Theorem islidlm 14627

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 14623	. 2 |- ( I e. U -> R e. _V )
3		eleq1w 2293	. . . . . 6 |- ( j = k -> ( j e. I <-> k e. I ) )
4	3	cbvexv 1968	. . . . 5 |- ( E. j j e. I <-> E. k k e. I )
5		ssel 3232	. . . . . . 7 |- ( I C_ B -> ( k e. I -> k e. B ) )
6		islidl.b	. . . . . . . 8 |- B = ( Base ` R )
7	6	basmex 13272	. . . . . . 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 1868	. . . . 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 1044	. 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 2232	. . . 4 |- (Scalar ` (ringLMod ` R ) ) = (Scalar ` (ringLMod ` R ) )
14		eqid 2232	. . . 4 |- ( Base ` (Scalar ` (ringLMod ` R ) ) ) = ( Base ` (Scalar ` (ringLMod ` R ) ) )
15		eqid 2232	. . . 4 |- ( Base ` (ringLMod ` R ) ) = ( Base ` (ringLMod ` R ) )
16		eqid 2232	. . . 4 |- ( +g ` (ringLMod ` R ) ) = ( +g ` (ringLMod ` R ) )
17		eqid 2232	. . . 4 |- ( .s ` (ringLMod ` R ) ) = ( .s ` (ringLMod ` R ) )
18		eqid 2232	. . . 4 |- ( LSubSp ` (ringLMod ` R ) ) = ( LSubSp ` (ringLMod ` R ) )
19	13, 14, 15, 16, 17, 18	islssm 14505	. . 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 14619	. . . . . 6 |- ( R e. _V -> (LIdeal ` R ) = ( LSubSp ` (ringLMod ` R ) ) )
21	1, 20	eqtrid 2277	. . . . 5 |- ( R e. _V -> U = ( LSubSp ` (ringLMod ` R ) ) )
22	21	eleq2d 2302	. . . 4 |- ( R e. _V -> ( I e. U <-> I e. ( LSubSp ` (ringLMod ` R ) ) ) )
23		rlmbasg 14603	. . . . . . 7 |- ( R e. _V -> ( Base ` R ) = ( Base ` (ringLMod ` R ) ) )
24	6, 23	eqtrid 2277	. . . . . 6 |- ( R e. _V -> B = ( Base ` (ringLMod ` R ) ) )
25	24	sseq2d 3268	. . . . 5 |- ( R e. _V -> ( I C_ B <-> I C_ ( Base ` (ringLMod ` R ) ) ) )
26		rlmscabas 14608	. . . . . . 7 |- ( R e. _V -> ( Base ` R ) = ( Base ` (Scalar ` (ringLMod ` R ) ) ) )
27	6, 26	eqtrid 2277	. . . . . 6 |- ( R e. _V -> B = ( Base ` (Scalar ` (ringLMod ` R ) ) ) )
28		islidl.p	. . . . . . . . . 10 |- .+ = ( +g ` R )
29		rlmplusgg 14604	. . . . . . . . . 10 |- ( R e. _V -> ( +g ` R ) = ( +g ` (ringLMod ` R ) ) )
30	28, 29	eqtrid 2277	. . . . . . . . 9 |- ( R e. _V -> .+ = ( +g ` (ringLMod ` R ) ) )
31		islidl.t	. . . . . . . . . . 11 |- .x. = ( .r ` R )
32		rlmvscag 14609	. . . . . . . . . . 11 |- ( R e. _V -> ( .r ` R ) = ( .s ` (ringLMod ` R ) ) )
33	31, 32	eqtrid 2277	. . . . . . . . . 10 |- ( R e. _V -> .x. = ( .s ` (ringLMod ` R ) ) )
34	33	oveqd 6067	. . . . . . . . 9 |- ( R e. _V -> ( x .x. a ) = ( x ( .s ` (ringLMod ` R ) ) a ) )
35		eqidd 2233	. . . . . . . . 9 |- ( R e. _V -> b = b )
36	30, 34, 35	oveq123d 6071	. . . . . . . 8 |- ( R e. _V -> ( ( x .x. a ) .+ b ) = ( ( x ( .s ` (ringLMod ` R ) ) a ) ( +g ` (ringLMod ` R ) ) b ) )
37	36	eleq1d 2301	. . . . . . 7 |- ( R e. _V -> ( ( ( x .x. a ) .+ b ) e. I <-> ( ( x ( .s ` (ringLMod ` R ) ) a ) ( +g ` (ringLMod ` R ) ) b ) e. I ) )
38	37	2ralbidv 2566	. . . . . 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 2757	. . . . 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 1351	. . . 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 712	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 1005   = wceq 1398   E.wex 1541   e. wcel 2203   A.wral 2520   _Vcvv 2813   C_ wss 3211   ` cfv 5352  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   .rcmulr 13291  Scalarcsca 13293   .scvsca 13294   LSubSpclss 14500  ringLModcrglmod 14582  LIdealclidl 14615

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220  df-plusg 13303  df-mulr 13304  df-sca 13306  df-vsca 13307  df-ip 13308  df-lssm 14501  df-sra 14583  df-rgmod 14584  df-lidl 14617

This theorem is referenced by:  rnglidlmcl  14628  dflidl2rng  14629