Theorem lidldomn1 41921

Description: If a (left) ideal (which is not the zero ideal) of a domain has a multiplicative identity element, the identity element is the identity of the domain. (Contributed by AV, 17-Feb-2020.)

Hypotheses

Ref	Expression
lidldomn1.l	|- L = (LIdeal ` R )
lidldomn1.t	|- .x. = ( .r ` R )
lidldomn1.1	|- .1. = ( 1r ` R )
lidldomn1.0	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
lidldomn1	|- ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) -> ( A. x e. U ( ( I .x. x ) = x /\ ( x .x. I ) = x ) -> I = .1. ) )

Distinct variable groups:   x, I   x, U   x, .x.

Allowed substitution hints:   R( x)   .1. ( x)   L( x)   .0. ( x)

Proof of Theorem lidldomn1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		domnring 19296	. . . 4 |- ( R e. Domn -> R e. Ring )
2	1	3ad2ant1 1082	. . 3 |- ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) -> R e. Ring )
3		simp2l 1087	. . 3 |- ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) -> U e. L )
4		simp2r 1088	. . 3 |- ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) -> U =/= { .0. } )
5		lidldomn1.l	. . . 4 |- L = (LIdeal ` R )
6		lidldomn1.0	. . . 4 |- .0. = ( 0g ` R )
7	5, 6	lidlnz 19228	. . 3 |- ( ( R e. Ring /\ U e. L /\ U =/= { .0. } ) -> E. y e. U y =/= .0. )
8	2, 3, 4, 7	syl3anc 1326	. 2 |- ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) -> E. y e. U y =/= .0. )
9		oveq2 6658	. . . . . . . . . . 11 |- ( x = y -> ( I .x. x ) = ( I .x. y ) )
10		id 22	. . . . . . . . . . 11 |- ( x = y -> x = y )
11	9, 10	eqeq12d 2637	. . . . . . . . . 10 |- ( x = y -> ( ( I .x. x ) = x <-> ( I .x. y ) = y ) )
12		oveq1 6657	. . . . . . . . . . 11 |- ( x = y -> ( x .x. I ) = ( y .x. I ) )
13	12, 10	eqeq12d 2637	. . . . . . . . . 10 |- ( x = y -> ( ( x .x. I ) = x <-> ( y .x. I ) = y ) )
14	11, 13	anbi12d 747	. . . . . . . . 9 |- ( x = y -> ( ( ( I .x. x ) = x /\ ( x .x. I ) = x ) <-> ( ( I .x. y ) = y /\ ( y .x. I ) = y ) ) )
15	14	rspcva 3307	. . . . . . . 8 |- ( ( y e. U /\ A. x e. U ( ( I .x. x ) = x /\ ( x .x. I ) = x ) ) -> ( ( I .x. y ) = y /\ ( y .x. I ) = y ) )
16	2	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) -> R e. Ring )
17		eqid 2622	. . . . . . . . . . . . . . . . . . . . 21 |- ( Base ` R ) = ( Base ` R )
18	17, 5	lidlss 19210	. . . . . . . . . . . . . . . . . . . 20 |- ( U e. L -> U C_ ( Base ` R ) )
19	18	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( U e. L /\ U =/= { .0. } ) -> U C_ ( Base ` R ) )
20	19	3ad2ant2 1083	. . . . . . . . . . . . . . . . . 18 |- ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) -> U C_ ( Base ` R ) )
21	20	sseld 3602	. . . . . . . . . . . . . . . . 17 |- ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) -> ( y e. U -> y e. ( Base ` R ) ) )
22	21	com12 32	. . . . . . . . . . . . . . . 16 |- ( y e. U -> ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) -> y e. ( Base ` R ) ) )
23	22	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( y e. U /\ y =/= .0. ) -> ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) -> y e. ( Base ` R ) ) )
24	23	impcom 446	. . . . . . . . . . . . . 14 |- ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) -> y e. ( Base ` R ) )
25		lidldomn1.t	. . . . . . . . . . . . . . 15 |- .x. = ( .r ` R )
26		lidldomn1.1	. . . . . . . . . . . . . . 15 |- .1. = ( 1r ` R )
27	17, 25, 26	ringlidm 18571	. . . . . . . . . . . . . 14 |- ( ( R e. Ring /\ y e. ( Base ` R ) ) -> ( .1. .x. y ) = y )
28	16, 24, 27	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) -> ( .1. .x. y ) = y )
29		eqeq2 2633	. . . . . . . . . . . . . . . 16 |- ( y = ( .1. .x. y ) -> ( ( I .x. y ) = y <-> ( I .x. y ) = ( .1. .x. y ) ) )
30	29	eqcoms 2630	. . . . . . . . . . . . . . 15 |- ( ( .1. .x. y ) = y -> ( ( I .x. y ) = y <-> ( I .x. y ) = ( .1. .x. y ) ) )
31	30	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) /\ ( .1. .x. y ) = y ) -> ( ( I .x. y ) = y <-> ( I .x. y ) = ( .1. .x. y ) ) )
32		ringgrp 18552	. . . . . . . . . . . . . . . . . . . 20 |- ( R e. Ring -> R e. Grp )
33	1, 32	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( R e. Domn -> R e. Grp )
