Theorem islidlm 13792

Description: Predicate of being a (left) ideal. (Contributed by Stefan O'Rear, 1-Apr-2015.)

Hypotheses

Ref	Expression
islidl.s	|- U = (LIdeal ` R )
islidl.b	|- B = ( Base ` R )
islidl.p	|- .+ = ( +g ` R )
islidl.t	|- .x. = ( .r ` R )

Assertion

Ref	Expression
islidlm	|- ( I e. U <-> ( I C_ B /\ E. j j e. I /\ A. x e. B A. a e. I A. b e. I ( ( x .x. a ) .+ b ) e. I ) )

Distinct variable groups:   x, B   I, a, b, j, x   R, a, b, x

Allowed substitution hints:   B( j, a, b)   .+ ( x, j, a, b)   R( j)   .x. ( x, j, a, b)   U( x, j, a, b)

Proof of Theorem islidlm

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		islidl.s	. . 3 |- U = (LIdeal ` R )
2	1	lidlmex 13788	. 2 |- ( I e. U -> R e. _V )
3		eleq1w 2250	. . . . . 6 |- ( j = k -> ( j e. I <-> k e. I ) )
4	3	cbvexv 1930	. . . . 5 |- ( E. j j e. I <-> E. k k e. I )
5		ssel 3164	. . . . . . 7 |- ( I C_ B -> ( k e. I -> k e. B ) )
6		islidl.b	. . . . . . . 8 |- B = ( Base ` R )
7	6	basmex 12570	. . . . . . 7 |- ( k e. B -> R e. _V )
8	5, 7	syl6 33	. . . . . 6 |- ( I C_ B -> ( k e. I -> R e. _V ) )
9	8	exlimdv 1830	. . . . 5 |- ( I C_ B -> ( E. k k e. I -> R e. _V ) )
10	4, 9	biimtrid 152	. . . 4 |- ( I C_ B -> ( E. j j e. I -> R e. _V ) )
11	10	imp 124	. . 3 |- ( ( I C_ B /\ E. j j e. I ) -> R e. _V )
12	11	3adant3 1019	. 2 |- ( ( I C_ B /\ E. j j e. I /\ A. x e. B A. a e. I A. b e. I ( ( x .x. a ) .+ b ) e. I ) -> R e. _V )
13		eqid 2189	. . . 4 |- (Scalar ` (ringLMod ` R ) ) = (Scalar ` (ringLMod ` R ) )
14		eqid 2189	. . . 4 |- ( Base ` (Scalar ` (ringLMod ` R ) ) ) = ( Base ` (Scalar ` (ringLMod ` R ) ) )
15		eqid 2189	. . . 4 |- ( Base ` (ringLMod ` R ) ) = ( Base ` (ringLMod ` R ) )
16		eqid 2189	. . . 4 |- ( +g ` (ringLMod ` R ) ) = ( +g ` (ringLMod ` R ) )
17		eqid 2189	. . . 4 |- ( .s ` (ringLMod ` R ) ) = ( .s ` (ringLMod ` R ) )
18		eqid 2189	. . . 4 |- ( LSubSp ` (ringLMod ` R ) ) = ( LSubSp ` (ringLMod ` R ) )
19	13, 14, 15, 16, 17, 18	islssm 13670	. . 3 |- ( I e. ( LSubSp ` (ringLMod ` R ) ) <-> ( I C_ ( Base ` (ringLMod ` R ) ) /\ E. j j e. I /\ A. x e. ( Base ` (Scalar ` (ringLMod ` R ) ) ) A. a e. I A. b e. I ( ( x ( .s ` (ringLMod ` R ) ) a ) ( +g ` (ringLMod ` R ) ) b ) e. I ) )
20		lidlvalg 13784	. . . . . 6 |- ( R e. _V -> (LIdeal ` R ) = ( LSubSp ` (ringLMod ` R ) ) )
21	1, 20	eqtrid 2234	. . . . 5 |- ( R e. _V -> U = ( LSubSp ` (ringLMod ` R ) ) )
22	21	eleq2d 2259	. . . 4 |- ( R e. _V -> ( I e. U <-> I e. ( LSubSp ` (ringLMod ` R ) ) ) )
23		rlmbasg 13768	. . . . . . 7 |- ( R e. _V -> ( Base ` R ) = ( Base ` (ringLMod ` R ) ) )
24	6, 23	eqtrid 2234	. . . . . 6 |- ( R e. _V -> B = ( Base ` (ringLMod ` R ) ) )
25	24	sseq2d 3200	. . . . 5 |- ( R e. _V -> ( I C_ B <-> I C_ ( Base ` (ringLMod ` R ) ) ) )
26		rlmscabas 13773	. . . . . . 7 |- ( R e. _V -> ( Base ` R ) = ( Base ` (Scalar ` (ringLMod ` R ) ) ) )
27	6, 26	eqtrid 2234	. . . . . 6 |- ( R e. _V -> B = ( Base ` (Scalar ` (ringLMod ` R ) ) ) )
28		islidl.p	. . . . . . . . . 10 |- .+ = ( +g ` R )
29		rlmplusgg 13769	. . . . . . . . . 10 |- ( R e. _V -> ( +g ` R ) = ( +g ` (ringLMod ` R ) ) )
30	28, 29	eqtrid 2234	. . . . . . . . 9 |- ( R e. _V -> .+ = ( +g ` (ringLMod ` R ) ) )
31		islidl.t	. . . . . . . . . . 11 |- .x. = ( .r ` R )
32		rlmvscag 13774	. . . . . . . . . . 11 |- ( R e. _V -> ( .r ` R ) = ( .s ` (ringLMod ` R ) ) )
33	31, 32	eqtrid 2234	. . . . . . . . . 10 |- ( R e. _V -> .x. = ( .s ` (ringLMod ` R ) ) )
34	33	oveqd 5912	. . . . . . . . 9 |- ( R e. _V -> ( x .x. a ) = ( x ( .s ` (ringLMod ` R ) ) a ) )
35		eqidd 2190	. . . . . . . . 9 |- ( R e. _V -> b = b )
36	30, 34, 35	oveq123d 5916	. . . . . . . 8 |- ( R e. _V -> ( ( x .x. a ) .+ b ) = ( ( x ( .s ` (ringLMod ` R ) ) a ) ( +g ` (ringLMod ` R ) ) b ) )
37	36	eleq1d 2258	. . . . . . 7 |- ( R e. _V -> ( ( ( x .x. a ) .+ b ) e. I <-> ( ( x ( .s ` (ringLMod ` R ) ) a ) ( +g ` (ringLMod ` R ) ) b ) e. I ) )
38	37	2ralbidv 2514	. . . . . 6 |- ( R e. _V -> ( A. a e. I A. b e. I ( ( x .x. a ) .+ b ) e. I <-> A. a e. I A. b e. I ( ( x ( .s ` (ringLMod ` R ) ) a ) ( +g ` (ringLMod ` R ) ) b ) e. I ) )
39	27, 38	raleqbidv 2698	. . . . 5 |- ( R e. _V -> ( A. x e. B A. a e. I A. b e. I ( ( x .x. a ) .+ b ) e. I <-> A. x e. ( Base ` (Scalar ` (ringLMod ` R ) ) ) A. a e. I A. b e. I ( ( x ( .s ` (ringLMod ` R ) ) a ) ( +g ` (ringLMod ` R ) ) b ) e. I ) )
40	25, 39	3anbi13d 1325	. . . 4 |- ( R e. _V -> ( ( I C_ B /\ E. j j e. I /\ A. x e. B A. a e. I A. b e. I ( ( x .x. a ) .+ b ) e. I ) <-> ( I C_ ( Base ` (ringLMod ` R ) ) /\ E. j j e. I /\ A. x e. ( Base ` (Scalar ` (ringLMod ` R ) ) ) A. a e. I A. b e. I ( ( x ( .s ` (ringLMod ` R ) ) a ) ( +g ` (ringLMod ` R ) ) b ) e. I ) ) )
41	22, 40	bibi12d 235	. . 3 |- ( R e. _V -> ( ( I e. U <-> ( I C_ B /\ E. j j e. I /\ A. x e. B A. a e. I A. b e. I ( ( x .x. a ) .+ b ) e. I ) ) <-> ( I e. ( LSubSp ` (ringLMod ` R ) ) <-> ( I C_ ( Base ` (ringLMod ` R ) ) /\ E. j j e. I /\ A. x e. ( Base ` (Scalar ` (ringLMod ` R ) ) ) A. a e. I A. b e. I ( ( x ( .s ` (ringLMod ` R ) ) a ) ( +g ` (ringLMod ` R ) ) b ) e. I ) ) ) )
42	19, 41	mpbiri 168	. 2 |- ( R e. _V -> ( I e. U <-> ( I C_ B /\ E. j j e. I /\ A. x e. B A. a e. I A. b e. I ( ( x .x. a ) .+ b ) e. I ) ) )
43	2, 12, 42	pm5.21nii 705	1 |- ( I e. U <-> ( I C_ B /\ E. j j e. I /\ A. x e. B A. a e. I A. b e. I ( ( x .x. a ) .+ b ) e. I ) )

