .0. -> ( ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) <-> ( ( .0. ( .r ` R ) y ) = y /\ ( y ( .r ` R ) .0. ) = y ) ) )
41	40	ralbidv 2986	. . . . . . . . . . . 12 |- ( x = .0. -> ( A. y e. { .0. } ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) <-> A. y e. { .0. } ( ( .0. ( .r ` R ) y ) = y /\ ( y ( .r ` R ) .0. ) = y ) ) )
42	41	rexsng 4219	. . . . . . . . . . 11 |- ( .0. e. _V -> ( E. x e. { .0. } A. y e. { .0. } ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) <-> A. y e. { .0. } ( ( .0. ( .r ` R ) y ) = y /\ ( y ( .r ` R ) .0. ) = y ) ) )
43	26, 42	syl 17	. . . . . . . . . 10 |- ( R e. Ring -> ( E. x e. { .0. } A. y e. { .0. } ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) <-> A. y e. { .0. } ( ( .0. ( .r ` R ) y ) = y /\ ( y ( .r ` R ) .0. ) = y ) ) )
44	35, 43	mpbird 247	. . . . . . . . 9 |- ( R e. Ring -> E. x e. { .0. } A. y e. { .0. } ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) )
45	44	adantr 481	. . . . . . . 8 |- ( ( R e. Ring /\ U = { .0. } ) -> E. x e. { .0. } A. y e. { .0. } ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) )
46	45	adantr 481	. . . . . . 7 |- ( ( ( R e. Ring /\ U = { .0. } ) /\ U e. L ) -> E. x e. { .0. } A. y e. { .0. } ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) )
47		simpr 477	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ U = { .0. } ) /\ U e. L ) -> U e. L )
48	2, 10	lidlbas 41923	. . . . . . . . . 10 |- ( U e. L -> ( Base ` I ) = U )
49	47, 48	syl 17	. . . . . . . . 9 |- ( ( ( R e. Ring /\ U = { .0. } ) /\ U e. L ) -> ( Base ` I ) = U )
50		simpr 477	. . . . . . . . . 10 |- ( ( R e. Ring /\ U = { .0. } ) -> U = { .0. } )
51	50	adantr 481	. . . . . . . . 9 |- ( ( ( R e. Ring /\ U = { .0. } ) /\ U e. L ) -> U = { .0. } )
52	49, 51	eqtrd 2656	. . . . . . . 8 |- ( ( ( R e. Ring /\ U = { .0. } ) /\ U e. L ) -> ( Base ` I ) = { .0. } )
53	10, 20	ressmulr 16006	. . . . . . . . . . . . . 14 |- ( U e. L -> ( .r ` R ) = ( .r ` I ) )
54	53	eqcomd 2628	. . . . . . . . . . . . 13 |- ( U e. L -> ( .r ` I ) = ( .r ` R ) )
55	54	adantl 482	. . . . . . . . . . . 12 |- ( ( ( R e. Ring /\ U = { .0. } ) /\ U e. L ) -> ( .r ` I ) = ( .r ` R ) )
56	55	oveqd 6667	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ U = { .0. } ) /\ U e. L ) -> ( x ( .r ` I ) y ) = ( x ( .r ` R ) y ) )
57	56	eqeq1d 2624	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ U = { .0. } ) /\ U e. L ) -> ( ( x ( .r ` I ) y ) = y <-> ( x ( .r ` R ) y ) = y ) )
58	55	oveqd 6667	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ U = { .0. } ) /\ U e. L ) -> ( y ( .r ` I ) x ) = ( y ( .r ` R ) x ) )
59	58	eqeq1d 2624	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ U = { .0. } ) /\ U e. L ) -> ( ( y ( .r ` I ) x ) = y <-> ( y ( .r ` R ) x ) = y ) )
60	57, 59	anbi12d 747	. . . . . . . . 9 |- ( ( ( R e. Ring /\ U = { .0. } ) /\ U e. L ) -> ( ( ( x ( .r ` I ) y ) = y /\ ( y ( .r ` I ) x ) = y ) <-> ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) ) )
61	52, 60	raleqbidv 3152	. . . . . . . 8 |- ( ( ( R e. Ring /\ U = { .0. } ) /\ U e. L ) -> ( A. y e. ( Base ` I ) ( ( x ( .r ` I ) y ) = y /\ ( y ( .r ` I ) x ) = y ) <-> A. y e. { .0. } ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) ) )
62	52, 61	rexeqbidv 3153	. . . . . . 7 |- ( ( ( R e. Ring /\ U = { .0. } ) /\ U e. L ) -> ( E. x e. ( Base ` I ) A. y e. ( Base ` I ) ( ( x ( .r ` I ) y ) = y /\ ( y ( .r ` I ) x ) = y ) <-> E. x e. { .0. } A. y e. { .0. } ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) ) )
63	46, 62	mpbird 247	. . . . . 6 |- ( ( ( R e. Ring /\ U = { .0. } ) /\ U e. L ) -> E. x e. ( Base ` I ) A. y e. ( Base ` I ) ( ( x ( .r ` I ) y ) = y /\ ( y ( .r ` I ) x ) = y ) )
64	63	ex 450	. . . . 5 |- ( ( R e. Ring /\ U = { .0. } ) -> ( U e. L -> E. x e. ( Base ` I ) A. y e. ( Base ` I ) ( ( x ( .r ` I ) y ) = y /\ ( y ( .r ` I ) x ) = y ) ) )
65	15, 64	sylbid 230	. . . 4 |- ( ( R e. Ring /\ U = { .0. } ) -> ( { .0. } e. L -> E. x e. ( Base ` I ) A. y e. ( Base ` I ) ( ( x ( .r ` I ) y ) = y /\ ( y ( .r ` I ) x ) = y ) ) )
66	5, 65	mpd 15	. . 3 |- ( ( R e. Ring /\ U = { .0. } ) -> E. x e. ( Base ` I ) A. y e. ( Base ` I ) ( ( x ( .r ` I ) y ) = y /\ ( y ( .r ` I ) x ) = y ) )
67	12, 66	jca 554	. 2 |- ( ( R e. Ring /\ U = { .0. } ) -> ( I e. Rng /\ E. x e. ( Base ` I ) A. y e. ( Base ` I ) ( ( x ( .r ` I ) y ) = y /\ ( y ( .r ` I ) x ) = y ) ) )
68		eqid 2622	. . 3 |- ( Base ` I ) = ( Base ` I )
69		eqid 2622	. . 3 |- ( .r ` I ) = ( .r ` I )
70	68, 69	isringrng 41881	. 2 |- ( I e. Ring <-> ( I e. Rng /\ E. x e. ( Base ` I ) A. y e. ( Base ` I ) ( ( x ( .r ` I ) y ) = y /\ ( y ( .r ` I ) x ) = y ) ) )
71	67, 70	sylibr 224	1 |- ( ( R e. Ring /\ U = { .0. } ) -> I e. Ring )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   {csn 4177   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   .rcmulr 15942   0gc0g 16100   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:  uzlidlring  41929