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Theorem 2idlval 20003
 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 6646 . . . . . 6 (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅))
3 2idlval.i . . . . . 6 𝐼 = (LIdeal‘𝑅)
42, 3eqtr4di 2851 . . . . 5 (𝑟 = 𝑅 → (LIdeal‘𝑟) = 𝐼)
5 fveq2 6646 . . . . . . . 8 (𝑟 = 𝑅 → (oppr𝑟) = (oppr𝑅))
6 2idlval.o . . . . . . . 8 𝑂 = (oppr𝑅)
75, 6eqtr4di 2851 . . . . . . 7 (𝑟 = 𝑅 → (oppr𝑟) = 𝑂)
87fveq2d 6650 . . . . . 6 (𝑟 = 𝑅 → (LIdeal‘(oppr𝑟)) = (LIdeal‘𝑂))
9 2idlval.j . . . . . 6 𝐽 = (LIdeal‘𝑂)
108, 9eqtr4di 2851 . . . . 5 (𝑟 = 𝑅 → (LIdeal‘(oppr𝑟)) = 𝐽)
114, 10ineq12d 4140 . . . 4 (𝑟 = 𝑅 → ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr𝑟))) = (𝐼𝐽))
12 df-2idl 20002 . . . 4 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr𝑟))))
133fvexi 6660 . . . . 5 𝐼 ∈ V
1413inex1 5186 . . . 4 (𝐼𝐽) ∈ V
1511, 12, 14fvmpt 6746 . . 3 (𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼𝐽))
16 fvprc 6639 . . . 4 𝑅 ∈ V → (2Ideal‘𝑅) = ∅)
17 inss1 4155 . . . . 5 (𝐼𝐽) ⊆ 𝐼
18 fvprc 6639 . . . . . 6 𝑅 ∈ V → (LIdeal‘𝑅) = ∅)
193, 18syl5eq 2845 . . . . 5 𝑅 ∈ V → 𝐼 = ∅)
20 sseq0 4307 . . . . 5 (((𝐼𝐽) ⊆ 𝐼𝐼 = ∅) → (𝐼𝐽) = ∅)
2117, 19, 20sylancr 590 . . . 4 𝑅 ∈ V → (𝐼𝐽) = ∅)
2216, 21eqtr4d 2836 . . 3 𝑅 ∈ V → (2Ideal‘𝑅) = (𝐼𝐽))
2315, 22pm2.61i 185 . 2 (2Ideal‘𝑅) = (𝐼𝐽)
241, 23eqtri 2821 1 𝑇 = (𝐼𝐽)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  ‘cfv 6325  opprcoppr 19372  LIdealclidl 19939  2Idealc2idl 20001 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-iota 6284  df-fun 6327  df-fv 6333  df-2idl 20002 This theorem is referenced by:  2idlcpbl  20004  qus1  20005  qusrhm  20007  crng2idl  20009
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