Theorem lmod1 42281

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 2622	. . . . 5 |- { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } = { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. }
2	1	grp1 17522	. . . 4 |- ( I e. V -> { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } e. Grp )
3		fvex 6201	. . . . . . 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 4908	. . . . . . . . . . . . 13 |- { I } e. _V
6	1	grpbase 15991	. . . . . . . . . . . . 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 4406	. . . . . . . . . . 11 |- <. ( Base ` ndx ) , { I } >. = <. ( Base ` ndx ) , ( Base ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) >.
9		tpeq1 4277	. . . . . . . . . . 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 3763	. . . . . . . . 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 2644	. . . . . . . 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 16018	. . . . . . 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 2631	. . . . 5 |- ( Base ` M ) = ( Base ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } )
16		fvex 6201	. . . . . . 7 |- ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) e. _V
17		snex 4908	. . . . . . . . . . . . 13 |- { <. <. I , I >. , I >. } e. _V
18	1	grpplusg 15992	. . . . . . . . . . . . 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 4406	. . . . . . . . . . 11 |- <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. = <. ( +g ` ndx ) , ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) >.
21		tpeq2 4278	. . . . . . . . . . 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 3763	. . . . . . . . 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 2644	. . . . . . . 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 16019	. . . . . . 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 2631	. . . . 5 |- ( +g ` M ) = ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } )
28	15, 27	grpprop 17438	. . . 4 |- ( M e. Grp <-> { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } e. Grp )
29	2, 28	sylibr 224	. . 3 |- ( I e. V -> M e. Grp )
30	29	adantr 481	. 2 |- ( ( I e. V /\ R e. Ring ) -> M e. Grp )
31	4	lmodsca 16020	. . . . 5 |- ( R e. Ring -> R = (Scalar ` M ) )
32	31	eqcomd 2628	. . . 4 |- ( R e. Ring -> (Scalar ` M ) = R )
33	32	adantl 482	. . 3 |- ( ( I e. V /\ R e. Ring ) -> (Scalar ` M ) = R )
34		simpr 477	. . 3 |- ( ( I e. V /\ R e. Ring ) -> R e. Ring )
35	33, 34	eqeltrd 2701	. 2 |- ( ( I e. V /\ R e. Ring ) -> (Scalar ` M ) e. Ring )
36	33	fveq2d 6195	. . . . . . 7 |- ( ( I e. V /\ R e. Ring ) -> ( Base ` (Scalar ` M ) ) = ( Base ` R ) )
37	36	eleq2d 2687	. . . . . 6 |- ( ( I e. V /\ R e. Ring ) -> ( q e. ( Base ` (Scalar ` M ) ) <-> q e. ( Base ` R ) ) )
38	36	eleq2d 2687	. . . . . 6 |- ( ( I e. V /\ R e. Ring ) -> ( r e. ( Base ` (Scalar ` M ) ) <-> r e. ( Base ` R ) ) )
39	37, 38	anbi12d 747	. . . . 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 790	. . . . . . . . . 10 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> I e. V )
41		simplr 792	. . . . . . . . . 10 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> R e. Ring )
42		simprr 796	. . . . . . . . . 10 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> r e. ( Base ` R ) )
43	40, 41, 42	3jca 1242	. . . . . . . . 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 42276	. . . . . . . . 9 |- ( ( I e. V /\ R e. Ring /\ r e. ( Base ` R ) ) -> ( r ( .s ` M ) I ) e. { I } )
45	43, 44	syl 17	. . . . . . . 8 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( r ( .s ` M ) I ) e. { I } )
46	4	lmod1lem2 42277	. . . . . . . . 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 17	. . . . . . . 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 42278	. . . . . . . 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 1242	. . . . . . 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 42279	. . . . . . 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 42280	. . . . . . . 8 |- ( ( I e. V /\ R e. Ring ) -> ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) I ) = I )
52	51	adantr 481	. . . . . . 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 558	. . . . . 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 450	. . . . 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 230	. . . 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 2970	. . 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 6658	. . . . . . . . . . . 12 |- ( x = I -> ( w ( +g ` M ) x ) = ( w ( +g ` M ) I ) )
