Theorem lmod1 32803

Description: The (smallest) structure representing a zero module over an arbitrary ring. (Contributed by AV, 29-Apr-2019.)

Hypothesis

Ref	Expression
lmod1.m	|- M = ( { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , ( x e. ( Base ` R ) , y e. { I } |-> y ) >. } )

Assertion

Ref	Expression
lmod1	|- ( ( I e. V /\ R e. Ring ) -> M e. LMod )

Distinct variable groups:   x, I, y   x, R, y   x, V, y   x, M, y

Proof of Theorem lmod1

Dummy variables r q w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2441	. . . . 5 |- { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } = { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. }
2	1	grp1 16011	. . . 4 |- ( I e. V -> { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } e. Grp )
3		fvex 5862	. . . . . . 7 |- ( Base ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) e. _V
4		lmod1.m	. . . . . . . . 9 |- M = ( { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , ( x e. ( Base ` R ) , y e. { I } |-> y ) >. } )
5		snex 4674	. . . . . . . . . . . . 13 |- { I } e. _V
6	1	grpbase 14609	. . . . . . . . . . . . 13 |- ( { I } e. _V -> { I } = ( Base ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) )
7	5, 6	ax-mp 5	. . . . . . . . . . . 12 |- { I } = ( Base ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } )
8	7	opeq2i 4202	. . . . . . . . . . 11 |- <. ( Base ` ndx ) , { I } >. = <. ( Base ` ndx ) , ( Base ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) >.
9		tpeq1 4099	. . . . . . . . . . 11 |- ( <. ( Base ` ndx ) , { I } >. = <. ( Base ` ndx ) , ( Base ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) >. -> { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. , <. (Scalar ` ndx ) , R >. } = { <. ( Base ` ndx ) , ( Base ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. , <. (Scalar ` ndx ) , R >. } )
10	8, 9	ax-mp 5	. . . . . . . . . 10 |- { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. , <. (Scalar ` ndx ) , R >. } = { <. ( Base ` ndx ) , ( Base ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. , <. (Scalar ` ndx ) , R >. }
11	10	uneq1i 3636	. . . . . . . . 9 |- ( { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , ( x e. ( Base ` R ) , y e. { I } |-> y ) >. } ) = ( { <. ( Base ` ndx ) , ( Base ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , ( x e. ( Base ` R ) , y e. { I } |-> y ) >. } )
12	4, 11	eqtri 2470	. . . . . . . 8 |- M = ( { <. ( Base ` ndx ) , ( Base ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , ( x e. ( Base ` R ) , y e. { I } |-> y ) >. } )
13	12	lmodbase 14634	. . . . . . 7 |- ( ( Base ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) e. _V -> ( Base ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) = ( Base ` M ) )
14	3, 13	ax-mp 5	. . . . . 6 |- ( Base ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) = ( Base ` M )
15	14	eqcomi 2454	. . . . 5 |- ( Base ` M ) = ( Base ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } )
16		fvex 5862	. . . . . . 7 |- ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) e. _V
17		snex 4674	. . . . . . . . . . . . 13 |- { <. <. I , I >. , I >. } e. _V
18	1	grpplusg 14610	. . . . . . . . . . . . 13 |- ( { <. <. I , I >. , I >. } e. _V -> { <. <. I , I >. , I >. } = ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) )
19	17, 18	ax-mp 5	. . . . . . . . . . . 12 |- { <. <. I , I >. , I >. } = ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } )
20	19	opeq2i 4202	. . . . . . . . . . 11 |- <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. = <. ( +g ` ndx ) , ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) >.
