Theorem lmod1 32049

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 2460	. . . . 5 |- { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } = { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. }
2	1	grp1 32037	. . . 4 |- ( I e. V -> { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } e. Grp )
3		fvex 5867	. . . . . . 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 4681	. . . . . . . . . . . . 13 |- { I } e. _V
6	1	grpbase 14584	. . . . . . . . . . . . 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 4210	. . . . . . . . . . 11 |- <. ( Base ` ndx ) , { I } >. = <. ( Base ` ndx ) , ( Base ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) >.
9		tpeq1 4108	. . . . . . . . . . 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 3647	. . . . . . . . 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 2489	. . . . . . . 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 14609	. . . . . . 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 2473	. . . . 5 |- ( Base ` M ) = ( Base ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } )
16		fvex 5867	. . . . . . 7 |- ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) e. _V
17		snex 4681	. . . . . . . . . . . . 13 |- { <. <. I , I >. , I >. } e. _V
18	1	grpplusg 14585	. . . . . . . . . . . . 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 4210	. . . . . . . . . . 11 |- <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. = <. ( +g ` ndx ) , ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) >.
21		tpeq2 4109	. . . . . . . . . . 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 3647	. . . . . . . . 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 2489	. . . . . . . 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 14610	. . . . . . 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 2473	. . . . 5 |- ( +g ` M ) = ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } )
28	15, 27	grpprop 15863	. . . 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 14611	. . . . 5 |- ( R e. Ring -> R = (Scalar ` M ) )
32	31	eqcomd 2468	. . . 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 2548	. 2 |- ( ( I e. V /\ R e. Ring ) -> (Scalar ` M ) e. Ring )
36	33	fveq2d 5861	. . . . . . . 8 |- ( ( I e. V /\ R e. Ring ) -> ( Base ` (Scalar ` M ) ) = ( Base ` R ) )
37	36	eleq2d 2530	. . . . . . 7 |- ( ( I e. V /\ R e. Ring ) -> ( q e. ( Base ` (Scalar ` M ) ) <-> q e. ( Base ` R ) ) )
38	36	eleq2d 2530	. . . . . . 7 |- ( ( I e. V /\ R e. Ring ) -> ( r e. ( Base ` (Scalar ` M ) ) <-> r e. ( Base ` R ) ) )
39	37, 38	anbi12d 710	. . . . . 6 |- ( ( 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	. . . . . . . . . . 11 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> I e. V )
41		simplr 754	. . . . . . . . . . 11 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> R e. Ring )
42		simprr 756	. . . . . . . . . . 11 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> r e. ( Base ` R ) )
43	40, 41, 42	3jca 1171	. . . . . . . . . 10 |- ( ( ( 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 32044	. . . . . . . . . 10 |- ( ( I e. V /\ R e. Ring /\ r e. ( Base ` R ) ) -> ( r ( .s ` M ) I ) e. { I } )
45	43, 44	syl 16	. . . . . . . . 9 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> ( r ( .s ` M ) I ) e. { I } )
46	4	lmod1lem2 32045	. . . . . . . . . 10 |- ( ( 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	. . . . . . . . 9 |- ( ( ( 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 32046	. . . . . . . . 9 |- ( ( ( 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 1171	. . . . . . . 8 |- ( ( ( 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 32047	. . . . . . . 8 |- ( ( ( 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 32048	. . . . . . . . 9 |- ( ( I e. V /\ R e. Ring ) -> ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) I ) = I )
52	51	adantr 465	. . . . . . . 8 |- ( ( ( 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	. . . . . . 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 ) ) ) /\ ( ( ( 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	. . . . . 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 ) ) ) )
55	39, 54	sylbid 215	. . . . 5 |- ( ( 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	imp 429	. . . 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 ) ) )
57	56	ralrimivva 2878	. . 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 ) ) )
58		oveq2 6283	. . . . . . . . . . . . 13 |- ( x = I -> ( w ( +g ` M ) x ) = ( w ( +g ` M ) I ) )
59	58	oveq2d 6291	. . . . . . . . . . . 12 |- ( x = I -> ( r ( .s ` M ) ( w ( +g ` M ) x ) ) = ( r ( .s ` M ) ( w ( +g ` M ) I ) ) )
60		oveq2 6283	. . . . . . . . . . . . 13 |- ( x = I -> ( r ( .s ` M ) x ) = ( r ( .s ` M ) I ) )
61	60	oveq2d 6291	. . . . . . . . . . . 12 |- ( x = I -> ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) x ) ) = ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) I ) ) )
62	59, 61	eqeq12d 2482	. . . . . . . . . . 11 |- ( 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 ) ) ) )
