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Theorem 2idlval 18007
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 5872 . . . . . 6  |-  ( r  =  R  ->  (LIdeal `  r )  =  (LIdeal `  R ) )
3 2idlval.i . . . . . 6  |-  I  =  (LIdeal `  R )
42, 3syl6eqr 2516 . . . . 5  |-  ( r  =  R  ->  (LIdeal `  r )  =  I )
5 fveq2 5872 . . . . . . . 8  |-  ( r  =  R  ->  (oppr `  r
)  =  (oppr `  R
) )
6 2idlval.o . . . . . . . 8  |-  O  =  (oppr
`  R )
75, 6syl6eqr 2516 . . . . . . 7  |-  ( r  =  R  ->  (oppr `  r
)  =  O )
87fveq2d 5876 . . . . . 6  |-  ( r  =  R  ->  (LIdeal `  (oppr
`  r ) )  =  (LIdeal `  O
) )
9 2idlval.j . . . . . 6  |-  J  =  (LIdeal `  O )
108, 9syl6eqr 2516 . . . . 5  |-  ( r  =  R  ->  (LIdeal `  (oppr
`  r ) )  =  J )
114, 10ineq12d 3697 . . . 4  |-  ( r  =  R  ->  (
(LIdeal `  r )  i^i  (LIdeal `  (oppr
`  r ) ) )  =  ( I  i^i  J ) )
12 df-2idl 18006 . . . 4  |- 2Ideal  =  ( r  e.  _V  |->  ( (LIdeal `  r )  i^i  (LIdeal `  (oppr
`  r ) ) ) )
13 fvex 5882 . . . . . 6  |-  (LIdeal `  R )  e.  _V
143, 13eqeltri 2541 . . . . 5  |-  I  e. 
_V
1514inex1 4597 . . . 4  |-  ( I  i^i  J )  e. 
_V
1611, 12, 15fvmpt 5956 . . 3  |-  ( R  e.  _V  ->  (2Ideal `  R )  =  ( I  i^i  J ) )
17 fvprc 5866 . . . 4  |-  ( -.  R  e.  _V  ->  (2Ideal `  R )  =  (/) )
18 inss1 3714 . . . . 5  |-  ( I  i^i  J )  C_  I
19 fvprc 5866 . . . . . 6  |-  ( -.  R  e.  _V  ->  (LIdeal `  R )  =  (/) )
203, 19syl5eq 2510 . . . . 5  |-  ( -.  R  e.  _V  ->  I  =  (/) )
21 sseq0 3826 . . . . 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 2501 . . 3  |-  ( -.  R  e.  _V  ->  (2Ideal `  R )  =  ( I  i^i  J ) )
2416, 23pm2.61i 164 . 2  |-  (2Ideal `  R )  =  ( I  i^i  J )
251, 24eqtri 2486 1  |-  T  =  ( I  i^i  J
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1395    e. wcel 1819   _Vcvv 3109    i^i cin 3470    C_ wss 3471   (/)c0 3793   ` cfv 5594  opprcoppr 17397  LIdealclidl 17942  2Idealc2idl 18005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-2idl 18006
This theorem is referenced by:  2idlcpbl  18008  qus1  18009  qusrhm  18011  crng2idl  18013
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