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Theorem 2idlval 18201
 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 5849 . . . . . 6 LIdeal LIdeal
3 2idlval.i . . . . . 6 LIdeal
42, 3syl6eqr 2461 . . . . 5 LIdeal
5 fveq2 5849 . . . . . . . 8 oppr oppr
6 2idlval.o . . . . . . . 8 oppr
75, 6syl6eqr 2461 . . . . . . 7 oppr
87fveq2d 5853 . . . . . 6 LIdealoppr LIdeal
9 2idlval.j . . . . . 6 LIdeal
108, 9syl6eqr 2461 . . . . 5 LIdealoppr
114, 10ineq12d 3642 . . . 4 LIdeal LIdealoppr
12 df-2idl 18200 . . . 4 2Ideal LIdeal LIdealoppr
13 fvex 5859 . . . . . 6 LIdeal
143, 13eqeltri 2486 . . . . 5
1514inex1 4535 . . . 4
1611, 12, 15fvmpt 5932 . . 3 2Ideal
17 fvprc 5843 . . . 4 2Ideal
18 inss1 3659 . . . . 5
19 fvprc 5843 . . . . . 6 LIdeal
203, 19syl5eq 2455 . . . . 5
21 sseq0 3771 . . . . 5
2218, 20, 21sylancr 661 . . . 4
2317, 22eqtr4d 2446 . . 3 2Ideal
2416, 23pm2.61i 164 . 2 2Ideal
251, 24eqtri 2431 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wceq 1405   wcel 1842  cvv 3059   cin 3413   wss 3414  c0 3738  cfv 5569  opprcoppr 17591  LIdealclidl 18136  2Idealc2idl 18199 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577  df-2idl 18200 This theorem is referenced by:  2idlcpbl  18202  qus1  18203  qusrhm  18205  crng2idl  18207
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