Theorem islidlm 13978

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 13974	. 2 |- ( I e. U -> R e. _V )
3		eleq1w 2254	. . . . . 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 3174	. . . . . . 7 |- ( I C_ B -> ( k e. I -> k e. B ) )
6		islidl.b	. . . . . . . 8 |- B = ( Base ` R )
7	6	basmex 12680	. . . . . . 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 2193	. . . 4 |- (Scalar ` (ringLMod ` R ) ) = (Scalar ` (ringLMod ` R ) )
14		eqid 2193	. . . 4 |- ( Base ` (Scalar ` (ringLMod ` R ) ) ) = ( Base ` (Scalar ` (ringLMod ` R ) ) )
15		eqid 2193	. . . 4 |- ( Base ` (ringLMod ` R ) ) = ( Base ` (ringLMod ` R ) )
16		eqid 2193	. . . 4 |- ( +g ` (ringLMod ` R ) ) = ( +g ` (ringLMod ` R ) )
17		eqid 2193	. . . 4 |- ( .s ` (ringLMod ` R ) ) = ( .s ` (ringLMod ` R ) )
18		eqid 2193	. . . 4 |- ( LSubSp ` (ringLMod ` R ) ) = ( LSubSp ` (ringLMod ` R ) )
19	13, 14, 15, 16, 17, 18	islssm 13856	. . 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 13970	. . . . . 6 |- ( R e. _V -> (LIdeal ` R ) = ( LSubSp ` (ringLMod ` R ) ) )
21	1, 20	eqtrid 2238	. . . . 5 |- ( R e. _V -> U = ( LSubSp ` (ringLMod ` R ) ) )
22	21	eleq2d 2263	. . . 4 |- ( R e. _V -> ( I e. U <-> I e. ( LSubSp ` (ringLMod ` R ) ) ) )
23		rlmbasg 13954	. . . . . . 7 |- ( R e. _V -> ( Base ` R ) = ( Base ` (ringLMod ` R ) ) )
24	6, 23	eqtrid 2238	. . . . . 6 |- ( R e. _V -> B = ( Base ` (ringLMod ` R ) ) )
25	24	sseq2d 3210	. . . . 5 |- ( R e. _V -> ( I C_ B <-> I C_ ( Base ` (ringLMod ` R ) ) ) )
26		rlmscabas 13959	. . . . . . 7 |- ( R e. _V -> ( Base ` R ) = ( Base ` (Scalar ` (ringLMod ` R ) ) ) )
27	6, 26	eqtrid 2238	. . . . . 6 |- ( R e. _V -> B = ( Base ` (Scalar ` (ringLMod ` R ) ) ) )
28		islidl.p	. . . . . . . . . 10 |- .+ = ( +g ` R )
29		rlmplusgg 13955	. . . . . . . . . 10 |- ( R e. _V -> ( +g ` R ) = ( +g ` (ringLMod ` R ) ) )
30	28, 29	eqtrid 2238	. . . . . . . . 9 |- ( R e. _V -> .+ = ( +g ` (ringLMod ` R ) ) )
31		islidl.t	. . . . . . . . . . 11 |- .x. = ( .r ` R )
32		rlmvscag 13960	. . . . . . . . . . 11 |- ( R e. _V -> ( .r ` R ) = ( .s ` (ringLMod ` R ) ) )
33	31, 32	eqtrid 2238	. . . . . . . . . 10 |- ( R e. _V -> .x. = ( .s ` (ringLMod ` R ) ) )
34	33	oveqd 5936	. . . . . . . . 9 |- ( R e. _V -> ( x .x. a ) = ( x ( .s ` (ringLMod ` R ) ) a ) )
35		eqidd 2194	. . . . . . . . 9 |- ( R e. _V -> b = b )
36	30, 34, 35	oveq123d 5940	. . . . . . . 8 |- ( R e. _V -> ( ( x .x. a ) .+ b ) = ( ( x ( .s ` (ringLMod ` R ) ) a ) ( +g ` (ringLMod ` R ) ) b ) )
37	36	eleq1d 2262	. . . . . . 7 |- ( R e. _V -> ( ( ( x .x. a ) .+ b ) e. I <-> ( ( x ( .s ` (ringLMod ` R ) ) a ) ( +g ` (ringLMod ` R ) ) b ) e. I ) )
38	37	2ralbidv 2518	. . . . . 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 2706	. . . . 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 2164   A.wral 2472   _Vcvv 2760   C_ wss 3154   ` cfv 5255  (class class class)co 5919   Basecbs 12621   +g cplusg 12698   .rcmulr 12699  Scalarcsca 12701   .scvsca 12702   LSubSpclss 13851  ringLModcrglmod 13933  LIdealclidl 13966

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 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-lttrn 7988  ax-pre-ltadd 7990

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-5 9046  df-6 9047  df-7 9048  df-8 9049  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-iress 12629  df-plusg 12711  df-mulr 12712  df-sca 12714  df-vsca 12715  df-ip 12716  df-lssm 13852  df-sra 13934  df-rgmod 13935  df-lidl 13968

This theorem is referenced by:  rnglidlmcl  13979  dflidl2rng  13980