34	33	3ad2ant1 1082	. . . . . . . . . . . . . . . . . 18 |- ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) -> R e. Grp )
35	34	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) -> R e. Grp )
36	19	sseld 3602	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( U e. L /\ U =/= { .0. } ) -> ( I e. U -> I e. ( Base ` R ) ) )
37	36	a1i 11	. . . . . . . . . . . . . . . . . . . 20 |- ( R e. Domn -> ( ( U e. L /\ U =/= { .0. } ) -> ( I e. U -> I e. ( Base ` R ) ) ) )
38	37	3imp 1256	. . . . . . . . . . . . . . . . . . 19 |- ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) -> I e. ( Base ` R ) )
39	38	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) -> I e. ( Base ` R ) )
40	17, 25	ringcl 18561	. . . . . . . . . . . . . . . . . 18 |- ( ( R e. Ring /\ I e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( I .x. y ) e. ( Base ` R ) )
41	16, 39, 24, 40	syl3anc 1326	. . . . . . . . . . . . . . . . 17 |- ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) -> ( I .x. y ) e. ( Base ` R ) )
42	17, 26	ringidcl 18568	. . . . . . . . . . . . . . . . . . . . 21 |- ( R e. Ring -> .1. e. ( Base ` R ) )
43	1, 42	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( R e. Domn -> .1. e. ( Base ` R ) )
44	43	3ad2ant1 1082	. . . . . . . . . . . . . . . . . . 19 |- ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) -> .1. e. ( Base ` R ) )
45	44	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) -> .1. e. ( Base ` R ) )
46	17, 25	ringcl 18561	. . . . . . . . . . . . . . . . . 18 |- ( ( R e. Ring /\ .1. e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( .1. .x. y ) e. ( Base ` R ) )
47	16, 45, 24, 46	syl3anc 1326	. . . . . . . . . . . . . . . . 17 |- ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) -> ( .1. .x. y ) e. ( Base ` R ) )
48		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- ( -g ` R ) = ( -g ` R )
49	17, 6, 48	grpsubeq0 17501	. . . . . . . . . . . . . . . . 17 |- ( ( R e. Grp /\ ( I .x. y ) e. ( Base ` R ) /\ ( .1. .x. y ) e. ( Base ` R ) ) -> ( ( ( I .x. y ) ( -g ` R ) ( .1. .x. y ) ) = .0. <-> ( I .x. y ) = ( .1. .x. y ) ) )
50	35, 41, 47, 49	syl3anc 1326	. . . . . . . . . . . . . . . 16 |- ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) -> ( ( ( I .x. y ) ( -g ` R ) ( .1. .x. y ) ) = .0. <-> ( I .x. y ) = ( .1. .x. y ) ) )
51	17, 25, 48, 16, 39, 45, 24	rngsubdir 18600	. . . . . . . . . . . . . . . . . 18 |- ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) -> ( ( I ( -g ` R ) .1. ) .x. y ) = ( ( I .x. y ) ( -g ` R ) ( .1. .x. y ) ) )
52	51	eqeq1d 2624	. . . . . . . . . . . . . . . . 17 |- ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) -> ( ( ( I ( -g ` R ) .1. ) .x. y ) = .0. <-> ( ( I .x. y ) ( -g ` R ) ( .1. .x. y ) ) = .0. ) )
53		simpl1 1064	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) -> R e. Domn )
54	34, 38, 44	3jca 1242	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) -> ( R e. Grp /\ I e. ( Base ` R ) /\ .1. e. ( Base ` R ) ) )
55	54	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) -> ( R e. Grp /\ I e. ( Base ` R ) /\ .1. e. ( Base ` R ) ) )
56	17, 48	grpsubcl 17495	. . . . . . . . . . . . . . . . . . . 20 |- ( ( R e. Grp /\ I e. ( Base ` R ) /\ .1. e. ( Base ` R ) ) -> ( I ( -g ` R ) .1. ) e. ( Base ` R ) )
57	55, 56	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) -> ( I ( -g ` R ) .1. ) e. ( Base ` R ) )
58	17, 25, 6	domneq0 19297	. . . . . . . . . . . . . . . . . . 19 |- ( ( R e. Domn /\ ( I ( -g ` R ) .1. ) e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( ( ( I ( -g ` R ) .1. ) .x. y ) = .0. <-> ( ( I ( -g ` R ) .1. ) = .0. \/ y = .0. ) ) )
59	53, 57, 24, 58	syl3anc 1326	. . . . . . . . . . . . . . . . . 18 |- ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) -> ( ( ( I ( -g ` R ) .1. ) .x. y ) = .0. <-> ( ( I ( -g ` R ) .1. ) = .0. \/ y = .0. ) ) )
60	17, 6, 48	grpsubeq0 17501	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( R e. Grp /\ I e. ( Base ` R ) /\ .1. e. ( Base ` R ) ) -> ( ( I ( -g ` R ) .1. ) = .0. <-> I = .1. ) )
61	55, 60	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) -> ( ( I ( -g ` R ) .1. ) = .0. <-> I = .1. ) )