63	62	3anbi2d 1299	. . . . . . . . . 10 |- ( 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 ) ) ) ) )
64	63	anbi1d 704	. . . . . . . . 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 ) ) ) /\ ( ( ( 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 ) ) ) )
65	64	ralbidv 2896	. . . . . . . 8 |- ( 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 ) ) ) )
66	65	ralsng 4055	. . . . . . 7 |- ( 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 ) ) ) )
67	66	adantr 465	. . . . . 6 |- ( ( 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 ) ) ) )
68		oveq2 6283	. . . . . . . . . . 11 |- ( w = I -> ( r ( .s ` M ) w ) = ( r ( .s ` M ) I ) )
69	68	eleq1d 2529	. . . . . . . . . 10 |- ( w = I -> ( ( r ( .s ` M ) w ) e. { I } <-> ( r ( .s ` M ) I ) e. { I } ) )
70		oveq1 6282	. . . . . . . . . . . 12 |- ( w = I -> ( w ( +g ` M ) I ) = ( I ( +g ` M ) I ) )
71	70	oveq2d 6291	. . . . . . . . . . 11 |- ( w = I -> ( r ( .s ` M ) ( w ( +g ` M ) I ) ) = ( r ( .s ` M ) ( I ( +g ` M ) I ) ) )
72	68	oveq1d 6290	. . . . . . . . . . 11 |- ( w = I -> ( ( r ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) I ) ) = ( ( r ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) )
73	71, 72	eqeq12d 2482	. . . . . . . . . 10 |- ( 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 ) ) ) )
74		oveq2 6283	. . . . . . . . . . 11 |- ( w = I -> ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) I ) )
75		oveq2 6283	. . . . . . . . . . . 12 |- ( w = I -> ( q ( .s ` M ) w ) = ( q ( .s ` M ) I ) )
76	75, 68	oveq12d 6293	. . . . . . . . . . 11 |- ( w = I -> ( ( q ( .s ` M ) w ) ( +g ` M ) ( r ( .s ` M ) w ) ) = ( ( q ( .s ` M ) I ) ( +g ` M ) ( r ( .s ` M ) I ) ) )
77	74, 76	eqeq12d 2482	. . . . . . . . . 10 |- ( 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 ) ) ) )
78	69, 73, 77	3anbi123d 1294	. . . . . . . . 9 |- ( 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 ) ) ) ) )
79		oveq2 6283	. . . . . . . . . . 11 |- ( w = I -> ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) I ) )
80	68	oveq2d 6291	. . . . . . . . . . 11 |- ( w = I -> ( q ( .s ` M ) ( r ( .s ` M ) w ) ) = ( q ( .s ` M ) ( r ( .s ` M ) I ) ) )
81	79, 80	eqeq12d 2482	. . . . . . . . . 10 |- ( 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 ) ) ) )
82		oveq2 6283	. . . . . . . . . . 11 |- ( w = I -> ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) w ) = ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) I ) )
83		id 22	. . . . . . . . . . 11 |- ( w = I -> w = I )
84	82, 83	eqeq12d 2482	. . . . . . . . . 10 |- ( w = I -> ( ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) w ) = w <-> ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) I ) = I ) )
85	81, 84	anbi12d 710	. . . . . . . . 9 |- ( 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 ) ) )
86	78, 85	anbi12d 710	. . . . . . . 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 ) ) ) /\ ( ( ( 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	ralsng 4055	. . . . . . 7 |- ( 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 ) ) ) )
88	87	adantr 465	. . . . . 6 |- ( ( 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 ) ) ) )
89	67, 88	bitrd 253	. . . . 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 ) ) <-> ( ( ( 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	89	ralbidv 2896	. . . 4 |- ( ( I e. V /\ R e. Ring ) -> ( 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. 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 ) ) ) )
91	90	ralbidv 2896	. . 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 ) ) ) )
92	57, 91	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 ) ) )
93	4	lmodbase 14609	. . . 4 |- ( { I } e. _V -> { I } = ( Base ` M ) )
94	5, 93	ax-mp 5	. . 3 |- { I } = ( Base ` M )
95		eqid 2460	. . 3 |- ( +g ` M ) = ( +g ` M )
96		eqid 2460	. . 3 |- ( .s ` M ) = ( .s ` M )
97		eqid 2460	. . 3 |- (Scalar ` M ) = (Scalar ` M )
98		eqid 2460	. . 3 |- ( Base ` (Scalar ` M ) ) = ( Base ` (Scalar ` M ) )
99		eqid 2460	. . 3 |- ( +g ` (Scalar ` M ) ) = ( +g ` (Scalar ` M ) )
100		eqid 2460	. . 3 |- ( .r ` (Scalar ` M ) ) = ( .r ` (Scalar ` M ) )
101		eqid 2460	. . 3 |- ( 1r ` (Scalar ` M ) ) = ( 1r ` (Scalar ` M ) )
102	94, 95, 96, 97, 98, 99, 100, 101	islmod 17292	. 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 ) ) ) )
103	30, 35, 92, 102	syl3anbrc 1175	1 |- ( ( I e. V /\ R e. Ring ) -> M e. LMod )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 968   = wceq 1374   e. wcel 1762   A.wral 2807   _Vcvv 3106   u. cun 3467   {csn 4020   {cpr 4022   {ctp 4024   <.cop 4026   ` cfv 5579  (class class class)co 6275   |-> cmpt2 6277   ndxcnx 14476   Basecbs 14479   +g cplusg 14544   .rcmulr 14545  Scalarcsca 14547   .scvsca 14548   Grpcgrp 15716   1rcur 16936   Ringcrg 16979   LModclmod 17288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-plusg 14557  df-sca 14560  df-vsca 14561  df-0g 14686  df-mnd 15721  df-grp 15851  df-mgp 16925  df-ur 16937  df-rng 16981  df-lmod 17290

This theorem is referenced by:  lmod1zr  32050