Theorem lmod1 40793

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 2471	. . . . 5 |- { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } = { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. }
2	1	grp1 16836	. . . 4 |- ( I e. V -> { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } e. Grp )
3		fvex 5889	. . . . . . 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 4641	. . . . . . . . . . . . 13 |- { I } e. _V
6	1	grpbase 15315	. . . . . . . . . . . . 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 4162	. . . . . . . . . . 11 |- <. ( Base ` ndx ) , { I } >. = <. ( Base ` ndx ) , ( Base ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) >.
9		tpeq1 4051	. . . . . . . . . . 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 3575	. . . . . . . . 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 2493	. . . . . . . 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 15340	. . . . . . 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 2480	. . . . 5 |- ( Base ` M ) = ( Base ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } )
16		fvex 5889	. . . . . . 7 |- ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) e. _V
17		snex 4641	. . . . . . . . . . . . 13 |- { <. <. I , I >. , I >. } e. _V
18	1	grpplusg 15316	. . . . . . . . . . . . 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 4162	. . . . . . . . . . 11 |- <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. = <. ( +g ` ndx ) , ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } ) >.
21		tpeq2 4052	. . . . . . . . . . 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 3575	. . . . . . . . 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 2493	. . . . . . . 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 15341	. . . . . . 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 2480	. . . . 5 |- ( +g ` M ) = ( +g ` { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } )
28	15, 27	grpprop 16763	. . . 4 |- ( M e. Grp <-> { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. } e. Grp )
29	2, 28	sylibr 217	. . 3 |- ( I e. V -> M e. Grp )
30	29	adantr 472	. 2 |- ( ( I e. V /\ R e. Ring ) -> M e. Grp )
31	4	lmodsca 15342	. . . . 5 |- ( R e. Ring -> R = (Scalar ` M ) )
32	31	eqcomd 2477	. . . 4 |- ( R e. Ring -> (Scalar ` M ) = R )
33	32	adantl 473	. . 3 |- ( ( I e. V /\ R e. Ring ) -> (Scalar ` M ) = R )
34		simpr 468	. . 3 |- ( ( I e. V /\ R e. Ring ) -> R e. Ring )
35	33, 34	eqeltrd 2549	. 2 |- ( ( I e. V /\ R e. Ring ) -> (Scalar ` M ) e. Ring )
36	33	fveq2d 5883	. . . . . . 7 |- ( ( I e. V /\ R e. Ring ) -> ( Base ` (Scalar ` M ) ) = ( Base ` R ) )
37	36	eleq2d 2534	. . . . . 6 |- ( ( I e. V /\ R e. Ring ) -> ( q e. ( Base ` (Scalar ` M ) ) <-> q e. ( Base ` R ) ) )
38	36	eleq2d 2534	. . . . . 6 |- ( ( I e. V /\ R e. Ring ) -> ( r e. ( Base ` (Scalar ` M ) ) <-> r e. ( Base ` R ) ) )
39	37, 38	anbi12d 725	. . . . 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 768	. . . . . . . . . 10 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> I e. V )
41		simplr 770	. . . . . . . . . 10 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> R e. Ring )
42		simprr 774	. . . . . . . . . 10 |- ( ( ( I e. V /\ R e. Ring ) /\ ( q e. ( Base ` R ) /\ r e. ( Base ` R ) ) ) -> r e. ( Base ` R ) )
43	40, 41, 42	3jca 1210	. . . . . . . . 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 40788	. . . . . . . . 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 40789	. . . . . . . . 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 40790	. . . . . . . 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 1210	. . . . . . 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 40791	. . . . . . 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 40792	. . . . . . . 8 |- ( ( I e. V /\ R e. Ring ) -> ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) I ) = I )
52	51	adantr 472	. . . . . . 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 544	. . . . . 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 441	. . . . 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 223	. . . 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 2813	. . 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 6316	. . . . . . . . . . . 12 |- ( x = I -> ( w ( +g ` M ) x ) = ( w ( +g ` M ) I ) )