21		tpeq2 4100	. . . . . . . . . . 11 |- ( <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. = <. ( +g ` ndx ) , ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) >. -> { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. , <. (Scalar ` ndx ) , R >. } = { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) >. , <. (Scalar ` ndx ) , R >. } )
22	20, 21	ax-mp 5	. . . . . . . . . 10 |- { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. , <. (Scalar ` ndx ) , R >. } = { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) >. , <. (Scalar ` ndx ) , R >. }
23	22	uneq1i 3636	. . . . . . . . 9 |- ( { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , ( x e. ( Base ` R ) , y e. { I } |-> y ) >. } ) = ( { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , ( x e. ( Base ` R ) , y e. { I } |-> y ) >. } )
24	4, 23	eqtri 2470	. . . . . . . 8 |- M = ( { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , ( x e. ( Base ` R ) , y e. { I } |-> y ) >. } )
25	24	lmodplusg 14635	. . . . . . 7 |- ( ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) e. _V -> ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) = ( +g ` M ) )
26	16, 25	ax-mp 5	. . . . . 6 |- ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) = ( +g ` M )
27	26	eqcomi 2454	. . . . 5 |- ( +g ` M ) = ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } )
28	15, 27	grpprop 15938	. . . 4 |- ( M e. Grp <-> { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } e. Grp )
29	2, 28	sylibr 212	. . 3 |- ( I e. V -> M e. Grp )
30	29	adantr 465	. 2 |- ( ( I e. V /\ R e. Ring ) -> M e. Grp )
31	4	lmodsca 14636	. . . . 5 |- ( R e. Ring -> R = (Scalar ` M ) )
32	31	eqcomd 2449	. . . 4 |- ( R e. Ring -> (Scalar ` M ) = R )
33	32	adantl 466	. . 3 |- ( ( I e. V /\ R e. Ring ) -> (Scalar ` M ) = R )
34		simpr 461	. . 3 |- ( ( I e. V /\ R e. Ring ) -> R e. Ring )
35	33, 34	eqeltrd 2529	. 2 |- ( ( I e. V /\ R e. Ring ) -> (Scalar ` M ) e. Ring )
36	33	fveq2d 5856	. . . . . . 7 |- ( ( I e. V /\ R e. Ring ) -> ( Base ` (Scalar ` M ) ) = ( Base ` R ) )
37	36	eleq2d 2511	. . . . . 6 |- ( ( I e. V /\ R e. Ring ) -> ( q e. ( Base ` (Scalar ` M ) ) <-> q e. ( Base ` R ) ) )
38	36	eleq2d 2511	. . . . . 6 |- ( ( I e. V /\ R e. Ring ) -> ( r e. ( Base ` (Scalar ` M ) ) <-> r e. ( Base ` R ) ) )
39	37, 38	anbi12d 710	. . . . 5 |- ( ( I e. V /\ R e. Ring ) -> ( ( q e. ( Base ` (Scalar ` M ) ) /\ r e. ( Base ` (Scalar ` M ) ) ) <-> ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) )
40		simpll 753	. . . . . . . . . 10 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> I e. V )
41		simplr 754	. . . . . . . . . 10 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> R e. Ring )
42		simprr 756	. . . . . . . . . 10 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> r e. ( Base ` R ) )
43	40, 41, 42	3jca 1175	. . . . . . . . 9 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( I e. V /\ R e. Ring /\ r e. ( Base ` R ) ) )
44	4	lmod1lem1 32798	. . . . . . . . 9 |- ( ( I e. V /\ R e. Ring /\ r e. ( Base ` R ) ) -> ( r ( .s ` M ) I ) e. { I } )
45	43, 44	syl 16	. . . . . . . 8 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( r ( .s ` M ) I ) e. { I } )
46	4	lmod1lem2 32799	. . . . . . . . 9 |- ( ( I e. V /\ R e. Ring /\ r e. ( Base ` R ) ) -> ( r ( .s ` M ) ( I ( +g ` M ) I ) ) = ( ( r ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) )
47	43, 46	syl 16	. . . . . . . 8 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( r ( .s ` M ) ( I ( +g ` M ) I ) ) = ( ( r ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) )
48	4	lmod1lem3 32800	. . . . . . . 8 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( ( q ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) )
49	45, 47, 48	3jca 1175	. . . . . . 7 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( ( r ( .s ` M ) I ) e. { I } /\ ( r ( .s ` M ) ( I ( +g ` M ) I ) ) = ( ( r ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( ( q ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) ) )
