Theorem islidlm 14316

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 14312	. 2 |- ( I e. U -> R e. _V )
3		eleq1w 2267	. . . . . 6 |- ( j = k -> ( j e. I <-> k e. I ) )
4	3	cbvexv 1943	. . . . 5 |- ( E. j j e. I <-> E. k k e. I )
5		ssel 3191	. . . . . . 7 |- ( I C_ B -> ( k e. I -> k e. B ) )
6		islidl.b	. . . . . . . 8 |- B = ( Base ` R )
7	6	basmex 12966	. . . . . . 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 1843	. . . . 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 1020	. 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 2206	. . . 4 |- (Scalar ` (ringLMod ` R ) ) = (Scalar ` (ringLMod ` R ) )
14		eqid 2206	. . . 4 |- ( Base ` (Scalar ` (ringLMod ` R ) ) ) = ( Base ` (Scalar ` (ringLMod ` R ) ) )
15		eqid 2206	. . . 4 |- ( Base ` (ringLMod ` R ) ) = ( Base ` (ringLMod ` R ) )
16		eqid 2206	. . . 4 |- ( +g ` (ringLMod ` R ) ) = ( +g ` (ringLMod ` R ) )
17		eqid 2206	. . . 4 |- ( .s ` (ringLMod ` R ) ) = ( .s ` (ringLMod ` R ) )
18		eqid 2206	. . . 4 |- ( LSubSp ` (ringLMod ` R ) ) = ( LSubSp ` (ringLMod ` R ) )
19	13, 14, 15, 16, 17, 18	islssm 14194	. . 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 14308	. . . . . 6 |- ( R e. _V -> (LIdeal ` R ) = ( LSubSp ` (ringLMod ` R ) ) )
21	1, 20	eqtrid 2251	. . . . 5 |- ( R e. _V -> U = ( LSubSp ` (ringLMod ` R ) ) )
22	21	eleq2d 2276	. . . 4 |- ( R e. _V -> ( I e. U <-> I e. ( LSubSp ` (ringLMod ` R ) ) ) )
23		rlmbasg 14292	. . . . . . 7 |- ( R e. _V -> ( Base ` R ) = ( Base ` (ringLMod ` R ) ) )
24	6, 23	eqtrid 2251	. . . . . 6 |- ( R e. _V -> B = ( Base ` (ringLMod ` R ) ) )
25	24	sseq2d 3227	. . . . 5 |- ( R e. _V -> ( I C_ B <-> I C_ ( Base ` (ringLMod ` R ) ) ) )
26		rlmscabas 14297	. . . . . . 7 |- ( R e. _V -> ( Base ` R ) = ( Base ` (Scalar ` (ringLMod ` R ) ) ) )
27	6, 26	eqtrid 2251	. . . . . 6 |- ( R e. _V -> B = ( Base ` (Scalar ` (ringLMod ` R ) ) ) )
28		islidl.p	. . . . . . . . . 10 |- .+ = ( +g ` R )
29		rlmplusgg 14293	. . . . . . . . . 10 |- ( R e. _V -> ( +g ` R ) = ( +g ` (ringLMod ` R ) ) )
30	28, 29	eqtrid 2251	. . . . . . . . 9 |- ( R e. _V -> .+ = ( +g ` (ringLMod ` R ) ) )
31		islidl.t	. . . . . . . . . . 11 |- .x. = ( .r ` R )
32		rlmvscag 14298	. . . . . . . . . . 11 |- ( R e. _V -> ( .r ` R ) = ( .s ` (ringLMod ` R ) ) )
33	31, 32	eqtrid 2251	. . . . . . . . . 10 |- ( R e. _V -> .x. = ( .s ` (ringLMod ` R ) ) )
34	33	oveqd 5974	. . . . . . . . 9 |- ( R e. _V -> ( x .x. a ) = ( x ( .s ` (ringLMod ` R ) ) a ) )
35		eqidd 2207	. . . . . . . . 9 |- ( R e. _V -> b = b )
36	30, 34, 35	oveq123d 5978	. . . . . . . 8 |- ( R e. _V -> ( ( x .x. a ) .+ b ) = ( ( x ( .s ` (ringLMod ` R ) ) a ) ( +g ` (ringLMod ` R ) ) b ) )
37	36	eleq1d 2275	. . . . . . 7 |- ( R e. _V -> ( ( ( x .x. a ) .+ b ) e. I <-> ( ( x ( .s ` (ringLMod ` R ) ) a ) ( +g ` (ringLMod ` R ) ) b ) e. I ) )
38	37	2ralbidv 2531	. . . . . 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 2719	. . . . 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 1327	. . . 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 706	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 981   = wceq 1373   E.wex 1516   e. wcel 2177   A.wral 2485   _Vcvv 2773   C_ wss 3170   ` cfv 5280  (class class class)co 5957   Basecbs 12907   +g cplusg 12984   .rcmulr 12985  Scalarcsca 12987   .scvsca 12988   LSubSpclss 14189  ringLModcrglmod 14271  LIdealclidl 14304

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-i2m1 8050  ax-0lt1 8051  ax-0id 8053  ax-rnegex 8054  ax-pre-ltirr 8057  ax-pre-lttrn 8059  ax-pre-ltadd 8061

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-ov 5960  df-oprab 5961  df-mpo 5962  df-pnf 8129  df-mnf 8130  df-ltxr 8132  df-inn 9057  df-2 9115  df-3 9116  df-4 9117  df-5 9118  df-6 9119  df-7 9120  df-8 9121  df-ndx 12910  df-slot 12911  df-base 12913  df-sets 12914  df-iress 12915  df-plusg 12997  df-mulr 12998  df-sca 13000  df-vsca 13001  df-ip 13002  df-lssm 14190  df-sra 14272  df-rgmod 14273  df-lidl 14306

This theorem is referenced by:  rnglidlmcl  14317  dflidl2rng  14318