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Theorem 2idlval 16001
 Description: Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
2idlval.i LIdeal
2idlval.o oppr
2idlval.j LIdeal
2idlval.t 2Ideal
Assertion
Ref Expression
2idlval

Proof of Theorem 2idlval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 2idlval.t . 2 2Ideal
2 fveq2 5541 . . . . . 6 LIdeal LIdeal
3 2idlval.i . . . . . 6 LIdeal
42, 3syl6eqr 2346 . . . . 5 LIdeal
5 fveq2 5541 . . . . . . . 8 oppr oppr
6 2idlval.o . . . . . . . 8 oppr
75, 6syl6eqr 2346 . . . . . . 7 oppr
87fveq2d 5545 . . . . . 6 LIdealoppr LIdeal
9 2idlval.j . . . . . 6 LIdeal
108, 9syl6eqr 2346 . . . . 5 LIdealoppr
114, 10ineq12d 3384 . . . 4 LIdeal LIdealoppr
12 df-2idl 16000 . . . 4 2Ideal LIdeal LIdealoppr
13 fvex 5555 . . . . . 6 LIdeal
143, 13eqeltri 2366 . . . . 5
1514inex1 4171 . . . 4
1611, 12, 15fvmpt 5618 . . 3 2Ideal
17 fvprc 5535 . . . 4 2Ideal
18 inss1 3402 . . . . 5
19 fvprc 5535 . . . . . 6 LIdeal
203, 19syl5eq 2340 . . . . 5
21 sseq0 3499 . . . . 5
2218, 20, 21sylancr 644 . . . 4
2317, 22eqtr4d 2331 . . 3 2Ideal
2416, 23pm2.61i 156 . 2 2Ideal
251, 24eqtri 2316 1
 Colors of variables: wff set class Syntax hints:   wn 3   wceq 1632   wcel 1696  cvv 2801   cin 3164   wss 3165  c0 3468  cfv 5271  opprcoppr 15420  LIdealclidl 15939  2Idealc2idl 15999 This theorem is referenced by:  2idlcpbl  16002  divs1  16003  divsrhm  16005  crng2idl  16007 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-2idl 16000
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