Theorem 2idlval 14650

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 14649	. . 3 |- ( x e. T -> R e. _V )
3		elinel1 3405	. . . 4 |- ( x e. ( I i^i J ) -> x e. I )
4		2idlval.i	. . . . 5 |- I = (LIdeal ` R )
5	4	lidlmex 14623	. . . 4 |- ( x e. I -> R e. _V )
6	3, 5	syl 14	. . 3 |- ( x e. ( I i^i J ) -> R e. _V )
7		lidlex 14621	. . . . . . . 8 |- ( R e. _V -> (LIdeal ` R ) e. _V )
8	4, 7	eqeltrid 2319	. . . . . . 7 |- ( R e. _V -> I e. _V )
9		inex1g 4246	. . . . . . 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 5670	. . . . . . . . 9 |- ( r = R -> (LIdeal ` r ) = (LIdeal ` R ) )
12	11, 4	eqtr4di 2283	. . . . . . . 8 |- ( r = R -> (LIdeal ` r ) = I )
13		fveq2 5670	. . . . . . . . . . 11 |- ( r = R -> (oppr ` r ) = (oppr ` R ) )
14		2idlval.o	. . . . . . . . . . 11 |- O = (oppr ` R )
15	13, 14	eqtr4di 2283	. . . . . . . . . 10 |- ( r = R -> (oppr ` r ) = O )
16	15	fveq2d 5674	. . . . . . . . 9 |- ( r = R -> (LIdeal ` (oppr ` r ) ) = (LIdeal ` O ) )
17		2idlval.j	. . . . . . . . 9 |- J = (LIdeal ` O )
18	16, 17	eqtr4di 2283	. . . . . . . 8 |- ( r = R -> (LIdeal ` (oppr ` r ) ) = J )
19	12, 18	ineq12d 3423	. . . . . . 7 |- ( r = R -> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) = ( I i^i J ) )
20		df-2idl 14648	. . . . . . 7 |- 2Ideal = ( r e. _V |-> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) )
21	19, 20	fvmptg 5753	. . . . . 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 2277	. . . 4 |- ( R e. _V -> T = ( I i^i J ) )
24	23	eleq2d 2302	. . 3 |- ( R e. _V -> ( x e. T <-> x e. ( I i^i J ) ) )
25	2, 6, 24	pm5.21nii 712	. 2 |- ( x e. T <-> x e. ( I i^i J ) )
26	25	eqriv 2229	1 |- T = ( I i^i J )

Syntax hints:   = wceq 1398   e. wcel 2203   _Vcvv 2813   i^i cin 3210   ` cfv 5352  opp_rcoppr 14211  LIdealclidl 14615  2Idealc2idl 14647

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-1re 8221  ax-addrcl 8224

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-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-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-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-mulr 13304  df-sca 13306  df-vsca 13307  df-ip 13308  df-lssm 14501  df-sra 14583  df-rgmod 14584  df-lidl 14617  df-2idl 14648

This theorem is referenced by: (None)