58	57	oveq2d 6324	. . . . . . . . . . 11 |- ( x = I -> ( r ( .s ` M ) ( w ( +g ` M ) x ) ) = ( r ( .s ` M ) ( w ( +g ` M ) I ) ) )
59		oveq2 6316	. . . . . . . . . . . 12 |- ( x = I -> ( r ( .s ` M ) x ) = ( r ( .s ` M ) I ) )
60	59	oveq2d 6324	. . . . . . . . . . 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 2486	. . . . . . . . . 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 1370	. . . . . . . . 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 719	. . . . . . . 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 2829	. . . . . . 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 3997	. . . . . 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 472	. . . . 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 6316	. . . . . . . . . 10 |- ( w = I -> ( r ( .s ` M ) w ) = ( r ( .s ` M ) I ) )
68	67	eleq1d 2533	. . . . . . . . 9 |- ( w = I -> ( ( r ( .s ` M ) w ) e. { I } <-> ( r ( .s ` M ) I ) e. { I } ) )
69		oveq1 6315	. . . . . . . . . . 11 |- ( w = I -> ( w ( +g ` M ) I ) = ( I ( +g ` M ) I ) )
70	69	oveq2d 6324	. . . . . . . . . 10 |- ( w = I -> ( r ( .s ` M ) ( w ( +g ` M ) I ) ) = ( r ( .s ` M ) ( I ( +g ` M ) I ) ) )
71	67	oveq1d 6323	. . . . . . . . . 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 2486	. . . . . . . . 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 6316	. . . . . . . . . 10 |- ( w = I -> ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( +g ` (Scalar ` M ) ) r ) ( .s ` M ) I ) )
74		oveq2 6316	. . . . . . . . . . 11 |- ( w = I -> ( q ( .s ` M ) w ) = ( q ( .s ` M ) I ) )
75	74, 67	oveq12d 6326	. . . . . . . . . 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 2486	. . . . . . . . 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 1365	. . . . . . . 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 6316	. . . . . . . . . 10 |- ( w = I -> ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) w ) = ( ( q ( .r ` (Scalar ` M ) ) r ) ( .s ` M ) I ) )
79	67	oveq2d 6324	. . . . . . . . . 10 |- ( w = I -> ( q ( .s ` M ) ( r ( .s ` M ) w ) ) = ( q ( .s ` M ) ( r ( .s ` M ) I ) ) )
80	78, 79	eqeq12d 2486	. . . . . . . . 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 6316	. . . . . . . . . 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 2486	. . . . . . . . 9 |- ( w = I -> ( ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) w ) = w <-> ( ( 1r ` (Scalar ` M ) ) ( .s ` M ) I ) = I ) )
84	80, 83	anbi12d 725	. . . . . . . 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 725	. . . . . . 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 3997	. . . . . 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 472	. . . . 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 261	. . . 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 2832	. . 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 240	. 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 15340	. . . 4 |- ( { I } e. _V -> { I } = ( Base ` M ) )
92	5, 91	ax-mp 5	. . 3 |- { I } = ( Base ` M )
93		eqid 2471	. . 3 |- ( +g ` M ) = ( +g ` M )
94		eqid 2471	. . 3 |- ( .s ` M ) = ( .s ` M )
95		eqid 2471	. . 3 |- (Scalar ` M ) = (Scalar ` M )
96		eqid 2471	. . 3 |- ( Base ` (Scalar ` M ) ) = ( Base ` (Scalar ` M ) )
97		eqid 2471	. . 3 |- ( +g ` (Scalar ` M ) ) = ( +g ` (Scalar ` M ) )
98		eqid 2471	. . 3 |- ( .r ` (Scalar ` M ) ) = ( .r ` (Scalar ` M ) )
99		eqid 2471	. . 3 |- ( 1r ` (Scalar ` M ) ) = ( 1r ` (Scalar ` M ) )
100	92, 93, 94, 95, 96, 97, 98, 99	islmod 18173	. 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 1214	1 |- ( ( I e. V /\ R e. Ring ) -> M e. LMod )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   A.wral 2756   _Vcvv 3031   u. cun 3388   {csn 3959   {cpr 3961   {ctp 3963   <.cop 3965   ` cfv 5589  (class class class)co 6308   |-> cmpt2 6310   ndxcnx 15196   Basecbs 15199   +g cplusg 15268   .rcmulr 15269  Scalarcsca 15271   .scvsca 15272   Grpcgrp 16747   1rcur 17813   Ringcrg 17858   LModclmod 18169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-plusg 15281  df-sca 15284  df-vsca 15285  df-0g 15418  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-grp 16751  df-mgp 17802  df-ur 17814  df-ring 17860  df-lmod 18171

This theorem is referenced by:  lmod1zr  40794