Theorem 2idlval 14460

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 14459	. . 3 |- ( x e. T -> R e. _V )
3		elinel1 3390	. . . 4 |- ( x e. ( I i^i J ) -> x e. I )
4		2idlval.i	. . . . 5 |- I = (LIdeal ` R )
5	4	lidlmex 14433	. . . 4 |- ( x e. I -> R e. _V )
6	3, 5	syl 14	. . 3 |- ( x e. ( I i^i J ) -> R e. _V )
7		lidlex 14431	. . . . . . . 8 |- ( R e. _V -> (LIdeal ` R ) e. _V )
8	4, 7	eqeltrid 2316	. . . . . . 7 |- ( R e. _V -> I e. _V )
9		inex1g 4219	. . . . . . 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 5626	. . . . . . . . 9 |- ( r = R -> (LIdeal ` r ) = (LIdeal ` R ) )
12	11, 4	eqtr4di 2280	. . . . . . . 8 |- ( r = R -> (LIdeal ` r ) = I )
13		fveq2 5626	. . . . . . . . . . 11 |- ( r = R -> (oppr ` r ) = (oppr ` R ) )
14		2idlval.o	. . . . . . . . . . 11 |- O = (oppr ` R )
15	13, 14	eqtr4di 2280	. . . . . . . . . 10 |- ( r = R -> (oppr ` r ) = O )
16	15	fveq2d 5630	. . . . . . . . 9 |- ( r = R -> (LIdeal ` (oppr ` r ) ) = (LIdeal ` O ) )
17		2idlval.j	. . . . . . . . 9 |- J = (LIdeal ` O )
18	16, 17	eqtr4di 2280	. . . . . . . 8 |- ( r = R -> (LIdeal ` (oppr ` r ) ) = J )
19	12, 18	ineq12d 3406	. . . . . . 7 |- ( r = R -> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) = ( I i^i J ) )
20		df-2idl 14458	. . . . . . 7 |- 2Ideal = ( r e. _V |-> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) )
21	19, 20	fvmptg 5709	. . . . . 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 2274	. . . 4 |- ( R e. _V -> T = ( I i^i J ) )
24	23	eleq2d 2299	. . 3 |- ( R e. _V -> ( x e. T <-> x e. ( I i^i J ) ) )
25	2, 6, 24	pm5.21nii 709	. 2 |- ( x e. T <-> x e. ( I i^i J ) )
26	25	eqriv 2226	1 |- T = ( I i^i J )

Syntax hints:   = wceq 1395   e. wcel 2200   _Vcvv 2799   i^i cin 3196   ` cfv 5317  opp_rcoppr 14025  LIdealclidl 14425  2Idealc2idl 14457

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-1re 8089  ax-addrcl 8092

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-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-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-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-mulr 13119  df-sca 13121  df-vsca 13122  df-ip 13123  df-lssm 14311  df-sra 14393  df-rgmod 14394  df-lidl 14427  df-2idl 14458

This theorem is referenced by: (None)