50	4	lmod1lem4 32801	. . . . . . 7 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( q ( .s ` M ) ( r ( .s ` M ) I ) ) )
51	4	lmod1lem5 32802	. . . . . . . 8 |- ( ( I e. V /\ R e. Ring ) -> ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) I ) = I )
52	51	adantr 465	. . . . . . 7 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) I ) = I )
53	49, 50, 52	jca32 535	. . . . . 6 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( ( ( r ( .s ` M ) I ) e. { I } /\ ( r ( .s ` M ) ( I ( +g ` M ) I ) ) = ( ( r ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( ( q ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) ) /\ ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( q ( .s ` M ) ( r ( .s ` M ) I ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) I ) = I ) ) )
54	53	ex 434	. . . . 5 |- ( ( I e. V /\ R e. Ring ) -> ( ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) -> ( ( ( r ( .s ` M ) I ) e. { I } /\ ( r ( .s ` M ) ( I ( +g ` M ) I ) ) = ( ( r ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( ( q ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) ) /\ ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( q ( .s ` M ) ( r ( .s ` M ) I ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) I ) = I ) ) ) )
55	39, 54	sylbid 215	. . . 4 |- ( ( I e. V /\ R e. Ring ) -> ( ( q e. ( Base ` (Scalar ` M ) ) /\ r e. ( Base ` (Scalar ` M ) ) ) -> ( ( ( r ( .s ` M ) I ) e. { I } /\ ( r ( .s ` M ) ( I ( +g ` M ) I ) ) = ( ( r ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( ( q ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) ) /\ ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( q ( .s ` M ) ( r ( .s ` M ) I ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) I ) = I ) ) ) )
56	55	ralrimivv 2861	. . 3 |- ( ( I e. V /\ R e. Ring ) -> A. q e. ( Base ` (Scalar ` M ) ) A. r e. ( Base ` (Scalar ` M ) ) ( ( ( r ( .s ` M ) I ) e. { I } /\ ( r ( .s ` M ) ( I ( +g ` M ) I ) ) = ( ( r ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( ( q ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) ) /\ ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( q ( .s ` M ) ( r ( .s ` M ) I ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) I ) = I ) ) )
57		oveq2 6285	. . . . . . . . . . . 12 |- ( x = I -> ( w ( +g ` M ) x ) = ( w ( +g ` M ) I ) )
58	57	oveq2d 6293	. . . . . . . . . . 11 |- ( x = I -> ( r ( .s ` M ) ( w ( +g ` M ) x ) ) = ( r ( .s ` M ) ( w ( +g ` M ) I ) ) )
59		oveq2 6285	. . . . . . . . . . . 12 |- ( x = I -> ( r ( .s ` M ) x ) = ( r ( .s ` M ) I ) )
60	59	oveq2d 6293	. . . . . . . . . . 11 |- ( x = I -> ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) x ) ) = ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) I ) ) )
61	58, 60	eqeq12d 2463	. . . . . . . . . 10 |- ( x = I -> ( ( r ( .s ` M ) ( w ( +g ` M ) x ) ) = ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) x ) ) <-> ( r ( .s ` M ) ( w ( +g ` M ) I ) ) = ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) I ) ) ) )
62	61	3anbi2d 1303	. . . . . . . . 9 |- ( x = I -> ( ( ( r ( .s ` M ) w ) e. { I } /\ ( r ( .s ` M ) ( w ( +g ` M ) x ) ) = ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) x ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) w ) ) ) <-> ( ( r ( .s ` M ) w ) e. { I } /\ ( r ( .s ` M ) ( w ( +g ` M ) I ) ) = ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) I ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) w ) ) ) ) )
63	62	anbi1d 704	. . . . . . . 8 |- ( x = I -> ( ( ( ( r ( .s ` M ) w ) e. { I } /\ ( r ( .s ` M ) ( w ( +g ` M ) x ) ) = ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) x ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( q ( .s ` M ) ( r ( .s ` M ) w ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) w ) = w ) ) <-> ( ( ( r ( .s ` M ) w ) e. { I } /\ ( r ( .s ` M ) ( w ( +g ` M ) I ) ) = ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) I ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( q ( .s ` M ) ( r ( .s ` M ) w ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) w ) = w ) ) ) )
