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Theorem 2idlval 21279
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 6907 . . . . . 6 (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅))
3 2idlval.i . . . . . 6 𝐼 = (LIdeal‘𝑅)
42, 3eqtr4di 2793 . . . . 5 (𝑟 = 𝑅 → (LIdeal‘𝑟) = 𝐼)
5 fveq2 6907 . . . . . . . 8 (𝑟 = 𝑅 → (oppr𝑟) = (oppr𝑅))
6 2idlval.o . . . . . . . 8 𝑂 = (oppr𝑅)
75, 6eqtr4di 2793 . . . . . . 7 (𝑟 = 𝑅 → (oppr𝑟) = 𝑂)
87fveq2d 6911 . . . . . 6 (𝑟 = 𝑅 → (LIdeal‘(oppr𝑟)) = (LIdeal‘𝑂))
9 2idlval.j . . . . . 6 𝐽 = (LIdeal‘𝑂)
108, 9eqtr4di 2793 . . . . 5 (𝑟 = 𝑅 → (LIdeal‘(oppr𝑟)) = 𝐽)
114, 10ineq12d 4229 . . . 4 (𝑟 = 𝑅 → ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr𝑟))) = (𝐼𝐽))
12 df-2idl 21278 . . . 4 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr𝑟))))
133fvexi 6921 . . . . 5 𝐼 ∈ V
1413inex1 5323 . . . 4 (𝐼𝐽) ∈ V
1511, 12, 14fvmpt 7016 . . 3 (𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼𝐽))
16 fvprc 6899 . . . 4 𝑅 ∈ V → (2Ideal‘𝑅) = ∅)
17 inss1 4245 . . . . 5 (𝐼𝐽) ⊆ 𝐼
18 fvprc 6899 . . . . . 6 𝑅 ∈ V → (LIdeal‘𝑅) = ∅)
193, 18eqtrid 2787 . . . . 5 𝑅 ∈ V → 𝐼 = ∅)
20 sseq0 4409 . . . . 5 (((𝐼𝐽) ⊆ 𝐼𝐼 = ∅) → (𝐼𝐽) = ∅)
2117, 19, 20sylancr 587 . . . 4 𝑅 ∈ V → (𝐼𝐽) = ∅)
2216, 21eqtr4d 2778 . . 3 𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼𝐽))
2315, 22pm2.61i 182 . 2 (2Ideal‘𝑅) = (𝐼𝐽)
241, 23eqtri 2763 1 𝑇 = (𝐼𝐽)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962  wss 3963  c0 4339  cfv 6563  opprcoppr 20350  LIdealclidl 21234  2Idealc2idl 21277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-2idl 21278
This theorem is referenced by:  2idlelb  21281  2idllidld  21282  2idlridld  21283  2idl0  21288  2idl1  21289  qus1  21302  qusrhm  21304  crng2idl  21309  oppr2idl  33494  qsdrngilem  33502  qsdrngi  33503
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