Theorem 2idlvalg 14812

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 14809	. . 3 |- 2Ideal = ( r e. _V |-> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) )
3		fveq2 5690	. . . . 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 5690	. . . . . . 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 5694	. . . . 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 14782	. . . . 5 |- ( R e. V -> (LIdeal ` R ) e. _V )
15	4, 14	eqeltrid 2325	. . . 4 |- ( R e. V -> I e. _V )
16		inex1g 4264	. . . 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 5793	. 2 |- ( R e. V -> (2Ideal ` R ) = ( I i^i J ) )
19	1, 18	eqtrid 2283	1 |- ( R e. V -> T = ( I i^i J ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   i^i cin 3219   ` cfv 5372  opp_rcoppr 14345  LIdealclidl 14776  2Idealc2idl 14808

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 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 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-mulr 13422  df-sca 13424  df-vsca 13425  df-ip 13426  df-lssm 14662  df-sra 14744  df-rgmod 14745  df-lidl 14778  df-2idl 14809

This theorem is referenced by:  2idlelb  14814  2idllidld  14815  2idlridld  14816  2idl0  14821  2idl1  14822  qus1  14835  crng2idl  14840