Syntax hints:   <-> wb 105   /\ w3a 980   = wceq 1364   E.wex 1503   e. wcel 2160   A.wral 2468   _Vcvv 2752   C_ wss 3144   ` cfv 5235  (class class class)co 5895   Basecbs 12511   +g cplusg 12586   .rcmulr 12587  Scalarcsca 12589   .scvsca 12590   LSubSpclss 13665  ringLModcrglmod 13747  LIdealclidl 13780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-addcom 7940  ax-addass 7942  ax-i2m1 7945  ax-0lt1 7946  ax-0id 7948  ax-rnegex 7949  ax-pre-ltirr 7952  ax-pre-lttrn 7954  ax-pre-ltadd 7956

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-pnf 8023  df-mnf 8024  df-ltxr 8026  df-inn 8949  df-2 9007  df-3 9008  df-4 9009  df-5 9010  df-6 9011  df-7 9012  df-8 9013  df-ndx 12514  df-slot 12515  df-base 12517  df-sets 12518  df-iress 12519  df-plusg 12599  df-mulr 12600  df-sca 12602  df-vsca 12603  df-ip 12604  df-lssm 13666  df-sra 13748  df-rgmod 13749  df-lidl 13782

This theorem is referenced by:  rnglidlmcl  13793  dflidl2rng  13794