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Theorem 2idlval 15985
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 5525 . . . . . 6  |-  ( r  =  R  ->  (LIdeal `  r )  =  (LIdeal `  R ) )
3 2idlval.i . . . . . 6  |-  I  =  (LIdeal `  R )
42, 3syl6eqr 2333 . . . . 5  |-  ( r  =  R  ->  (LIdeal `  r )  =  I )
5 fveq2 5525 . . . . . . . 8  |-  ( r  =  R  ->  (oppr `  r
)  =  (oppr `  R
) )
6 2idlval.o . . . . . . . 8  |-  O  =  (oppr
`  R )
75, 6syl6eqr 2333 . . . . . . 7  |-  ( r  =  R  ->  (oppr `  r
)  =  O )
87fveq2d 5529 . . . . . 6  |-  ( r  =  R  ->  (LIdeal `  (oppr
`  r ) )  =  (LIdeal `  O
) )
9 2idlval.j . . . . . 6  |-  J  =  (LIdeal `  O )
108, 9syl6eqr 2333 . . . . 5  |-  ( r  =  R  ->  (LIdeal `  (oppr
`  r ) )  =  J )
114, 10ineq12d 3371 . . . 4  |-  ( r  =  R  ->  (
(LIdeal `  r )  i^i  (LIdeal `  (oppr
`  r ) ) )  =  ( I  i^i  J ) )
12 df-2idl 15984 . . . 4  |- 2Ideal  =  ( r  e.  _V  |->  ( (LIdeal `  r )  i^i  (LIdeal `  (oppr
`  r ) ) ) )
13 fvex 5539 . . . . . 6  |-  (LIdeal `  R )  e.  _V
143, 13eqeltri 2353 . . . . 5  |-  I  e. 
_V
1514inex1 4155 . . . 4  |-  ( I  i^i  J )  e. 
_V
1611, 12, 15fvmpt 5602 . . 3  |-  ( R  e.  _V  ->  (2Ideal `  R )  =  ( I  i^i  J ) )
17 fvprc 5519 . . . 4  |-  ( -.  R  e.  _V  ->  (2Ideal `  R )  =  (/) )
18 inss1 3389 . . . . 5  |-  ( I  i^i  J )  C_  I
19 fvprc 5519 . . . . . 6  |-  ( -.  R  e.  _V  ->  (LIdeal `  R )  =  (/) )
203, 19syl5eq 2327 . . . . 5  |-  ( -.  R  e.  _V  ->  I  =  (/) )
21 sseq0 3486 . . . . 5  |-  ( ( ( I  i^i  J
)  C_  I  /\  I  =  (/) )  -> 
( I  i^i  J
)  =  (/) )
2218, 20, 21sylancr 644 . . . 4  |-  ( -.  R  e.  _V  ->  ( I  i^i  J )  =  (/) )
2317, 22eqtr4d 2318 . . 3  |-  ( -.  R  e.  _V  ->  (2Ideal `  R )  =  ( I  i^i  J ) )
2416, 23pm2.61i 156 . 2  |-  (2Ideal `  R )  =  ( I  i^i  J )
251, 24eqtri 2303 1  |-  T  =  ( I  i^i  J
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152   (/)c0 3455   ` cfv 5255  opprcoppr 15404  LIdealclidl 15923  2Idealc2idl 15983
This theorem is referenced by:  2idlcpbl  15986  divs1  15987  divsrhm  15989  crng2idl  15991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-2idl 15984
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