64	63	ralbidv 2880	. . . . . . 7 |- ( x = I -> ( A. w e. { I } ( ( ( r ( .s ` M ) w ) e. { I } /\ ( r ( .s ` M ) ( w ( +g ` M ) x ) ) = ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) x ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( q ( .s ` M ) ( r ( .s ` M ) w ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) w ) = w ) ) <-> A. w e. { I } ( ( ( r ( .s ` M ) w ) e. { I } /\ ( r ( .s ` M ) ( w ( +g ` M ) I ) ) = ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) I ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( q ( .s ` M ) ( r ( .s ` M ) w ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) w ) = w ) ) ) )
65	64	ralsng 4045	. . . . . 6 |- ( I e. V -> ( A. x e. { I } A. w e. { I } ( ( ( r ( .s ` M ) w ) e. { I } /\ ( r ( .s ` M ) ( w ( +g ` M ) x ) ) = ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) x ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( q ( .s ` M ) ( r ( .s ` M ) w ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) w ) = w ) ) <-> A. w e. { I } ( ( ( r ( .s ` M ) w ) e. { I } /\ ( r ( .s ` M ) ( w ( +g ` M ) I ) ) = ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) I ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( q ( .s ` M ) ( r ( .s ` M ) w ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) w ) = w ) ) ) )
66	65	adantr 465	. . . . 5 |- ( ( I e. V /\ R e. Ring ) -> ( A. x e. { I } A. w e. { I } ( ( ( r ( .s ` M ) w ) e. { I } /\ ( r ( .s ` M ) ( w ( +g ` M ) x ) ) = ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) x ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( q ( .s ` M ) ( r ( .s ` M ) w ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) w ) = w ) ) <-> A. w e. { I } ( ( ( r ( .s ` M ) w ) e. { I } /\ ( r ( .s ` M ) ( w ( +g ` M ) I ) ) = ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) I ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( q ( .s ` M ) ( r ( .s ` M ) w ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) w ) = w ) ) ) )
67		oveq2 6285	. . . . . . . . . 10 |- ( w = I -> ( r ( .s ` M ) w ) = ( r ( .s ` M ) I ) )
68	67	eleq1d 2510	. . . . . . . . 9 |- ( w = I -> ( ( r ( .s ` M ) w ) e. { I } <-> ( r ( .s ` M ) I ) e. { I } ) )
69		oveq1 6284	. . . . . . . . . . 11 |- ( w = I -> ( w ( +g ` M ) I ) = ( I ( +g ` M ) I ) )
70	69	oveq2d 6293	. . . . . . . . . 10 |- ( w = I -> ( r ( .s ` M ) ( w ( +g ` M ) I ) ) = ( r ( .s ` M ) ( I ( +g ` M ) I ) ) )
71	67	oveq1d 6292	. . . . . . . . . 10 |- ( w = I -> ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) I ) ) = ( ( r ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) )
72	70, 71	eqeq12d 2463	. . . . . . . . 9 |- ( w = I -> ( ( r ( .s ` M ) ( w ( +g ` M ) I ) ) = ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) I ) ) <-> ( r ( .s ` M ) ( I ( +g ` M ) I ) ) = ( ( r ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) ) )
73		oveq2 6285	. . . . . . . . . 10 |- ( w = I -> ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) I ) )
74		oveq2 6285	. . . . . . . . . . 11 |- ( w = I -> ( q ( .s ` M ) w ) = ( q ( .s ` M ) I ) )
75	74, 67	oveq12d 6295	. . . . . . . . . 10 |- ( w = I -> ( ( q ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) w ) ) = ( ( q ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) )
76	73, 75	eqeq12d 2463	. . . . . . . . 9 |- ( w = I -> ( ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) w ) ) <-> ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( ( q ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) ) )
