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Theorem 2idlval 14034
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.
StepHypRef Expression
1 2idlval.t . . . 4  |-  T  =  (2Ideal `  R )
212idlmex 14033 . . 3  |-  ( x  e.  T  ->  R  e.  _V )
3 elinel1 3349 . . . 4  |-  ( x  e.  ( I  i^i 
J )  ->  x  e.  I )
4 2idlval.i . . . . 5  |-  I  =  (LIdeal `  R )
54lidlmex 14007 . . . 4  |-  ( x  e.  I  ->  R  e.  _V )
63, 5syl 14 . . 3  |-  ( x  e.  ( I  i^i 
J )  ->  R  e.  _V )
7 lidlex 14005 . . . . . . . 8  |-  ( R  e.  _V  ->  (LIdeal `  R )  e.  _V )
84, 7eqeltrid 2283 . . . . . . 7  |-  ( R  e.  _V  ->  I  e.  _V )
9 inex1g 4169 . . . . . . 7  |-  ( I  e.  _V  ->  (
I  i^i  J )  e.  _V )
108, 9syl 14 . . . . . 6  |-  ( R  e.  _V  ->  (
I  i^i  J )  e.  _V )
11 fveq2 5558 . . . . . . . . 9  |-  ( r  =  R  ->  (LIdeal `  r )  =  (LIdeal `  R ) )
1211, 4eqtr4di 2247 . . . . . . . 8  |-  ( r  =  R  ->  (LIdeal `  r )  =  I )
13 fveq2 5558 . . . . . . . . . . 11  |-  ( r  =  R  ->  (oppr `  r
)  =  (oppr `  R
) )
14 2idlval.o . . . . . . . . . . 11  |-  O  =  (oppr
`  R )
1513, 14eqtr4di 2247 . . . . . . . . . 10  |-  ( r  =  R  ->  (oppr `  r
)  =  O )
1615fveq2d 5562 . . . . . . . . 9  |-  ( r  =  R  ->  (LIdeal `  (oppr
`  r ) )  =  (LIdeal `  O
) )
17 2idlval.j . . . . . . . . 9  |-  J  =  (LIdeal `  O )
1816, 17eqtr4di 2247 . . . . . . . 8  |-  ( r  =  R  ->  (LIdeal `  (oppr
`  r ) )  =  J )
1912, 18ineq12d 3365 . . . . . . 7  |-  ( r  =  R  ->  (
(LIdeal `  r )  i^i  (LIdeal `  (oppr
`  r ) ) )  =  ( I  i^i  J ) )
20 df-2idl 14032 . . . . . . 7  |- 2Ideal  =  ( r  e.  _V  |->  ( (LIdeal `  r )  i^i  (LIdeal `  (oppr
`  r ) ) ) )
2119, 20fvmptg 5637 . . . . . 6  |-  ( ( R  e.  _V  /\  ( I  i^i  J )  e.  _V )  -> 
(2Ideal `  R )  =  ( I  i^i 
J ) )
2210, 21mpdan 421 . . . . 5  |-  ( R  e.  _V  ->  (2Ideal `  R )  =  ( I  i^i  J ) )
231, 22eqtrid 2241 . . . 4  |-  ( R  e.  _V  ->  T  =  ( I  i^i 
J ) )
2423eleq2d 2266 . . 3  |-  ( R  e.  _V  ->  (
x  e.  T  <->  x  e.  ( I  i^i  J ) ) )
252, 6, 24pm5.21nii 705 . 2  |-  ( x  e.  T  <->  x  e.  ( I  i^i  J ) )
2625eqriv 2193 1  |-  T  =  ( I  i^i  J
)
Colors of variables: wff set class
Syntax hints:    = wceq 1364    e. wcel 2167   _Vcvv 2763    i^i cin 3156   ` cfv 5258  opprcoppr 13599  LIdealclidl 13999  2Idealc2idl 14031
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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7968  ax-resscn 7969  ax-1re 7971  ax-addrcl 7974
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-inn 8988  df-2 9046  df-3 9047  df-4 9048  df-5 9049  df-6 9050  df-7 9051  df-8 9052  df-ndx 12657  df-slot 12658  df-base 12660  df-sets 12661  df-iress 12662  df-mulr 12745  df-sca 12747  df-vsca 12748  df-ip 12749  df-lssm 13885  df-sra 13967  df-rgmod 13968  df-lidl 14001  df-2idl 14032
This theorem is referenced by: (None)
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