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Theorem 2idlval 17313
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 5689 . . . . . 6  |-  ( r  =  R  ->  (LIdeal `  r )  =  (LIdeal `  R ) )
3 2idlval.i . . . . . 6  |-  I  =  (LIdeal `  R )
42, 3syl6eqr 2491 . . . . 5  |-  ( r  =  R  ->  (LIdeal `  r )  =  I )
5 fveq2 5689 . . . . . . . 8  |-  ( r  =  R  ->  (oppr `  r
)  =  (oppr `  R
) )
6 2idlval.o . . . . . . . 8  |-  O  =  (oppr
`  R )
75, 6syl6eqr 2491 . . . . . . 7  |-  ( r  =  R  ->  (oppr `  r
)  =  O )
87fveq2d 5693 . . . . . 6  |-  ( r  =  R  ->  (LIdeal `  (oppr
`  r ) )  =  (LIdeal `  O
) )
9 2idlval.j . . . . . 6  |-  J  =  (LIdeal `  O )
108, 9syl6eqr 2491 . . . . 5  |-  ( r  =  R  ->  (LIdeal `  (oppr
`  r ) )  =  J )
114, 10ineq12d 3551 . . . 4  |-  ( r  =  R  ->  (
(LIdeal `  r )  i^i  (LIdeal `  (oppr
`  r ) ) )  =  ( I  i^i  J ) )
12 df-2idl 17312 . . . 4  |- 2Ideal  =  ( r  e.  _V  |->  ( (LIdeal `  r )  i^i  (LIdeal `  (oppr
`  r ) ) ) )
13 fvex 5699 . . . . . 6  |-  (LIdeal `  R )  e.  _V
143, 13eqeltri 2511 . . . . 5  |-  I  e. 
_V
1514inex1 4431 . . . 4  |-  ( I  i^i  J )  e. 
_V
1611, 12, 15fvmpt 5772 . . 3  |-  ( R  e.  _V  ->  (2Ideal `  R )  =  ( I  i^i  J ) )
17 fvprc 5683 . . . 4  |-  ( -.  R  e.  _V  ->  (2Ideal `  R )  =  (/) )
18 inss1 3568 . . . . 5  |-  ( I  i^i  J )  C_  I
19 fvprc 5683 . . . . . 6  |-  ( -.  R  e.  _V  ->  (LIdeal `  R )  =  (/) )
203, 19syl5eq 2485 . . . . 5  |-  ( -.  R  e.  _V  ->  I  =  (/) )
21 sseq0 3667 . . . . 5  |-  ( ( ( I  i^i  J
)  C_  I  /\  I  =  (/) )  -> 
( I  i^i  J
)  =  (/) )
2218, 20, 21sylancr 663 . . . 4  |-  ( -.  R  e.  _V  ->  ( I  i^i  J )  =  (/) )
2317, 22eqtr4d 2476 . . 3  |-  ( -.  R  e.  _V  ->  (2Ideal `  R )  =  ( I  i^i  J ) )
2416, 23pm2.61i 164 . 2  |-  (2Ideal `  R )  =  ( I  i^i  J )
251, 24eqtri 2461 1  |-  T  =  ( I  i^i  J
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1369    e. wcel 1756   _Vcvv 2970    i^i cin 3325    C_ wss 3326   (/)c0 3635   ` cfv 5416  opprcoppr 16712  LIdealclidl 17249  2Idealc2idl 17311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-iota 5379  df-fun 5418  df-fv 5424  df-2idl 17312
This theorem is referenced by:  2idlcpbl  17314  divs1  17315  divsrhm  17317  crng2idl  17319
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