77	68, 72, 76	3anbi123d 1298	. . . . . . . 8 |- ( w = I -> ( ( ( r ( .s ` M ) w ) e. { I } /\ ( r ( .s ` M ) ( w ( +g ` M ) I ) ) = ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) I ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) w ) ) ) <-> ( ( r ( .s ` M ) I ) e. { I } /\ ( r ( .s ` M ) ( I ( +g ` M ) I ) ) = ( ( r ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( ( q ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) ) ) )
78		oveq2 6285	. . . . . . . . . 10 |- ( w = I -> ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) I ) )
79	67	oveq2d 6293	. . . . . . . . . 10 |- ( w = I -> ( q ( .s ` M ) ( r ( .s ` M ) w ) ) = ( q ( .s ` M ) ( r ( .s ` M ) I ) ) )
80	78, 79	eqeq12d 2463	. . . . . . . . 9 |- ( w = I -> ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( q ( .s ` M ) ( r ( .s ` M ) w ) ) <-> ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( q ( .s ` M ) ( r ( .s ` M ) I ) ) ) )
81		oveq2 6285	. . . . . . . . . 10 |- ( w = I -> ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) w ) = ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) I ) )
82		id 22	. . . . . . . . . 10 |- ( w = I -> w = I )
83	81, 82	eqeq12d 2463	. . . . . . . . 9 |- ( w = I -> ( ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) w ) = w <-> ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) I ) = I ) )
84	80, 83	anbi12d 710	. . . . . . . 8 |- ( w = I -> ( ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( q ( .s ` M ) ( r ( .s ` M ) w ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) w ) = w ) <-> ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( q ( .s ` M ) ( r ( .s ` M ) I ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) I ) = I ) ) )
85	77, 84	anbi12d 710	. . . . . . 7 |- ( w = I -> ( ( ( ( r ( .s ` M ) w ) e. { I } /\ ( r ( .s ` M ) ( w ( +g ` M ) I ) ) = ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) I ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( q ( .s ` M ) ( r ( .s ` M ) w ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) w ) = w ) ) <-> ( ( ( r ( .s ` M ) I ) e. { I } /\ ( r ( .s ` M ) ( I ( +g ` M ) I ) ) = ( ( r ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( ( q ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) ) /\ ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( q ( .s ` M ) ( r ( .s ` M ) I ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) I ) = I ) ) ) )
86	85	ralsng 4045	. . . . . 6 |- ( I e. V -> ( A. w e. { I } ( ( ( r ( .s ` M ) w ) e. { I } /\ ( r ( .s ` M ) ( w ( +g ` M ) I ) ) = ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) I ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( q ( .s ` M ) ( r ( .s ` M ) w ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) w ) = w ) ) <-> ( ( ( r ( .s ` M ) I ) e. { I } /\ ( r ( .s ` M ) ( I ( +g ` M ) I ) ) = ( ( r ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( ( q ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) ) /\ ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( q ( .s ` M ) ( r ( .s ` M ) I ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) I ) = I ) ) ) )
87	86	adantr 465	. . . . 5 |- ( ( I e. V /\ R e. Ring ) -> ( A. w e. { I } ( ( ( r ( .s ` M ) w ) e. { I } /\ ( r ( .s ` M ) ( w ( +g ` M ) I ) ) = ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) I ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( q ( .s ` M ) ( r ( .s ` M ) w ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) w ) = w ) ) <-> ( ( ( r ( .s ` M ) I ) e. { I } /\ ( r ( .s ` M ) ( I ( +g ` M ) I ) ) = ( ( r ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( ( q ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) ) /\ ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( q ( .s ` M ) ( r ( .s ` M ) I ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) I ) = I ) ) ) )
