Theorem 2idlvalg 14840

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
2idlvalg	|- ( R e. V -> T = ( I i^i J ) )

Proof of Theorem 2idlvalg

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2idlval.t	. 2 |- T = (2Ideal ` R )
2		df-2idl 14837	. . 3 |- 2Ideal = ( r e. _V |-> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) )
3		fveq2 5695	. . . . 5 |- ( r = R -> (LIdeal ` r ) = (LIdeal ` R ) )
4		2idlval.i	. . . . 5 |- I = (LIdeal ` R )
5	3, 4	eqtr4di 2289	. . . 4 |- ( r = R -> (LIdeal ` r ) = I )
6		fveq2 5695	. . . . . . 7 |- ( r = R -> (oppr ` r ) = (oppr ` R ) )
7		2idlval.o	. . . . . . 7 |- O = (oppr ` R )
8	6, 7	eqtr4di 2289	. . . . . 6 |- ( r = R -> (oppr ` r ) = O )
9	8	fveq2d 5699	. . . . 5 |- ( r = R -> (LIdeal ` (oppr ` r ) ) = (LIdeal ` O ) )
10		2idlval.j	. . . . 5 |- J = (LIdeal ` O )
11	9, 10	eqtr4di 2289	. . . 4 |- ( r = R -> (LIdeal ` (oppr ` r ) ) = J )
12	5, 11	ineq12d 3433	. . 3 |- ( r = R -> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) = ( I i^i J ) )
13		elex 2833	. . 3 |- ( R e. V -> R e. _V )
14		lidlex 14810	. . . . 5 |- ( R e. V -> (LIdeal ` R ) e. _V )
15	4, 14	eqeltrid 2325	. . . 4 |- ( R e. V -> I e. _V )
16		inex1g 4269	. . . 4 |- ( I e. _V -> ( I i^i J ) e. _V )
17	15, 16	syl 14	. . 3 |- ( R e. V -> ( I i^i J ) e. _V )
18	2, 12, 13, 17	fvmptd3 5799	. 2 |- ( R e. V -> (2Ideal ` R ) = ( I i^i J ) )
19	1, 18	eqtrid 2283	1 |- ( R e. V -> T = ( I i^i J ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   i^i cin 3219   ` cfv 5377  opp_rcoppr 14372  LIdealclidl 14804  2Idealc2idl 14836

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-mulr 13445  df-sca 13447  df-vsca 13448  df-ip 13449  df-lssm 14690  df-sra 14772  df-rgmod 14773  df-lidl 14806  df-2idl 14837

This theorem is used by:  2idlelb  14842  2idllidld  14843  2idlridld  14844  2idl0  14849  2idl1  14850  qus1  14863  crng2idl  14868