Theorem islidlm 14437

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 14433	. 2 |- ( I e. U -> R e. _V )
3		eleq1w 2290	. . . . . 6 |- ( j = k -> ( j e. I <-> k e. I ) )
4	3	cbvexv 1965	. . . . 5 |- ( E. j j e. I <-> E. k k e. I )
5		ssel 3218	. . . . . . 7 |- ( I C_ B -> ( k e. I -> k e. B ) )
6		islidl.b	. . . . . . . 8 |- B = ( Base ` R )
7	6	basmex 13087	. . . . . . 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 1865	. . . . 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 1041	. 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 2229	. . . 4 |- (Scalar ` (ringLMod ` R ) ) = (Scalar ` (ringLMod ` R ) )
14		eqid 2229	. . . 4 |- ( Base ` (Scalar ` (ringLMod ` R ) ) ) = ( Base ` (Scalar ` (ringLMod ` R ) ) )
15		eqid 2229	. . . 4 |- ( Base ` (ringLMod ` R ) ) = ( Base ` (ringLMod ` R ) )
16		eqid 2229	. . . 4 |- ( +g ` (ringLMod ` R ) ) = ( +g ` (ringLMod ` R ) )
17		eqid 2229	. . . 4 |- ( .s ` (ringLMod ` R ) ) = ( .s ` (ringLMod ` R ) )
18		eqid 2229	. . . 4 |- ( LSubSp ` (ringLMod ` R ) ) = ( LSubSp ` (ringLMod ` R ) )
19	13, 14, 15, 16, 17, 18	islssm 14315	. . 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 14429	. . . . . 6 |- ( R e. _V -> (LIdeal ` R ) = ( LSubSp ` (ringLMod ` R ) ) )
21	1, 20	eqtrid 2274	. . . . 5 |- ( R e. _V -> U = ( LSubSp ` (ringLMod ` R ) ) )
22	21	eleq2d 2299	. . . 4 |- ( R e. _V -> ( I e. U <-> I e. ( LSubSp ` (ringLMod ` R ) ) ) )
23		rlmbasg 14413	. . . . . . 7 |- ( R e. _V -> ( Base ` R ) = ( Base ` (ringLMod ` R ) ) )
24	6, 23	eqtrid 2274	. . . . . 6 |- ( R e. _V -> B = ( Base ` (ringLMod ` R ) ) )
25	24	sseq2d 3254	. . . . 5 |- ( R e. _V -> ( I C_ B <-> I C_ ( Base ` (ringLMod ` R ) ) ) )
26		rlmscabas 14418	. . . . . . 7 |- ( R e. _V -> ( Base ` R ) = ( Base ` (Scalar ` (ringLMod ` R ) ) ) )
27	6, 26	eqtrid 2274	. . . . . 6 |- ( R e. _V -> B = ( Base ` (Scalar ` (ringLMod ` R ) ) ) )
28		islidl.p	. . . . . . . . . 10 |- .+ = ( +g ` R )
29		rlmplusgg 14414	. . . . . . . . . 10 |- ( R e. _V -> ( +g ` R ) = ( +g ` (ringLMod ` R ) ) )
30	28, 29	eqtrid 2274	. . . . . . . . 9 |- ( R e. _V -> .+ = ( +g ` (ringLMod ` R ) ) )
31		islidl.t	. . . . . . . . . . 11 |- .x. = ( .r ` R )
32		rlmvscag 14419	. . . . . . . . . . 11 |- ( R e. _V -> ( .r ` R ) = ( .s ` (ringLMod ` R ) ) )
33	31, 32	eqtrid 2274	. . . . . . . . . 10 |- ( R e. _V -> .x. = ( .s ` (ringLMod ` R ) ) )
34	33	oveqd 6017	. . . . . . . . 9 |- ( R e. _V -> ( x .x. a ) = ( x ( .s ` (ringLMod ` R ) ) a ) )
35		eqidd 2230	. . . . . . . . 9 |- ( R e. _V -> b = b )
36	30, 34, 35	oveq123d 6021	. . . . . . . 8 |- ( R e. _V -> ( ( x .x. a ) .+ b ) = ( ( x ( .s ` (ringLMod ` R ) ) a ) ( +g ` (ringLMod ` R ) ) b ) )
37	36	eleq1d 2298	. . . . . . 7 |- ( R e. _V -> ( ( ( x .x. a ) .+ b ) e. I <-> ( ( x ( .s ` (ringLMod ` R ) ) a ) ( +g ` (ringLMod ` R ) ) b ) e. I ) )
38	37	2ralbidv 2554	. . . . . 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 2744	. . . . 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 1348	. . . 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 709	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 1002   = wceq 1395   E.wex 1538   e. wcel 2200   A.wral 2508   _Vcvv 2799   C_ wss 3197   ` cfv 5317  (class class class)co 6000   Basecbs 13027   +g cplusg 13105   .rcmulr 13106  Scalarcsca 13108   .scvsca 13109   LSubSpclss 14310  ringLModcrglmod 14392  LIdealclidl 14425

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-pre-ltirr 8107  ax-pre-lttrn 8109  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-ltxr 8182  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-5 9168  df-6 9169  df-7 9170  df-8 9171  df-ndx 13030  df-slot 13031  df-base 13033  df-sets 13034  df-iress 13035  df-plusg 13118  df-mulr 13119  df-sca 13121  df-vsca 13122  df-ip 13123  df-lssm 14311  df-sra 14393  df-rgmod 14394  df-lidl 14427

This theorem is referenced by:  rnglidlmcl  14438  dflidl2rng  14439