62	61	biimpd 219	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) -> ( ( I ( -g ` R ) .1. ) = .0. -> I = .1. ) )
63		eqneqall 2805	. . . . . . . . . . . . . . . . . . . . . 22 |- ( y = .0. -> ( y =/= .0. -> I = .1. ) )
64	63	com12 32	. . . . . . . . . . . . . . . . . . . . 21 |- ( y =/= .0. -> ( y = .0. -> I = .1. ) )
65	64	adantl 482	. . . . . . . . . . . . . . . . . . . 20 |- ( ( y e. U /\ y =/= .0. ) -> ( y = .0. -> I = .1. ) )
66	65	adantl 482	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) -> ( y = .0. -> I = .1. ) )
67	62, 66	jaod 395	. . . . . . . . . . . . . . . . . 18 |- ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) -> ( ( ( I ( -g ` R ) .1. ) = .0. \/ y = .0. ) -> I = .1. ) )
68	59, 67	sylbid 230	. . . . . . . . . . . . . . . . 17 |- ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) -> ( ( ( I ( -g ` R ) .1. ) .x. y ) = .0. -> I = .1. ) )
69	52, 68	sylbird 250	. . . . . . . . . . . . . . . 16 |- ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) -> ( ( ( I .x. y ) ( -g ` R ) ( .1. .x. y ) ) = .0. -> I = .1. ) )
70	50, 69	sylbird 250	. . . . . . . . . . . . . . 15 |- ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) -> ( ( I .x. y ) = ( .1. .x. y ) -> I = .1. ) )
71	70	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) /\ ( .1. .x. y ) = y ) -> ( ( I .x. y ) = ( .1. .x. y ) -> I = .1. ) )
72	31, 71	sylbid 230	. . . . . . . . . . . . 13 |- ( ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) /\ ( .1. .x. y ) = y ) -> ( ( I .x. y ) = y -> I = .1. ) )
73	28, 72	mpdan 702	. . . . . . . . . . . 12 |- ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ ( y e. U /\ y =/= .0. ) ) -> ( ( I .x. y ) = y -> I = .1. ) )
74	73	ex 450	. . . . . . . . . . 11 |- ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) -> ( ( y e. U /\ y =/= .0. ) -> ( ( I .x. y ) = y -> I = .1. ) ) )
75	74	com13 88	. . . . . . . . . 10 |- ( ( I .x. y ) = y -> ( ( y e. U /\ y =/= .0. ) -> ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) -> I = .1. ) ) )
76	75	expd 452	. . . . . . . . 9 |- ( ( I .x. y ) = y -> ( y e. U -> ( y =/= .0. -> ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) -> I = .1. ) ) ) )
77	76	adantr 481	. . . . . . . 8 |- ( ( ( I .x. y ) = y /\ ( y .x. I ) = y ) -> ( y e. U -> ( y =/= .0. -> ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) -> I = .1. ) ) ) )
78	15, 77	syl 17	. . . . . . 7 |- ( ( y e. U /\ A. x e. U ( ( I .x. x ) = x /\ ( x .x. I ) = x ) ) -> ( y e. U -> ( y =/= .0. -> ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) -> I = .1. ) ) ) )
79	78	ex 450	. . . . . 6 |- ( y e. U -> ( A. x e. U ( ( I .x. x ) = x /\ ( x .x. I ) = x ) -> ( y e. U -> ( y =/= .0. -> ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) -> I = .1. ) ) ) ) )
80	79	pm2.43b 55	. . . . 5 |- ( A. x e. U ( ( I .x. x ) = x /\ ( x .x. I ) = x ) -> ( y e. U -> ( y =/= .0. -> ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) -> I = .1. ) ) ) )
81	80	com14 96	. . . 4 |- ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) -> ( y e. U -> ( y =/= .0. -> ( A. x e. U ( ( I .x. x ) = x /\ ( x .x. I ) = x ) -> I = .1. ) ) ) )
82	81	imp 445	. . 3 |- ( ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) /\ y e. U ) -> ( y =/= .0. -> ( A. x e. U ( ( I .x. x ) = x /\ ( x .x. I ) = x ) -> I = .1. ) ) )
83	82	rexlimdva 3031	. 2 |- ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) -> ( E. y e. U y =/= .0. -> ( A. x e. U ( ( I .x. x ) = x /\ ( x .x. I ) = x ) -> I = .1. ) ) )
84	8, 83	mpd 15	1 |- ( ( R e. Domn /\ ( U e. L /\ U =/= { .0. } ) /\ I e. U ) -> ( A. x e. U ( ( I .x. x ) = x /\ ( x .x. I ) = x ) -> I = .1. ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   {csn 4177   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .rcmulr 15942   0gc0g 16100   Grpcgrp 17422   -gcsg 17424   1rcur 18501   Ringcrg 18547  LIdealclidl 19170  Domncdomn 19280

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-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-nzr 19258  df-domn 19284

This theorem is referenced by:  uzlidlring  41929