Theorem 2idlval 14515

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 14514	. . 3 |- ( x e. T -> R e. _V )
3		elinel1 3393	. . . 4 |- ( x e. ( I i^i J ) -> x e. I )
4		2idlval.i	. . . . 5 |- I = (LIdeal ` R )
5	4	lidlmex 14488	. . . 4 |- ( x e. I -> R e. _V )
6	3, 5	syl 14	. . 3 |- ( x e. ( I i^i J ) -> R e. _V )
7		lidlex 14486	. . . . . . . 8 |- ( R e. _V -> (LIdeal ` R ) e. _V )
8	4, 7	eqeltrid 2318	. . . . . . 7 |- ( R e. _V -> I e. _V )
9		inex1g 4225	. . . . . . 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 5639	. . . . . . . . 9 |- ( r = R -> (LIdeal ` r ) = (LIdeal ` R ) )
12	11, 4	eqtr4di 2282	. . . . . . . 8 |- ( r = R -> (LIdeal ` r ) = I )
13		fveq2 5639	. . . . . . . . . . 11 |- ( r = R -> (oppr ` r ) = (oppr ` R ) )
14		2idlval.o	. . . . . . . . . . 11 |- O = (oppr ` R )
15	13, 14	eqtr4di 2282	. . . . . . . . . 10 |- ( r = R -> (oppr ` r ) = O )
16	15	fveq2d 5643	. . . . . . . . 9 |- ( r = R -> (LIdeal ` (oppr ` r ) ) = (LIdeal ` O ) )
17		2idlval.j	. . . . . . . . 9 |- J = (LIdeal ` O )
18	16, 17	eqtr4di 2282	. . . . . . . 8 |- ( r = R -> (LIdeal ` (oppr ` r ) ) = J )
19	12, 18	ineq12d 3409	. . . . . . 7 |- ( r = R -> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) = ( I i^i J ) )
20		df-2idl 14513	. . . . . . 7 |- 2Ideal = ( r e. _V |-> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) )
21	19, 20	fvmptg 5722	. . . . . 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 2276	. . . 4 |- ( R e. _V -> T = ( I i^i J ) )
24	23	eleq2d 2301	. . 3 |- ( R e. _V -> ( x e. T <-> x e. ( I i^i J ) ) )
25	2, 6, 24	pm5.21nii 711	. 2 |- ( x e. T <-> x e. ( I i^i J ) )
26	25	eqriv 2228	1 |- T = ( I i^i J )

Syntax hints:   = wceq 1397   e. wcel 2202   _Vcvv 2802   i^i cin 3199   ` cfv 5326  opp_rcoppr 14079  LIdealclidl 14480  2Idealc2idl 14512

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-iress 13089  df-mulr 13173  df-sca 13175  df-vsca 13176  df-ip 13177  df-lssm 14366  df-sra 14448  df-rgmod 14449  df-lidl 14482  df-2idl 14513

This theorem is referenced by: (None)