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Theorem 2idlval 19054
 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 = (oppr𝑅)
2idlval.j 𝐽 = (LIdeal‘𝑂)
2idlval.t 𝑇 = (2Ideal‘𝑅)
Assertion
Ref Expression
2idlval 𝑇 = (𝐼𝐽)

Proof of Theorem 2idlval
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 2idlval.t . 2 𝑇 = (2Ideal‘𝑅)
2 fveq2 6103 . . . . . 6 (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅))
3 2idlval.i . . . . . 6 𝐼 = (LIdeal‘𝑅)
42, 3syl6eqr 2662 . . . . 5 (𝑟 = 𝑅 → (LIdeal‘𝑟) = 𝐼)
5 fveq2 6103 . . . . . . . 8 (𝑟 = 𝑅 → (oppr𝑟) = (oppr𝑅))
6 2idlval.o . . . . . . . 8 𝑂 = (oppr𝑅)
75, 6syl6eqr 2662 . . . . . . 7 (𝑟 = 𝑅 → (oppr𝑟) = 𝑂)
87fveq2d 6107 . . . . . 6 (𝑟 = 𝑅 → (LIdeal‘(oppr𝑟)) = (LIdeal‘𝑂))
9 2idlval.j . . . . . 6 𝐽 = (LIdeal‘𝑂)
108, 9syl6eqr 2662 . . . . 5 (𝑟 = 𝑅 → (LIdeal‘(oppr𝑟)) = 𝐽)
114, 10ineq12d 3777 . . . 4 (𝑟 = 𝑅 → ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr𝑟))) = (𝐼𝐽))
12 df-2idl 19053 . . . 4 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr𝑟))))
13 fvex 6113 . . . . . 6 (LIdeal‘𝑅) ∈ V
143, 13eqeltri 2684 . . . . 5 𝐼 ∈ V
1514inex1 4727 . . . 4 (𝐼𝐽) ∈ V
1611, 12, 15fvmpt 6191 . . 3 (𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼𝐽))
17 fvprc 6097 . . . 4 𝑅 ∈ V → (2Ideal‘𝑅) = ∅)
18 inss1 3795 . . . . 5 (𝐼𝐽) ⊆ 𝐼
19 fvprc 6097 . . . . . 6 𝑅 ∈ V → (LIdeal‘𝑅) = ∅)
203, 19syl5eq 2656 . . . . 5 𝑅 ∈ V → 𝐼 = ∅)
21 sseq0 3927 . . . . 5 (((𝐼𝐽) ⊆ 𝐼𝐼 = ∅) → (𝐼𝐽) = ∅)
2218, 20, 21sylancr 694 . . . 4 𝑅 ∈ V → (𝐼𝐽) = ∅)
2317, 22eqtr4d 2647 . . 3 𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼𝐽))
2416, 23pm2.61i 175 . 2 (2Ideal‘𝑅) = (𝐼𝐽)
251, 24eqtri 2632 1 𝑇 = (𝐼𝐽)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ‘cfv 5804  opprcoppr 18445  LIdealclidl 18991  2Idealc2idl 19052 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-2idl 19053 This theorem is referenced by:  2idlcpbl  19055  qus1  19056  qusrhm  19058  crng2idl  19060
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