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Theorem 2idlval 16304
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 5728 . . . . . 6  |-  ( r  =  R  ->  (LIdeal `  r )  =  (LIdeal `  R ) )
3 2idlval.i . . . . . 6  |-  I  =  (LIdeal `  R )
42, 3syl6eqr 2486 . . . . 5  |-  ( r  =  R  ->  (LIdeal `  r )  =  I )
5 fveq2 5728 . . . . . . . 8  |-  ( r  =  R  ->  (oppr `  r
)  =  (oppr `  R
) )
6 2idlval.o . . . . . . . 8  |-  O  =  (oppr
`  R )
75, 6syl6eqr 2486 . . . . . . 7  |-  ( r  =  R  ->  (oppr `  r
)  =  O )
87fveq2d 5732 . . . . . 6  |-  ( r  =  R  ->  (LIdeal `  (oppr
`  r ) )  =  (LIdeal `  O
) )
9 2idlval.j . . . . . 6  |-  J  =  (LIdeal `  O )
108, 9syl6eqr 2486 . . . . 5  |-  ( r  =  R  ->  (LIdeal `  (oppr
`  r ) )  =  J )
114, 10ineq12d 3543 . . . 4  |-  ( r  =  R  ->  (
(LIdeal `  r )  i^i  (LIdeal `  (oppr
`  r ) ) )  =  ( I  i^i  J ) )
12 df-2idl 16303 . . . 4  |- 2Ideal  =  ( r  e.  _V  |->  ( (LIdeal `  r )  i^i  (LIdeal `  (oppr
`  r ) ) ) )
13 fvex 5742 . . . . . 6  |-  (LIdeal `  R )  e.  _V
143, 13eqeltri 2506 . . . . 5  |-  I  e. 
_V
1514inex1 4344 . . . 4  |-  ( I  i^i  J )  e. 
_V
1611, 12, 15fvmpt 5806 . . 3  |-  ( R  e.  _V  ->  (2Ideal `  R )  =  ( I  i^i  J ) )
17 fvprc 5722 . . . 4  |-  ( -.  R  e.  _V  ->  (2Ideal `  R )  =  (/) )
18 inss1 3561 . . . . 5  |-  ( I  i^i  J )  C_  I
19 fvprc 5722 . . . . . 6  |-  ( -.  R  e.  _V  ->  (LIdeal `  R )  =  (/) )
203, 19syl5eq 2480 . . . . 5  |-  ( -.  R  e.  _V  ->  I  =  (/) )
21 sseq0 3659 . . . . 5  |-  ( ( ( I  i^i  J
)  C_  I  /\  I  =  (/) )  -> 
( I  i^i  J
)  =  (/) )
2218, 20, 21sylancr 645 . . . 4  |-  ( -.  R  e.  _V  ->  ( I  i^i  J )  =  (/) )
2317, 22eqtr4d 2471 . . 3  |-  ( -.  R  e.  _V  ->  (2Ideal `  R )  =  ( I  i^i  J ) )
2416, 23pm2.61i 158 . 2  |-  (2Ideal `  R )  =  ( I  i^i  J )
251, 24eqtri 2456 1  |-  T  =  ( I  i^i  J
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   _Vcvv 2956    i^i cin 3319    C_ wss 3320   (/)c0 3628   ` cfv 5454  opprcoppr 15727  LIdealclidl 16242  2Idealc2idl 16302
This theorem is referenced by:  2idlcpbl  16305  divs1  16306  divsrhm  16308  crng2idl  16310
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-2idl 16303
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