58	57	oveq2d 6666	. . . . . . . . . . 11 |- ( x = I -> ( r ( .s ` M ) ( w ( +g ` M ) x ) ) = ( r ( .s ` M ) ( w ( +g ` M ) I ) ) )
59		oveq2 6658	. . . . . . . . . . . 12 |- ( x = I -> ( r ( .s ` M ) x ) = ( r ( .s ` M ) I ) )
60	59	oveq2d 6666	. . . . . . . . . . 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 2637	. . . . . . . . . 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 1404	. . . . . . . . 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 741	. . . . . . . 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 2986	. . . . . . 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 4218	. . . . . 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 481	. . . . 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 6658	. . . . . . . . . 10 |- ( w = I -> ( r ( .s ` M ) w ) = ( r ( .s ` M ) I ) )
68	67	eleq1d 2686	. . . . . . . . 9 |- ( w = I -> ( ( r ( .s ` M ) w ) e. { I } <-> ( r ( .s ` M ) I ) e. { I } ) )
69		oveq1 6657	. . . . . . . . . . 11 |- ( w = I -> ( w ( +g ` M ) I ) = ( I ( +g ` M ) I ) )
70	69	oveq2d 6666	. . . . . . . . . 10 |- ( w = I -> ( r ( .s ` M ) ( w ( +g ` M ) I ) ) = ( r ( .s ` M ) ( I ( +g ` M ) I ) ) )
71	67	oveq1d 6665	. . . . . . . . . 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 2637	. . . . . . . . 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 6658	. . . . . . . . . 10 |- ( w = I -> ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) I ) )
74		oveq2 6658	. . . . . . . . . . 11 |- ( w = I -> ( q ( .s ` M ) w ) = ( q ( .s ` M ) I ) )
75	74, 67	oveq12d 6668	. . . . . . . . . 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 2637	. . . . . . . . 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 1399	. . . . . . . 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 6658	. . . . . . . . . 10 |- ( w = I -> ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) I ) )
79	67	oveq2d 6666	. . . . . . . . . 10 |- ( w = I -> ( q ( .s ` M ) ( r ( .s ` M ) w ) ) = ( q ( .s ` M ) ( r ( .s ` M ) I ) ) )
80	78, 79	eqeq12d 2637	. . . . . . . . 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 6658	. . . . . . . . . 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 2637	. . . . . . . . 9 |- ( w = I -> ( ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) w ) = w <-> ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) I ) = I ) )
84	80, 83	anbi12d 747	. . . . . . . 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 747	. . . . . . 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 4218	. . . . . 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 481	. . . . 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 268	. . . 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 2989	. . 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 247	. 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 16018	. . . 4 |- ( { I } e. _V -> { I } = ( Base ` M ) )
92	5, 91	ax-mp 5	. . 3 |- { I } = ( Base ` M )
93		eqid 2622	. . 3 |- ( +g ` M ) = ( +g ` M )
94		eqid 2622	. . 3 |- ( .s ` M ) = ( .s ` M )
95		eqid 2622	. . 3 |- (Scalar ` M ) = (Scalar ` M )
96		eqid 2622	. . 3 |- ( Base ` (Scalar ` M ) ) = ( Base ` (Scalar ` M ) )
97		eqid 2622	. . 3 |- ( +g ` (Scalar ` M ) ) = ( +g ` (Scalar ` M ) )
98		eqid 2622	. . 3 |- ( .r ` (Scalar ` M ) ) = ( .r ` (Scalar ` M ) )
99		eqid 2622	. . 3 |- ( 1r ` (Scalar ` M ) ) = ( 1r ` (Scalar ` M ) )
100	92, 93, 94, 95, 96, 97, 98, 99	islmod 18867	. 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 1246	1 |- ( ( I e. V /\ R e. Ring ) -> M e. LMod )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   u. cun 3572   {csn 4177   {cpr 4179   {ctp 4181   <.cop 4183   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ndxcnx 15854   Basecbs 15857   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   Grpcgrp 17422   1rcur 18501   Ringcrg 18547   LModclmod 18863

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-int 4476  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-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-sca 15957  df-vsca 15958  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865

This theorem is referenced by:  lmod1zr  42282