Theorem 2idlval 13982

Description: Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)

Hypotheses

Ref	Expression
2idlval.i	|- I = (LIdeal ` R )
2idlval.o	|- O = (oppr ` R )
2idlval.j	|- J = (LIdeal ` O )
2idlval.t	|- T = (2Ideal ` R )

Assertion

Ref	Expression
2idlval	|- T = ( I i^i J )

Proof of Theorem 2idlval

Dummy variables x r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2idlval.t	. . . 4 |- T = (2Ideal ` R )
2	1	2idlmex 13981	. . 3 |- ( x e. T -> R e. _V )
3		elinel1 3345	. . . 4 |- ( x e. ( I i^i J ) -> x e. I )
4		2idlval.i	. . . . 5 |- I = (LIdeal ` R )
5	4	lidlmex 13955	. . . 4 |- ( x e. I -> R e. _V )
6	3, 5	syl 14	. . 3 |- ( x e. ( I i^i J ) -> R e. _V )
7		lidlex 13953	. . . . . . . 8 |- ( R e. _V -> (LIdeal ` R ) e. _V )
8	4, 7	eqeltrid 2280	. . . . . . 7 |- ( R e. _V -> I e. _V )
9		inex1g 4165	. . . . . . 7 |- ( I e. _V -> ( I i^i J ) e. _V )
10	8, 9	syl 14	. . . . . 6 |- ( R e. _V -> ( I i^i J ) e. _V )
11		fveq2 5546	. . . . . . . . 9 |- ( r = R -> (LIdeal ` r ) = (LIdeal ` R ) )
12	11, 4	eqtr4di 2244	. . . . . . . 8 |- ( r = R -> (LIdeal ` r ) = I )
13		fveq2 5546	. . . . . . . . . . 11 |- ( r = R -> (oppr ` r ) = (oppr ` R ) )
14		2idlval.o	. . . . . . . . . . 11 |- O = (oppr ` R )
15	13, 14	eqtr4di 2244	. . . . . . . . . 10 |- ( r = R -> (oppr ` r ) = O )
16	15	fveq2d 5550	. . . . . . . . 9 |- ( r = R -> (LIdeal ` (oppr ` r ) ) = (LIdeal ` O ) )
17		2idlval.j	. . . . . . . . 9 |- J = (LIdeal ` O )
18	16, 17	eqtr4di 2244	. . . . . . . 8 |- ( r = R -> (LIdeal ` (oppr ` r ) ) = J )
19	12, 18	ineq12d 3361	. . . . . . 7 |- ( r = R -> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) = ( I i^i J ) )
20		df-2idl 13980	. . . . . . 7 |- 2Ideal = ( r e. _V |-> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) )
21	19, 20	fvmptg 5625	. . . . . 6 |- ( ( R e. _V /\ ( I i^i J ) e. _V ) -> (2Ideal ` R ) = ( I i^i J ) )
22	10, 21	mpdan 421	. . . . 5 |- ( R e. _V -> (2Ideal ` R ) = ( I i^i J ) )
23	1, 22	eqtrid 2238	. . . 4 |- ( R e. _V -> T = ( I i^i J ) )
24	23	eleq2d 2263	. . 3 |- ( R e. _V -> ( x e. T <-> x e. ( I i^i J ) ) )
25	2, 6, 24	pm5.21nii 705	. 2 |- ( x e. T <-> x e. ( I i^i J ) )
26	25	eqriv 2190	1 |- T = ( I i^i J )

Syntax hints:   = wceq 1364   e. wcel 2164   _Vcvv 2760   i^i cin 3152   ` cfv 5246  opp_rcoppr 13547  LIdealclidl 13947  2Idealc2idl 13979

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 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4462  ax-setind 4565  ax-cnex 7953  ax-resscn 7954  ax-1re 7956  ax-addrcl 7959

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-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-iota 5207  df-fun 5248  df-fn 5249  df-f 5250  df-f1 5251  df-fo 5252  df-f1o 5253  df-fv 5254  df-ov 5913  df-oprab 5914  df-mpo 5915  df-inn 8973  df-2 9031  df-3 9032  df-4 9033  df-5 9034  df-6 9035  df-7 9036  df-8 9037  df-ndx 12611  df-slot 12612  df-base 12614  df-sets 12615  df-iress 12616  df-mulr 12699  df-sca 12701  df-vsca 12702  df-ip 12703  df-lssm 13833  df-sra 13915  df-rgmod 13916  df-lidl 13949  df-2idl 13980

This theorem is referenced by: (None)