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Theorem 2idlval 18201
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 variable  r is distinct from all other variables.
StepHypRef Expression
1 2idlval.t . 2  |-  T  =  (2Ideal `  R )
2 fveq2 5849 . . . . . 6  |-  ( r  =  R  ->  (LIdeal `  r )  =  (LIdeal `  R ) )
3 2idlval.i . . . . . 6  |-  I  =  (LIdeal `  R )
42, 3syl6eqr 2461 . . . . 5  |-  ( r  =  R  ->  (LIdeal `  r )  =  I )
5 fveq2 5849 . . . . . . . 8  |-  ( r  =  R  ->  (oppr `  r
)  =  (oppr `  R
) )
6 2idlval.o . . . . . . . 8  |-  O  =  (oppr
`  R )
75, 6syl6eqr 2461 . . . . . . 7  |-  ( r  =  R  ->  (oppr `  r
)  =  O )
87fveq2d 5853 . . . . . 6  |-  ( r  =  R  ->  (LIdeal `  (oppr
`  r ) )  =  (LIdeal `  O
) )
9 2idlval.j . . . . . 6  |-  J  =  (LIdeal `  O )
108, 9syl6eqr 2461 . . . . 5  |-  ( r  =  R  ->  (LIdeal `  (oppr
`  r ) )  =  J )
114, 10ineq12d 3642 . . . 4  |-  ( r  =  R  ->  (
(LIdeal `  r )  i^i  (LIdeal `  (oppr
`  r ) ) )  =  ( I  i^i  J ) )
12 df-2idl 18200 . . . 4  |- 2Ideal  =  ( r  e.  _V  |->  ( (LIdeal `  r )  i^i  (LIdeal `  (oppr
`  r ) ) ) )
13 fvex 5859 . . . . . 6  |-  (LIdeal `  R )  e.  _V
143, 13eqeltri 2486 . . . . 5  |-  I  e. 
_V
1514inex1 4535 . . . 4  |-  ( I  i^i  J )  e. 
_V
1611, 12, 15fvmpt 5932 . . 3  |-  ( R  e.  _V  ->  (2Ideal `  R )  =  ( I  i^i  J ) )
17 fvprc 5843 . . . 4  |-  ( -.  R  e.  _V  ->  (2Ideal `  R )  =  (/) )
18 inss1 3659 . . . . 5  |-  ( I  i^i  J )  C_  I
19 fvprc 5843 . . . . . 6  |-  ( -.  R  e.  _V  ->  (LIdeal `  R )  =  (/) )
203, 19syl5eq 2455 . . . . 5  |-  ( -.  R  e.  _V  ->  I  =  (/) )
21 sseq0 3771 . . . . 5  |-  ( ( ( I  i^i  J
)  C_  I  /\  I  =  (/) )  -> 
( I  i^i  J
)  =  (/) )
2218, 20, 21sylancr 661 . . . 4  |-  ( -.  R  e.  _V  ->  ( I  i^i  J )  =  (/) )
2317, 22eqtr4d 2446 . . 3  |-  ( -.  R  e.  _V  ->  (2Ideal `  R )  =  ( I  i^i  J ) )
2416, 23pm2.61i 164 . 2  |-  (2Ideal `  R )  =  ( I  i^i  J )
251, 24eqtri 2431 1  |-  T  =  ( I  i^i  J
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1405    e. wcel 1842   _Vcvv 3059    i^i cin 3413    C_ wss 3414   (/)c0 3738   ` cfv 5569  opprcoppr 17591  LIdealclidl 18136  2Idealc2idl 18199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577  df-2idl 18200
This theorem is referenced by:  2idlcpbl  18202  qus1  18203  qusrhm  18205  crng2idl  18207
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