Theorem lmod1 31143

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 2451	. . . . 5 |- { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } = { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. }
2	1	grp1 31131	. . . 4 |- ( I e. V -> { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } e. Grp )
3		fvex 5801	. . . . . . 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 4633	. . . . . . . . . . . . 13 |- { I } e. _V
6	1	grpbase 14382	. . . . . . . . . . . . 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 4163	. . . . . . . . . . 11 |- <. ( Base ` ndx ) , { I } >. = <. ( Base ` ndx ) , ( Base ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) >.
9		tpeq1 4063	. . . . . . . . . . 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 3606	. . . . . . . . 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 2480	. . . . . . . 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 14407	. . . . . . 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 2464	. . . . 5 |- ( Base ` M ) = ( Base ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } )
16		fvex 5801	. . . . . . 7 |- ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) e. _V
17		snex 4633	. . . . . . . . . . . . 13 |- { <. <. I , I >. , I >. } e. _V
18	1	grpplusg 14383	. . . . . . . . . . . . 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 4163	. . . . . . . . . . 11 |- <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. = <. ( +g ` ndx ) , ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) >.
21		tpeq2 4064	. . . . . . . . . . 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 3606	. . . . . . . . 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 2480	. . . . . . . 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 14408	. . . . . . 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 2464	. . . . 5 |- ( +g ` M ) = ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } )
28	15, 27	grpprop 15661	. . . 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 14409	. . . . 5 |- ( R e. Ring -> R = (Scalar ` M ) )
32	31	eqcomd 2459	. . . 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 2539	. 2 |- ( ( I e. V /\ R e. Ring ) -> (Scalar ` M ) e. Ring )
36	33	fveq2d 5795	. . . . . . . 8 |- ( ( I e. V /\ R e. Ring ) -> ( Base ` (Scalar ` M ) ) = ( Base ` R ) )
37	36	eleq2d 2521	. . . . . . 7 |- ( ( I e. V /\ R e. Ring ) -> ( q e. ( Base ` (Scalar ` M ) ) <-> q e. ( Base ` R ) ) )
38	36	eleq2d 2521	. . . . . . 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 1168	. . . . . . . . . 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 31138	. . . . . . . . . 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 31139	. . . . . . . . . 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 31140	. . . . . . . . 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 1168	. . . . . . . 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 31141	. . . . . . . 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 31142	. . . . . . . . 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 2906	. . 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 6200	. . . . . . . . . . . . 13 |- ( x = I -> ( w ( +g ` M ) x ) = ( w ( +g ` M ) I ) )
59	58	oveq2d 6208	. . . . . . . . . . . 12 |- ( x = I -> ( r ( .s ` M ) ( w ( +g ` M ) x ) ) = ( r ( .s ` M ) ( w ( +g ` M ) I ) ) )
60		oveq2 6200	. . . . . . . . . . . . 13 |- ( x = I -> ( r ( .s ` M ) x ) = ( r ( .s ` M ) I ) )
61	60	oveq2d 6208	. . . . . . . . . . . 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 2473	. . . . . . . . . . 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 1295	. . . . . . . . . 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 2838	. . . . . . . 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 4012	. . . . . . 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 6200	. . . . . . . . . . 11 |- ( w = I -> ( r ( .s ` M ) w ) = ( r ( .s ` M ) I ) )
69	68	eleq1d 2520	. . . . . . . . . 10 |- ( w = I -> ( ( r ( .s ` M ) w ) e. { I } <-> ( r ( .s ` M ) I ) e. { I } ) )
70		oveq1 6199	. . . . . . . . . . . 12 |- ( w = I -> ( w ( +g ` M ) I ) = ( I ( +g ` M ) I ) )
71	70	oveq2d 6208	. . . . . . . . . . 11 |- ( w = I -> ( r ( .s ` M ) ( w ( +g ` M ) I ) ) = ( r ( .s ` M ) ( I ( +g ` M ) I ) ) )
72	68	oveq1d 6207	. . . . . . . . . . 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 2473	. . . . . . . . . 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 6200	. . . . . . . . . . 11 |- ( w = I -> ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) I ) )
75		oveq2 6200	. . . . . . . . . . . 12 |- ( w = I -> ( q ( .s ` M ) w ) = ( q ( .s ` M ) I ) )
76	75, 68	oveq12d 6210	. . . . . . . . . . 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 2473	. . . . . . . . . 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 1290	. . . . . . . . 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 6200	. . . . . . . . . . 11 |- ( w = I -> ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) I ) )
80	68	oveq2d 6208	. . . . . . . . . . 11 |- ( w = I -> ( q ( .s ` M ) ( r ( .s ` M ) w ) ) = ( q ( .s ` M ) ( r ( .s ` M ) I ) ) )
81	79, 80	eqeq12d 2473	. . . . . . . . . 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 6200	. . . . . . . . . . 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 2473	. . . . . . . . . 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 4012	. . . . . . 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 2838	. . . 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 2838	. . 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 14407	. . . 4 |- ( { I } e. _V -> { I } = ( Base ` M ) )
94	5, 93	ax-mp 5	. . 3 |- { I } = ( Base ` M )
95		eqid 2451	. . 3 |- ( +g ` M ) = ( +g ` M )
96		eqid 2451	. . 3 |- ( .s ` M ) = ( .s ` M )
97		eqid 2451	. . 3 |- (Scalar ` M ) = (Scalar ` M )
98		eqid 2451	. . 3 |- ( Base ` (Scalar ` M ) ) = ( Base ` (Scalar ` M ) )
99		eqid 2451	. . 3 |- ( +g ` (Scalar ` M ) ) = ( +g ` (Scalar ` M ) )
100		eqid 2451	. . 3 |- ( .r ` (Scalar ` M ) ) = ( .r ` (Scalar ` M ) )
101		eqid 2451	. . 3 |- ( 1r ` (Scalar ` M ) ) = ( 1r ` (Scalar ` M ) )
102	94, 95, 96, 97, 98, 99, 100, 101	islmod 17060	. 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 1172	1 |- ( ( I e. V /\ R e. Ring ) -> M e. LMod )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   A.wral 2795   _Vcvv 3070   u. cun 3426   {csn 3977   {cpr 3979   {ctp 3981   <.cop 3983   ` cfv 5518  (class class class)co 6192   |-> cmpt2 6194   ndxcnx 14275   Basecbs 14278   +g cplusg 14342   .rcmulr 14343  Scalarcsca 14345   .scvsca 14346   Grpcgrp 15514   1rcur 16710   Ringcrg 16753   LModclmod 17056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-plusg 14355  df-sca 14358  df-vsca 14359  df-0g 14484  df-mnd 15519  df-grp 15649  df-mgp 16699  df-ur 16711  df-rng 16755  df-lmod 17058

This theorem is referenced by:  lmod1zr  31144