88	66, 87	bitrd 253	. . . 4 |- ( ( I e. V /\ R e. Ring ) -> ( A. x e. { I } A. w e. { I } ( ( ( r ( .s ` M ) w ) e. { I } /\ ( r ( .s ` M ) ( w ( +g ` M ) x ) ) = ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) x ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( q ( .s ` M ) ( r ( .s ` M ) w ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) w ) = w ) ) <-> ( ( ( r ( .s ` M ) I ) e. { I } /\ ( r ( .s ` M ) ( I ( +g ` M ) I ) ) = ( ( r ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( ( q ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) ) /\ ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( q ( .s ` M ) ( r ( .s ` M ) I ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) I ) = I ) ) ) )
89	88	2ralbidv 2885	. . 3 |- ( ( I e. V /\ R e. Ring ) -> ( A. q e. ( Base ` (Scalar ` M ) ) A. r e. ( Base ` (Scalar ` M ) ) A. x e. { I } A. w e. { I } ( ( ( r ( .s ` M ) w ) e. { I } /\ ( r ( .s ` M ) ( w ( +g ` M ) x ) ) = ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) x ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( q ( .s ` M ) ( r ( .s ` M ) w ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) w ) = w ) ) <-> A. q e. ( Base ` (Scalar ` M ) ) A. r e. ( Base ` (Scalar ` M ) ) ( ( ( r ( .s ` M ) I ) e. { I } /\ ( r ( .s ` M ) ( I ( +g ` M ) I ) ) = ( ( r ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( ( q ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) ) /\ ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) I ) = ( q ( .s ` M ) ( r ( .s ` M ) I ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) I ) = I ) ) ) )
90	56, 89	mpbird 232	. 2 |- ( ( I e. V /\ R e. Ring ) -> A. q e. ( Base ` (Scalar ` M ) ) A. r e. ( Base ` (Scalar ` M ) ) A. x e. { I } A. w e. { I } ( ( ( r ( .s ` M ) w ) e. { I } /\ ( r ( .s ` M ) ( w ( +g ` M ) x ) ) = ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) x ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( q ( .s ` M ) ( r ( .s ` M ) w ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) w ) = w ) ) )
91	4	lmodbase 14634	. . . 4 |- ( { I } e. _V -> { I } = ( Base ` M ) )
92	5, 91	ax-mp 5	. . 3 |- { I } = ( Base ` M )
93		eqid 2441	. . 3 |- ( +g ` M ) = ( +g ` M )
94		eqid 2441	. . 3 |- ( .s ` M ) = ( .s ` M )
95		eqid 2441	. . 3 |- (Scalar ` M ) = (Scalar ` M )
96		eqid 2441	. . 3 |- ( Base ` (Scalar ` M ) ) = ( Base ` (Scalar ` M ) )
97		eqid 2441	. . 3 |- ( +g ` (Scalar ` M ) ) = ( +g ` (Scalar ` M ) )
98		eqid 2441	. . 3 |- ( .r ` (Scalar ` M ) ) = ( .r ` (Scalar ` M ) )
99		eqid 2441	. . 3 |- ( 1r ` (Scalar ` M ) ) = ( 1r ` (Scalar ` M ) )
100	92, 93, 94, 95, 96, 97, 98, 99	islmod 17384	. 2 |- ( M e. LMod <-> ( M e. Grp /\ (Scalar ` M ) e. Ring /\ A. q e. ( Base ` (Scalar ` M ) ) A. r e. ( Base ` (Scalar ` M ) ) A. x e. { I } A. w e. { I } ( ( ( r ( .s ` M ) w ) e. { I } /\ ( r ( .s ` M ) ( w ( +g ` M ) x ) ) = ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) x ) ) /\ ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) w ) ) ) /\ ( ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( q ( .s ` M ) ( r ( .s ` M ) w ) ) /\ ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) w ) = w ) ) ) )
101	30, 35, 90, 100	syl3anbrc 1179	1 |- ( ( I e. V /\ R e. Ring ) -> M e. LMod )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 972   = wceq 1381   e. wcel 1802   A.wral 2791   _Vcvv 3093   u. cun 3456   {csn 4010   {cpr 4012   {ctp 4014   <.cop 4016   ` cfv 5574  (class class class)co 6277   |-> cmpt2 6279   ndxcnx 14501   Basecbs 14504   +g cplusg 14569   .rcmulr 14570  Scalarcsca 14572   .scvsca 14573   Grpcgrp 15922   1rcur 17021   Ringcrg 17066   LModclmod 17380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-n0 10797  df-z 10866  df-uz 11086  df-fz 11677  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-plusg 14582  df-sca 14585  df-vsca 14586  df-0g 14711  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-grp 15926  df-mgp 17010  df-ur 17022  df-ring 17068  df-lmod 17382

This theorem is referenced by:  lmod1zr  32804